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\title[Non-orientable branched coverings, $b$-Hurwitz numbers,  and positivity]{Non-orientable branched coverings, $b$-Hurwitz numbers,  and positivity for multiparametric Jack expansions (extended abstract)}

%% define your authors in the usual way
%% use \addressmark{1}, \addressmark{2} etc for the institutions, and use \thanks{} for contact details
\author{Guillaume Chapuy\addressmark{1} \thanks{\href{mailto:guillaume.chapuy@irif.fr}{guillaume.chapuy@irif.fr} GC has received funding from the European Research
  Council (ERC) under the European Union’s Horizon 2020 research and
  innovation programme (grant agreement No. ERC-2016-STG 716083).}
\and Maciej Dołęga\addressmark{2}
\thanks{\href{mailto:mdolega@impan.pl}{mdolega@impan.pl}. 
  MD is supported from \textit{Narodowe Centrum Nauki}, grant UMO-2017/26/D/ST1/00186.}
    }
  %\thanks{\href{}{mdolega@impan.pl}. MD is supported from \textit{Narodowe Centrum Nauki}, grant UMO-2017/26/D/ST1/00186. Emails: {\tt guillaume.chapuy@irif.fr,  mdolega@impan.pl}.  }}


%\author[Optional shorter names]{Longer names me\thanks{\href{mailto:hello@world.c}{hello@world.c}. Longer names me was partially supported by Grant 2017.11.14.$\partial$\;supp.}}

%% then use \addressmark to match authors to institutions here
\address{\addressmark{1}CNRS, IRIF UMR 8243, Universit\'e de Paris, \\ \addressmark{2}Institute of Mathematics, 
Polish Academy of Sciences, 
%ul. Śniadeckich 8, %% I commented this out to gain one line! 
00-956 Warszawa, Poland.}

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\received{\today}

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%% put your English abstract here, or comment this out if you don't have one yet
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\abstract{
	{ We introduce a one-parameter deformation of the 2-Toda
        tau-function %of (weighted) Hurwitz numbers,
	of maps (or more generally, constellations), obtained by deforming Schur functions into Jack symmetric functions. We show that its coefficients are polynomials in the deformation parameter $b$ with nonnegative integer coefficients.
	These coefficients count generalized constellations on an arbitrary surface, orientable or not, with an appropriate $b$-weighting that ``measures'' in some sense their non-orientability.
The particular case of bipartite maps gives the best progress so far towards the ``$b$-conjecture'' of Goulden and Jackson from 1996.

%
	%

%	Notable special cases include
%	%
%	non-orientable \textit{dessins d'enfants} for which we prove the
%        most general result so far towards the Matching-Jack conjecture
%        and the ``$b$-conjecture'' of Goulden and Jackson from 1996, expansions of the $\beta$-ensemble matrix model, deformations of the HCIZ integral, and $b$-Hurwitz numbers that we
%        introduce here and that are $b$-deformations of classical
 %       (single or double) Hurwitz numbers obtained for $b=0$.

	Our proof consists in showing that the partition function satisfies an infinite set of PDEs. These PDEs have two definitions, one given by Lax equations, the other one following an explicit combinatorial decomposition.
	}
}

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\begin{document}

\maketitle
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\section{Introduction}

\begin{quote}
\textit{This paper is an abbreviated version of~\cite{ChapuyDolega2020}, of which we present some of the main results and ideas. We refer to~\cite{ChapuyDolega2020} for complete proofs and many more developments.
%GC-26/11 space saving (we say later in the intro that coverings are not addressed here).
%some technical definitions, more background.
	%, and for the geometric viewpoint on coverings, which we omit here. }
	}
\end{quote}



\subsection{Constellations and Schur functions}

{\bf Hurwitz numbers and tau-functions.}
Hurwitz numbers, in their most general sense, count the number of combinatorially inequivalent branched coverings of the sphere by an orientable surface with a given number of ramification points and given ramification profiles. Hurwitz numbers and their variants (dessins d'enfants, weighted, monotone, orbifold Hurwitz numbers)  have numerous connections to mathematical physics, combinatorics, and the moduli spaces of curves, see e.g.%
%GC-26/11: 
% I don't think that citing Kontsevich or Okounkov is truly helpful here
% citing Harnard-Guay-Paquet, for example, is much more interesting for FPSAC audience.
% The pb is that we cannot have too many references!
% and I think it would be good to cite something about weighted Hurwitz numbers
% (either HGP or my paper about topological recursion with Eynard et al.)
% I would be happy to cite some general reference with many references to other
% papers, but I don't really know what to cite
% For now, I have chosen to remove Okounkov and Kontsevich here (as nobody cares, even if this is fancy), and to cite Harnad-Guay-Paquet later.
% This is a net win of 2-1 = 1 ref. and makes the paper 12 pages long.
%edit: OK, I moved them back, I just removed Tutte and Lehman-Walsh.
%     and I replaced Miwa-Jimbo by Harnad-Guay-Paquet
%
~\cite{Kontsevich1992,% GouldenJackson1997a,EkedahlLandoShapiroVainshtein2001,GraberVakil2003,GouldenJacksonVakil2005,
  OkounkovPandharipande2006, Guay-PaquetHarnad2017% Mirzakhani2007, Guay-PaquetHarnad2017
}.
%
%	Hurwitz himself~\cite{Hurwitz1891} showed that Hurwitz numbers can be expressed in terms of characters of the symmetric group. Equivalently,




Branched coverings can be realized as certain coloured graphs drawn on orientable surfaces called \emph{constellations}~\cite{LandoZvonkin2004}, a very general model of maps on surfaces which cover in particular \emph{bipartite maps}, or \emph{transposition factorisations}, as special cases.
%, the prominent example being \emph{bipartite maps}. %Hurwitz numbers can be realized
The enumeration of constellations is strongly linked to characters of the symmetric group (or Schur functions) which gives it a very rich structure.
A fundamental fact in the field
% , whose origins go back to 
 %        Pandariphande~\cite{Pandharipande2000},
 %        Okounkov~\cite{Okounkov2000a}, Orlov and Scherbin \cite{OrlovScherbin2000},
 % and now understood in a wide generality (see e.g.~\cite{GouldenJackson2008,Guay-PaquetHarnad2017}) 
 is that a certain generating function of weighted Hurwitz numbers (or constellations) 
 %can be used to define a formal power series which 
 is a tau-function of the KP/2-Toda hierarchies (see for example~\cite{Guay-PaquetHarnad2017} and references therein). Explicitly, in the case of $k$-constellations, this generating function has the form
\begin{align}\label{eq:SchurIntro}
	\tau^{(k)}(t;\pp,\qq,u_1,\dots,u_k) := \sum_{n \geq
	0}t^n \sum_{\la \vdash n} \left(\frac{f_\lambda}{n!}\right)^2 \tilde{s}_\la(\pp)
	\tilde{s}_\la(\qq)\tilde{s}_\la(\underline{u_1})\tilde{s}_\la(\underline{u_2})\dots \tilde{s}_\la(\underline{u_k}),
\end{align}
    where $\tilde{s}_\la=\frac{n!}{f_\la}\cdot s_\la$ %
    is the normalized Schur function indexed by the
    integer partition $\lambda$ of $n$, expressed as a polynomial in the
    power-sum variables $\mathbf{p}=(p_i)_{i\geq 1}$ or
    $\mathbf{q}=(q_i)_{i\geq 1}$,  where $\underline{u}=(u,u,\dots)$ is the specialization of all power-sum variables to the same parameter $u$,
    and where $f_\lambda$ is the dimension of the irreducible representation of the symmetric group indexed by $\lambda$.
%
From this function (or more precisely its logarithm) 
one can extract the generating functions for surfaces of genus $g$ with $n$ boundaries, which are one of the main subject of interest of the map-enumeration community.
%one can extract all the forms $\omega_{g,n}$ associated to the generating functions of constellations on surfaces of genus $g$ with $n$ boundaries, which obey the Chekhov-Eynard-Orantin topological recursion.
% ~\cite{ChekhovEynardOrantin2006, EynardOrantin2007, AlexandrovChapuyEynardHarnad2020}
%.
%Weighted Hurwitz numbers% ~\cite{Guay-PaquetHarnad2017}
%correspond in some sense to the case $k=\infty$, which contains the Okounkov-Pandariphande Hurwitz numbers as a special case % (see~\cref{sec:infinite})
In itself, the particular case $k=1$ of \emph{bipartite maps} has been extensively studied.
%and is still an subject of active study in the combinatorics community.
We refer to \cite{LandoZvonkin2004, ChapuyDolega2020} for references.


\subsection{Jack polynomials,  $b$-deformations, and our main result}
In this paper we consider the one-parameter deformation, or \emph{$b$-deformation}, of the function $\tau^{(k)}$ defined by 
\begin{align}\label{eq:JackIntro}
  \tau^{(k)}_b (t;\pp,\qq,u_1,\dots,u_k) :&=
	%
 %
	\sum_{n \geq 0}t^n\sum_{\la
	\vdash n}\frac{1}{j_\la^{(\alpha)}} J_\la^{(\alpha)}(\pp)J_\la^{(\alpha)}(\qq) J_\la^{(\alpha)}(\underline{u_1}) \dots J_\la^{(\alpha)}(\underline{u_k}) ,
   \end{align}
where $J^{(\alpha)}_\la$ is the Jack symmetric function of parameter
$\alpha = 1+b$, for a formal variable $b$, and where
$j_\la^{(\alpha)}:=\hook_{\a}(\la) \hook'_{\a}(\la)$ is a natural $b$-deformation of
$n!^2/{f_\lambda^2}$
, see Section~\ref{subsec:Jack}.
Jack functions are obtained as a one-parameter limit of Macdonald polynomials that interpolates between Schur and zonal polynomials, respectively for $b=0,1$~\cite{% Jack1970/1971,
  Macdonald1995}.
%
In particular the function~$\tau^{(k)}_b$ is equal to~$\tau^{(k)}$ for $b=0$.

%Many classical problems in algebraic combinatorics (dealing with symmetric functions, maps, coverings, tableaux, partitions) are connected to Schur or zonal polynomials. Understanding how to use Jack symmetric functions to build continuous deformations between them has become an important research goal in the last decades% , see~\cite{Stanley1989, HanlonStanleyStembridge1992,GouldenJackson1996, DolegaFeray2016,BorodinGorinGuionnet2017,GuionnetHuang2019}
%. It often requires to develop new methods that shed new light even on the most classical results.

%In our context,
%the deformation~\eqref{eq:JackIntro} was introduced by
%Goulden and Jackson~\cite{GouldenJackson1996} in the case $k=1$ of
%bipartite maps and is strongly related to the Matching-Jack
%conjecture and the $b$-conjecture of these authors, see the discussion below.
%{\bf Link with the Matching-Jack conjecture and the $b$-conjecture.}
The deformation~\eqref{eq:JackIntro} was introduced by Goulden and Jackson~\cite{GouldenJackson1996} in the case $k=1$ of bipartite maps (in fact \cite{GouldenJackson1996} considers a more general function where the sequence $\underline{u_1}$ is replaced by a third arbitrary sequence of parameters). Using the connection between zonal polynomials and representation theory of the Gelfand pair $(\mathfrak{S}_{2n},\mathbb{H}_n)$, they proved that for $b=1$ this function enumerates analogues of bipartite maps on general surfaces (orientable or not).
In the same paper they formulate the ``$b$--conjecture'' and the 
related ``Matching-Jack conjecture'', among the most remarkable open problems in algebraic combinatorics. They assert that the coefficients have an interpretation for arbitrary $b$:
they count bipartite maps on general surfaces, with a weight which is a
polynomial in $b$ with nonnegative coefficients.
%as in our main
%theorem. 
The representation theoretic tools used in the case $b\in\{0,1\}$ do not
apply for general $b$, and this conjecture is still wide open despite many
partial results~\cite{GouldenJackson1996,% BrownJackson2007,
  LaCroix2009,% KanunnikovVassilieva2016,DolegaFeray2017,
  Dolega2017a,KanunnikovPromyslovVassilieva2018}.
 
Our main result, in the case $k=1$, goes much further than these results and establishes the $b$-conjecture in the case of \emph{two full sets of variables} ($(p_i)$ and $(q_i)$  in our notation). In the case $k>1$, it leads us to introduce a non-orientable generalization of constellations, and to a more general result, not directly comparable with the conjectures of~\cite{GouldenJackson1996}.

\begin{figure}%
	\centering
\includegraphics[height=45mm]{vertex.pdf}
\includegraphics[height=45mm]{constellation2.pdf}
\includegraphics[height=35mm]{constellation3.pdf}
\caption{\textit{Left:} the local colour-constraints in a $k$-constellation. 
	\textit{Center:} a rooted $3$-constellation on the Klein bottle, in ribbon-graph representation. The right-path of the root (Section~\ref{sec:combinatorial}) is highlighted in red.
	\textit{Right:} A planar $3$-constellation. By adding the dotted edges and considering the greyed  areas as hyperedges, one recovers the usual (orientation-depending) representation of constellations~\cite{LandoZvonkin2004} familiar to connoisseurs of the orientable case. Moreover, what we call $k$-constellations here are often called $(k+1)$-constellations in the orientable literature.}
\label{fig:constellation}
\end{figure}


%{\bf Main result}
% Our
%main result is not strictly comparable with the Matching-Jack conjecture
%and the $b$-conjecture: the case $k=1$ of our result is a special case
%of both, and the case of general $k$ is incomparable with
%them. However, the case $k=1$ of our results
%is by
%far the most general progress towards them
%and covers, largely, all the previously proven cases. 
%See \cref{subsec:betaEnsembles} for
%more on the Matching-Jack conjecture and the $b$-conjecture.


%\smallskip
\subsection{Main result.}
Our main result gives a combinatorial
(and in fact, geometric)
meaning to the
coefficients of $\tau^{(k)}_b$.
%GC-26/11: I had to remove this for space reason
%in terms of certain maps on surfaces.
%
%that we call  \emph{$k$-constellations} which generalize the classical notion from orientable to arbitrary surfaces  (orientable or not).
%%In this extended abstract we will keep the viewpoint of maps, and we refer the reader to the long version for definitions and results concerning generalized branched
%coverings.
%The definition of constellations will be given in \cref{sec:Constellations}.
%
%GC-26/11: added simply connected faces (in case you get the same referee as last year!!!)
%
Define a $k$-constellation as a graph embedded on a compact surface (orientable or not), with simply connected faces, whose vertices are colored with colours in $[0..k]$ subject to the local constraints of Figure~\ref{fig:constellation}--Left (see Section~\ref{sec:Constellations} for more details). We let $F(\bM)$ and $V_i(\bM)$ be the sets of faces and $i$-coloured vertices of $\bM$, respectively.
\begin{theorem}[Main result]\label{thm:abbreviated}
For every $k\geq 1$, we have 
	\begin{align}\label{eq:logIntro}
		(1+b)\frac{t\partial }{\partial t} \ln \tau^{(k)}_b(t;\pp,\qq,u_1,&\dots,u_k) =\sum_{(\bM,c)}\kappa(\bM) t^{|\bM|}b^{\nu_\rho(\bM,c)}, 
	\end{align}
	where the sum is taken over all rooted $k$-constellations
        (orientable or not). Here $|\bM|$ is the
        size of $\bM$ (defined in \cref{sec:constellations}) and the monomial  $\kappa(\bM)$  keeps track of the degrees of faces,
$0$-coloured vertices, and of the number of $i$-coloured vertices for $1 \leq i \leq k$ as follows 
	\begin{align}\label{eq:kappa}
		\kappa(\bM):=\prod_{f\in F(\bM)} p_{\deg(f)}\prod_{v\in V_0(\bM)} q_{\deg(v)} \prod_{i=1}^k u_i^{|V_i(\bM)|}.
	\end{align}
	Moreover $\nu_\rho(\bM,c)$ is a nonnegative integer
%GC-26/11 removed to save space, I liked it but found no other way:
%attached to $(\bM,c)$ 
	which is zero if and only if $\bM$ is orientable.

	In particular, the coefficients of the LHS of~\eqref{eq:logIntro} are polynomials in $b$, and they have nonnegative integer coefficients. 
      \end{theorem}
   

{\bf Branched covers and other developments.}
% \Gnew{TODO: talk about what we do in the long paper, but not here (we introduce branched covers, we take the projective limit $k\rightarrow \infty$, we define Hurwitz numbers and establish the Cut-And-Join equation.).
% Maybe also import here some of the "possible developments".
% Finally, the following paragraph (which I deleted because I wanted to arrive more quickly at our results, could maybe find a place here: Many classical problems in algebraic combinatorics (dealing with symmetric functions, maps, coverings, tableaux, partitions) are connected to Schur or zonal polynomials. Understanding how to use Jack symmetric functions to build continuous deformations between them has become an important research goal in the last decades% , see~\cite{Stanley1989, HanlonStanleyStembridge1992,GouldenJackson1996, DolegaFeray2016,BorodinGorinGuionnet2017,GuionnetHuang2019}
% . It often requires to develop new methods that shed new light even on the most classical results.
% }
The full paper~\cite{ChapuyDolega2020}, of which we have chosen to keep the title, addresses several subjects which we omit in this extended abstract.
The main one is the introduction of generalized branched coverings of the sphere by a non-orientable surface, which are in bijection with our generalized constellations, thus endowing the $b$-deformed tau
function $\tau^{(k)}_b$ with a geometric meaning. 
In~\cite{ChapuyDolega2020} we also construct the projective limit ($k\rightarrow \infty$) of $\tau^{(k)}_b$ and use it to  introduce a non-orientable, $b$-weighted,
analogue of the (classical or weighted) Hurwitz numbers (see e.g.~\cite{Guay-PaquetHarnad2017}). 
In particular all results concerning constellations are
extended to this setting, including $b$-weights, decomposition
equations, and $b$-polynomiality,
%For the special cases of 
%(simple or double)
%$b$-weighted Hurwitz numbers,
%which correspond to the case where all
%ramification points except the first two are simple,
%we proved a
and some classical results such as the Cut-And-Join equation and piecewise
polynomiality for classsical Hurwitz numbers are generalized. A special case of the $b$-weighted Hurwitz
numbers enable us to introduce a $b$-deformation
of monotone Hurwitz numbers and the HCIZ integral. 
Finally, see~\cite{ChapuyDolega2020} for a more detailed discussion of the link of our results with 
%discussed dessins d'enfants and $\beta$-ensembles, and we made a detailed account of
the $b$- and Matching-Jack conjectures of Goulden and Jackson.
%and how they relate to our results. 
%{\bf $b$-Hurwitz numbers.} As a consequence of our work we introduce new $b$-deformations of
%weighted and classical Hurwitz numbers and we investigate their properties, including the Cut-And-Join equation and piecewise polynomiality.


{\bf Method of proof and plan of the paper}
% Partial differential equations, and the Lax structure of the Tutte equations.}
Our method of proof goes by showing that both sides of Equation~\eqref{eq:logIntro} satisfy the same PDEs. 
The differential operators defining these PDEs take two different forms.
For constellations, they follow from a combinatorial decomposition which is a vast generalization of the classical Tutte/Lehman-Walsh decomposition, in which the parameter ``$b$'' is appropriately inserted.
For the ``Jack polynomial'' side, these equations are defined by two companion \textit{Lax equations}, and they follow from algebro-combinatorial manipulations on Jack polynomials.
%
%while for coverings (or constellations), they follow explicitly from a combinatorial decomposition. 
Proving that the ``Lax'' and ``combinatorial'' forms are in fact equal is one of the hardest  tasks of~\cite{ChapuyDolega2020}. The proof relies on a combinatorial heuristic that can be made precise for $b\in \{0,1\}$. This heuristic is ``lifted'' towards an algebraic proof by developing a technical operator paradigm which occupies much of the paper~\cite{ChapuyDolega2020}.

\smallskip
This extended abstract is organized as follows. In Section~\ref{sec:constellations}, we introduce the (new) notion of constellations on general surfaces. In Section~\ref{sec:MonTutte} we present the notion of MON, the associated $b$-weights,  and the combinatorial decomposition equation. Finally in Section~\ref{sec:LaxJackProof}, we present the Lax version of the differential operators, the main technical result, we discuss Jack polynomials and give some ideas of the proof structure.
%The presence of this Lax structure, which holds for general $b$, indicates that traces of integrability remain present beyond the two classical points $b\in\{0,1\}$.

%
%
%
%
%
%
%
%
%
%
%
%
%



\section{Maps  and constellations}
\label{sec:constellations}

%In this section we \Gremoved{quickly review branched coverings before
%  introducing their non-orientable generalization. We then} introduce
%$k$-constellations as a combinatorial model for \Gnew{branched
%  coverings by all (orientable or non-orientable) surfaces}.
%\subsection{Maps and constellations}
\label{sec:Constellations}

For us a \emph{surface} is a compact, two dimensional, real
manifold. By the classification theorem a connected surface $\mathcal{S}$ is
uniquely determined by its Euler characteristic
$\chi_{\mathcal{S}} \leq 2$
 (or, equivalently, its \emph{genus} $g_{\mathcal{S}} \in \frac{1}{2}\N$  given by
$\chi_\mathcal{S} = 2-2 g_{\mathcal{S}}$) together with
the information whether $\mathcal{S}$ is orientable or not.

An embedding of a graph (possibly with multiple edges) into a surface which cuts it into simply
connected pieces (called \emph{faces}) is called \emph{a map}. We
consider maps up to homeomorphisms of surfaces. A small neighborhood of an edge around a vertex is called a
\emph{half-edge} and a small neighborhood of a vertex delimited by two
consecutive half-edges is called \emph{a corner}. It is convenient to represent a map by its \emph{ribbon graph}, which is the surface with boundary made by a small neighbourhood of the graph on the surface it embeds in (see \cref{fig:constellation}--Center).
Our first contribution is a notion of constellation that generalizes the classical notion from the orientable case~\cite{LandoZvonkin2004}.
\begin{definition}[Constellation, see Figure~\ref{fig:constellation}]
	Let $k\geq 1$ be an integer. A \emph{$k$-constellation} is a map, equipped with a coloring of its vertices with colors in $\{0,1,2,\dots,k\}$, such that
\begin{enumerate}[itemsep=0pt,topsep=0pt, parsep=0pt]
\item
each vertex colored by $0$ ($k$, respectively) has only neighbours of
color $1$ ($k-1$ respectively),
\item
for any $0 < i < k$ and for any vertex $v$ colored by $i$, each corner
of $v$ separates vertices colored by $i-1$ and $i+1$.%
\end{enumerate}
\end{definition}

	The \emph{degree} of a face in a $k$-constellation is the
        number of corners of colour $0$ it contains, which is the same
        as the number of corners of colour $k$, and as half the number
	of corners of any other colour. The \emph{size} of a constellation $\bM$ is its number of corners of
	colour $0$ and is denoted by $|\bM|$.
	A constellation of size $n$ is \emph{labelled} if its corners of colour $0$ are labelled with the integers from $1$ to $n$, and if each such corner carries an (arbitrary) orientation.
A constellation  is \emph{rooted} if it is equipped with a
distinguished oriented corner of colour $0$, called the \emph{root}
(if the constellation is already labelled, the orientation of the root
corner is already given, but for unlabelled maps, this orientation is
part of the information given by the rooting). The \emph{root vertex
  (or face, respectively)} is the vertex (or face) incident to the
root corner.
%The \emph{color} of an edge is the pair $\{i,j\}$ formed by
%        the colors of its two endpoints.

\emph{The 2-profile} of a $k$-constellation is the $k+2$-tuple 
$(\la,\mu, v_1(\bM),\dots, v_k(\bM))$, where $\la$ is the
partition encoding face degrees, $\mu$ is the partition encoding
degrees of vertices of colour~$0$ and $v_i(\bM)$ is the number of vertices of colour $i$ in $\bM$ for $i\geq 1$.
The Euler characteristic $\chi(\bM)$ of a constellation (and its genus if it is connected) can be recovered from the %full profile %$(\lambda^{-1}, \dots, \lambda^k)$ 
%via the
Riemann-Hurwitz/Euler formula:
\begin{align}\label{eq:EulerCharacteristic}
	\chi(\bM) =\ell(\lambda)+\ell(\mu) - \sum_{i=1}^k (n- v_i(\bM)). 
\end{align}
In this paper we will be able to enumerate connected
$k$-constellations fully controlling their $2$-profiles and
orientability, thus in particular we will always determine their underlying surface.


\section{MON's and the $b$-deformed decomposition equation}
\label{sec:MonTutte}


\subsection{MON's and weights}

\begin{figure}%
\center\includegraphics[width=\linewidth]{MON.pdf}
	\caption{The main axioms of MONs.}
\label{fig:MON}
\end{figure}

Our way to assign a $b$-dependent weight to a map proceeds by repeated edge-deletions. The weight attached to each deletion depends on a number of arbitrary choices subject to suitable axioms, encompassed by the concept of \emph{measure of non-orientability}. 
\begin{definition}[MON; see \cref{fig:MON}]\label{def:MON}
	A \emph{measure of non-orientability (MON)} is a function
        $\rho(\cdot,\cdot)$ with value in $\mathbb{Q}[b]$ that
        associates to a vertex-colored map  $\bM$ and an edge $e$ in $\bM$, some value $\rho(\bM,e)$ and that satisfies the following properties.
	\begin{enumerate}[itemsep=0pt,topsep=0pt, parsep=0pt]
		\item
			Let $\bN:=\bM\setminus \{e\}$ and let $c_1,c_2$ be the two corners delimited by the endpoints of $e$ in~$\bN$.
			\begin{enumerate}[itemsep=0pt,topsep=0pt, parsep=0pt]
				\item If $c_1,c_2$ belong to two distinct connected components of $\bN$, then  $\rho(\bM,e)=1$.
				\item If $c_1,c_2$ belong to the same connected component of $\bN$ but to two different faces, then let $\tilde e$ be the other edge that could be added to $\bN$ between these corners to form a new map $\tilde\bM$. Then
					$\rho(\bM,e)+\rho(\tilde\bM,\tilde{e})=1+b$.
									\item If $c_1,c_2$ belong to the same face of $\bN$, then $\rho(\bM,e)=1$ if $e$ splits this face into two faces (``untwisted diagonal)'' and $\rho(\bM,e)=b$ otherwise (``twisted diagonal'').
		\end{enumerate}
		\item the value of  $\rho(\bM,e)$ depends only on the connected component of $\bM$ containing $e$.
%
                \end{enumerate}
                A MON $\rho$ is \emph{integral} if $\rho(\bM,e)$ belongs to
	$\{1,b\}$ for any $\bM$ and $e$, and if the following is true:
        for every pair $(\bM,e)$ which is in case (b) above,  and such that $\bM$ is orientable, we have $\rho(\bM,e)=1$ and $\rho(\tilde{\bM},\tilde{e})=b$.
        \Gremoved{
          If $e_1,e_2,\dots,e_i$ are edges of $\bM$, we will denote
	$$
	\rho(\bM; e_1,\dots,e_i):=\rho(\bM,e_1) \rho(\bM\setminus\{e_1\},e_2)\dots \rho(\bM\setminus\{e_1,\dots,e_{i-1}\},e_i).
	$$
	This quantity in general depends on the ordering of the edges $e_1,\dots,e_i$. We will also use the notation $\rho(\bM;L)$ where $L=(e_1,\dots,e_i)$ is an ordered list of edges.}
\end{definition}

%\begin{figure}%
%\center\includegraphics[width=\linewidth]{niceMON.pdf}
%	\caption{A coherent MON: $\rho(\bM\cup\{e_i\},e_i)+\rho(\bM\cup\{\tilde e_{i+1}\},\tilde{e}_{i+1})=(1+b).$}
%\label{fig:niceMON}
%\end{figure}

The idea of using MON's or their variants already appeared in previous
works on the $b$-conjecture starting from~\cite{LaCroix2009}. % (see also~\cite{DolegaFeraySniady2014,Dolega2017a})
Here we have added Axiom (2) which is necessary for some of our arguments.
We also allow MON's to take values in $\mathbb{Q}[b]$ (previous authors only consider what we call here \emph{integral} MON's), which
%GC-26/11:
%is not needed strictly speaking for us, but 
we believe is
%more 
natural~\cite[Rem.~3]{ChapuyDolega2020}.
%the generating function arguments in the next section. 
%\Gremoved{
%Note also that previous authors only consider what we call here \emph{integral} MON's. Although considering non-integrals MON's is not needed strictly speaking for this paper, we believe that this is natural and may be useful for further developments (see \cref{rem:rhoSYM} below).
%

In~\cite{ChapuyDolega2020}, we also introduce the notion of \emph{coherent MON}, which is too technical to be stated here. A coherent MON satisfies an additional restriction that implies the following property, which we can take as an alternative definition for the purposes of this extended abstract: if $c$ is a corner of colour $j$ in $\bM$ and $f$ is a face in $\bM$, then the average value, among all possible additions of an edge $e$ between $c$ and some corner of colour $(j+1)$ in $f$, of the value $\rho(\bM \cup \{e\},e)$, is equal to $\frac{1+b}{2}$. It is crucial for us (but easy to see, \cite[Lem.~3.3]{ChapuyDolega2020}) that there exist MONs which are both coherent and integral.



\subsection{The combinatorial decomposition}
\label{sec:combinatorial}

We now present an algorithm that enables one to ``exhaust'' any rooted constellation  by repeated edge deletions. It is 
a generalization of the classical ``root-edge decompositions''
going back to Tutte
%GC-26/11: 
%~\cite{Tutte1962a}
for the planar case 
and to Lehman and Walsh
%GC-26/11: 
%~\cite{WalshLehman1972}
for higher genera. 

Define the \emph{right-path} of the corner $c$ in the map $\bM$ as the sequence of $k$ edges following the corner $c$ along the face it belongs to in $\bM$ (Figure~\ref{fig:constellation}-Center). It is easy to see that removing a right-path from a $k$-constellation, one obtains again a $k$-constellation. The \emph{combinatorial decomposition} of a connected rooted $k$-constellation $(\bM,c)$ is defined as follows: remove the right-path of the root, and iterate the operation on each corner incident to the root vertex until the root vertex is isolated; delete the root vertex; then, iterate the  construction on each remaining connected component, until no edge remains\footnote{We refer to the paper~\cite{ChapuyDolega2020} for the detailed construction, in which each remaining component is canonically rooted. For technical reasons, an operation of \emph{duality} has to be applied to each remaining connected component before iterating. This duality generalizes the classical map duality and it  exchanges the roles of vertices of colour $0$ and faces. We omit these subtle considerations here, see~\cite{ChapuyDolega2020} for more.\label{footnote:Duality}}.

Assuming that an underlying MON $\rho$ has been fixed, every time an edge $e$ is deleted by the algorithm, we say that the weight $\rho(\bN,e)$ is \emph{collected} by the algorithm (where $\bN$ is the current map at the time $e$ is deleted).
The following crucial property follows directly from the axioms of MON's.
\begin{definition-lemma}\label{deflemma:nu}
         Let $\rho$ be a MON, and let $(\bM,c)$ be a connected rooted
$k$-constellation. We define the \emph{weight} $\vec\rho(\bM,c)$ of
$(\bM,c)$ as the product of all the weights collected during the
combinatorial decomposition of $(\bM,c)$. If the MON $\rho$ is integral
then we have
         $$\vec\rho(\bM,c) = b^{\nu_{\rho}(\bM,c)},
         $$
         where $\nu_{\rho}(\bM,c)$ is a nonnegative integer, which is
zero if and only if $\bM$ is orientable.
\end{definition-lemma}
%\begin{proof}
%	The fact that $\vec\rho(\bM,c)$ is a monomial is a direct consequence of the definitions. The fact that $\nu_{\rho}(\bM,c)$ is zero if and only if $\bM$ is orientable follows from the fact that it is orientable if and only if the weight $1$ (instead of $b$) is collected at each step of the combinatorial decomposition, which is clear by inspecting all cases in the definition of a MON.
%      \end{proof}


   
\subsection{Generating functions of maps and the decomposition equations.}
\label{subsec:TutteEq}

	If $\bM$ is a constellation we denote by $cc(\bM)$ its number
        of connected components, and by $F(\bM)$ the set of its
        faces. For $i\geq 0$ we denote by $V_i(\bM)$ the set of
        vertices of color $i$ and by $v_i(\bM)$ its cardinality.
Recall also that its size $|\bM|$ is its number of corners
of colour $0$.

  For the rest of this paper we fix indeterminates $b$,
        $\pp=(p_i)_{i\geq 1}$, $\qq=(q_i)_{i\geq 1}$,
        $\yy=(y_i)_{i\geq 0}$, $\uu = (u_i)_{i\geq 1}$, which we will use to track combinatorial parameters of constellations in our generating functions.
Let $(\bM,c)$ be a rooted $k$-constellation, and $f_c$ be its root face. We associate to $(\bM,c)$ the monomial
		$$
		 \vec\kappa(\bM,c):= y_{\deg (f_c)}\prod_{f\in F(\bM)\setminus\{f_c\}} p_{\deg(f)}\prod_{v\in V_0(\bM)} q_{\deg(v)} \prod_{i=1}^k u_i^{v_i(\bM)}
		 = \frac{y_{\deg (f_c)}}{p_{\deg (f_c)}}\kappa(\bM),
		$$
		where  we recall that the monomial $\kappa(\bM)$ is defined by~\eqref{eq:kappa}.
%	\end{definition}

		\begin{definition}
Let $\rho$ be a MON.
Let \sloppy $\vec{H}_\rho(t;\pp,\qq, \yy,u_1,\dots,u_k) \in \QQ(b)[\yy,\pp,\qq,u_1,\dots,u_k][[t]] $ be the multivariate generating
function of rooted connected $k$-constellations given by
\begin{equation}
\label{eq:MultivariateGeneratingMapsRootedConnected}
	\vec{H}_\rho(t;\pp,\qq, \yy,u_1,\dots,u_k) := \sum_{n\geq 1}\sum_{(\bM,c)} t^{n} \vec{\rho}(\bM,c) \vec\kappa(\bM,c),
\end{equation}
where the second sum is taken over rooted connected (unlabelled) $k$-constellations of size~$n$.
%
%
%
%
%
%
%Formally $\vec{H}_\rho$ is viewed as formal power series
%in $t$, with coefficients that are polynomials in the
%variables $y_i,p_i,q_i,u_i$, with coefficients in $\mathbb{Q}(b)$, that is
%\[ \vec{H}_\rho \in \QQ(b)[\yy,\pp,\qq,u_1,\dots,u_k][[t]].\]
%\Gnew{
We also consider the multivariate generating
function of possibly disconnected \emph{labelled} $k$-constellations given by the formula
\begin{align}\label{eq:expansionUnconnected}
	F_\rho := 1+\sum_{\bM} \frac{t^{|\bM|}}{2^{|\bM|-cc(\bM)}|\bM|!} \frac{\tilde{\rho}(\bM) \kappa(\bM)}{(1+b)^{cc(\bM)}},
\end{align}
			where the $b$-weight $\tilde{\rho}(\bM)$ is defined from $\vec\rho(\bM,c)$, multiplicatively over connected components, and averaged over the choice of a random root in each component.
\end{definition}
%If we assume that MON satisfies some additional technical relation
%(that we call \emph{coherent} in our paper~\cite{ChapuyDolega2020})
%then we can show that the functions $F_\rho$ and $\vec{H}_\rho$ do not
%depend on the choice of $\rho$ and are related by the following formula
These functions are related by a version of the ``Exp-Log'' principle, namely
\begin{lemma}\label{lemma:explog} Introduce the operator $\Theta_Y := \sum_{i\geq 1}p_i\frac{\partial}{\partial y_i}$. Then we have
\begin{align}\label{eq:newDefFrho}
(1+b) t \frac{\partial}{\partial t}\ln F_\rho =\Theta_Y \vec H_\rho. 
\end{align}
\end{lemma}

%We will now state a set of equations (which we call ``decomposition equations'') that characterizes these functions. We first need to define some operators:
%
The function $F_\rho$ is characterized by the following ``decomposition equations'', which originates in the combinatorial decomposition of the previous section.
\begin{theorem}[Main PDE's -- combinatorial form]\label{thm:Tutte}
  Let $\rho$ be any coherent MON.
Then	
	the generating series $F_\rho \equiv F_\rho(t;\pp,\qq,\yy,u_1,\dots,u_k)$
	satisfies the following set of equations, for $m\geq 1$:
	\begin{align}\label{eq:TutteUnconnected}
		m \frac{q_m\partial}{\partial q_m} F_\rho = \Theta_Y t^m \cdot q_m \cdot
	\big(Y_+\prod_{l=1}^{k} (\Lambda_{Y}+u_l)\big)^{m}  \frac{y_0}{1+b} F_\rho,
	\end{align}
where
	\begin{align*}
	\Lambda_{Y} &:= (1+b)\sum_{i,j\geq 1}y_{i+j-1}\frac{i\partial^2}{\partial
  p_i \partial y_{j-1}} +\sum_{i,j\geq
  1}y_{i-1}p_j\frac{\partial}{\partial y_{i+j-1}}
+b\cdot\sum_{i\geq
			0}y_{i}\frac{i\partial}{\partial y_i },
	%
%
	\\
Y_+ &:= \sum_{i\geq 0}y_{i+1}\frac{\partial}{\partial
      y_i}.
	%
        %
    \end{align*}

      \end{theorem}

The proof of Theorem~\ref{thm:Tutte} closely follows the combinatorial
decomposition: roughly speaking, each term in the operator $\Lambda_Y$
corresponds to a way to add an edge from the root corner of a map in a
$k$-constellation. The product over $l=1..k$
in~\eqref{eq:TutteUnconnected} corresponds, very roughly, to the $k$ edges to be added to construct the right-path of the root, while the parameter $m$ controls the degree of the root vertex (so the operation of constructing a right-path has to be iterated $m$ times). That being said, there are considerable difficulties to overcome to make such an interpretation valid in the presence of $b$-weights.

%GC26/11 rewrote this paragraph:
% (my main point is: this is difficult; 
% but still,  I would like this paragraph to be understandable)
The main one is that the $b$-weight $\vec\rho(\bM,c)$ is defined for rooted objects,  while~\eqref{eq:TutteUnconnected} suggests a (randomized) decomposition applied to unrooted objects. 
%Thus one can not interpret, as is usual in map enumeration, the operator $\partial/\partial q_m$ cannot be directly interpreted as a ``rooting operator on a face of degree $m$'', since the $b$-weight and root-degree could be complicatedly correlated.
For $b\in\{0,1\}$ this would make no big difference, but since the $b$-weight and root-degree could be complicatedly correlated, here this introduces a difficulty.
Note moreover that  working with rooted objects (and with a deterministic, rather than random, decomposition) is necessary for applications to the integrality of coefficients, as averaging over the choice of a root may otherwise introduce rational numbers. In fact, we give in~\cite{ChapuyDolega2020} a ``connected'' version of~\eqref{eq:TutteUnconnected}, which is non-linear and on which integrality of coefficients is apparent. The subtle problem of correlations between degrees and $b$-weight is also the reason why we need to introduce duality as mentioned in  the footnote page~\pageref{footnote:Duality}.
%GC: here I just want to insist that this is NOT just a Tutte equation
% it, truly, is difficult!
% (I don't want to get a report saying that this part is trivial!)

\section{The Lax equations, Jack polynomials, and the proof}
\label{sec:LaxJackProof}

%GC-26/11:
%The two theorems below are the keystone of this paper. They show that
The theorem below shows that
the operators that appear in the decomposition equations can be
alternatively defined inductively by certain recurrence relations
involving commutators, which is the crucial link between Jack
polynomials and constellations. Their proof is the hardest part of the
paper~\cite{ChapuyDolega2020}.


\begin{definition}
The Laplace--Beltrami operator $D_{1+b}$ is the differential operator defined by
\begin{equation}
\label{eq:Laplace-Beltrami}
   D_{1+b} = \frac{1}{2}\left((1+b)\sum_{i,j\geq 1}p_{i+j}\frac{ij\partial^2}{\partial
  p_i \partial p_{j}} +\sum_{i,j\geq
  1}p_{i}p_j\frac{(i+j)\partial}{\partial p_{i+j}}+b\cdot\sum_{i\geq
  1}p_{i}\frac{i(i-1)\partial}{\partial p_i}\right).
\end{equation}
\end{definition}
%
Here and below $[\cdot,\cdot]$ denotes the algebra commutator,
$[A,B]=AB-BA$.
\begin{definition}\label{def:Lax}
	Define the three differential operators $(A_j)_{j\geq 1}$,
 $(B^{(k)}_m)_{m\geq 1}$, and $\Omega^{(k)}_Y$ on $\PPP:=\mathbb{Q}(b)[p_1,p_2,\dots]$ by the
	fact that the formal power series of operators 
	\begin{align*}
	A(s)&:=\sum_{j\geq
          0}\frac{s^j}{j!} A_{j+1} \\
    B^{(k)}(s)&:=\sum_{m\geq
          0}\frac{s^m}{m!} B^{(k)}_{m+1}
    \end{align*}
	  %and $C(s):=\sum_{j\geq
         % 0}\frac{s^j (j+1) \partial }{\partial q_{j+1}}$
	 each
        satisfies a Lax equation with respective Lax pairs
        $(A(s),D_{1+b}) and (B^{(k)}(s),\Omega^{(k)}_Y)$, % and  $(C(s),-C)$. 
	namely
	$$
	\frac{d}{ds} A(s) = [D_{1+b},A(s)]\ \ \ \mbox{and} \ \ \ 
	\frac{d}{ds} B^{(k)}(s) = [\Omega^{(k)}_Y,B^{(k)}(s)], \ \ \
%        \frac{d}{ds} C(s) = -[C,C(s)].
	$$
	with initial conditions $A_1  =  p_1/(1+b)$ and $B^{(k)}_1=\sum_{j=1}^{k+1}e_{k+1-j}(u_1,\dots,u_k)A_j$, and  with
	$$\Omega^{(k)}_Y:= \sum_{j=1}^{k+1}e_{k+1-j}(u_1,\dots,u_k)A_{j+1}.$$
\end{definition}

%\begin{align}\label{eq:EjCombi}
% A_{j}&:=\Theta_Y Y_+\Lambda_{Y}^{j-1} \frac{y_0}{1+b}, \qquad
%        \Omega^{(k)}_Y:= \sum_{j=1}^{k+1}e_{k+1-j}(u_1,\dots,u_k)A_{j+1},\\
%  B^{(k)}_{j}&:= (j-1)! \Theta_Y
%               \big(Y_+\prod_{i=1}^{k}(\Lambda_{Y}+u_i)\big)^j
%               \frac{y_0}{1+b},\qquad C := \sum_{j\geq 1}q_j\frac{(j+1)
%               \partial}{\partial q_{j+1}}
%\end{align}

The following theorem shows that the operators $A_{j}$ and $B^{(k)}_{m}$ defined by the Lax equation are in fact the ``building blocks'' which appear in the decomposition
equation~\eqref{eq:TutteUnconnected}. 
%We show that these operators
%satisfy Lax equations, that is they can
%be built recursively using commutators.
\begin{theorem}[Combinatorial operators solve the Lax equations]
  \label{prop:Lax}
	The operators $A_j, B^{(k)}_m$ defined above admit the explicit expression, on $\PPP$, 
\begin{align}\label{eq:EjCombi}
 A_{j} =\Theta_Y Y_+\Lambda_{Y}^{j-1} \frac{y_0}{1+b},
	%\qquad
        %\Omega^{(k)}_Y:= \sum_{j=1}^{k+1}e_{k+1-j}(u_1,\dots,u_k)A_{j+1},
\ \ \ \ \ \ \ \  	
  B^{(k)}_{m} = (m-1)! \Theta_Y
               \big(Y_+\prod_{i=1}^{k}(\Lambda_{Y}+u_i)\big)^m
               \frac{y_0}{1+b}.
	       %,\qquad C := \sum_{j\geq 1}q_j\frac{(j+1)
               %\partial}{\partial q_{j+1}}
\end{align}
%	\color{red}
%with solutions
%
%$$
%	A(s) = e^{s D_{1+b}} A_1 e^{-s D_{1+b}}\ \ \
%	B^{(k)}(s) = e^{s \Omega^{(k)}_Y} B^{(k)}_1 e^{-s
%          \Omega^{(k)}_Y}\ \ \  \mbox{and} \ \ \ C(s) = e^{-s
%          C} \frac{\partial}{\partial q_1} e^{sC}.
%$$
%	\color{red}
\end{theorem}

%This gives us an equivalent form of our main PDE.
%
%\begin{theorem}[Main PDE's -- algebraic form]\label{thm:TutteCommut}
%  Let $\rho$ be any coherent MON.
%Then	
%	the generating series $F_\rho \equiv F_\rho(t;\pp,\qq,\yy,u_1,\dots,u_k)$
%	satisfies the following equations:
%	\begin{align}\label{eq:TutteCommut}
%		\frac{\partial}{\partial q_1} F_\rho = t B^{(k)}_1
%                                                       F_\rho,\qquad C F_\rho = \Omega^{(k)}_Y F_\rho.
%	\end{align}
%      \end{theorem}
%
 The proof
 %the Lax equation for the series $C(s)$ is straightforward, but the remaining part of
 of~\cref{prop:Lax} is the crucial
 part of~\cite{ChapuyDolega2020} occupying most of its
 content.
 The technical algebraic proof has its origin in
 the following combinatorial ideas.
 %interpretation of the operators $A_j$ and
 %$B_j$ which leads to the valid proof in the special cases $b=0,1$.

 \begin{proof}[Sketch of the proof of \cref{prop:Lax} for $b\in \{0,1\}$]
	 The Lax equation for $A(s)$ is equivalent to the recurrence relation 
 \begin{align}\label{eq:commut1}
	 A_1 &= p_1/(1+b) \ \ \ ,\ \ A_{j+1} = [D_{1+b},A_j], \mbox { , for } j\geq 1.
\end{align}
%	 We now sketch a proof of~\eqref{eq:Commut1} for $b\in \{0,1\}$.
	 By induction, assume~\eqref{eq:EjCombi} for some $j\geq 1$. By analyzing carefully the root-edge decomposition, one can see that the operator $A_{j}$ has the following combinatorial interpretation: its creates an isolated root vertex (operator $y_0/(1+b)$), then a path of length $j\!-\!1$ from this root (iteration of the operator $\Lambda_{Y}$, where the $y$-variables keep track of the root face degree), and it ``anonymizes'' the root face (operator $\Theta_Y$).
% 
%
%	 The operator $A_{j+1}:=\Theta_Y
%	\Lambda_{Y}^{j} \frac{y_0}{1+b}$  can be interpreted as follows. First,
%create a new isolated vertex of color $0$, considered as the root
%vertex and counted by the factor . Then connect it
%successively to existing vertices $1,\dots,j$ ). Finally restore the marking of the root face from the $y$ to the $p$ variable (operator $\Theta_Y$). Thus this operator has the effect of creating a root and a partial path of length $j$ from this root, at the level of the $p$ variables.
%	
	Similarly, it can be shown that the operator $D_{1+b}$ can be interpreted as adding an edge of given color, at an arbitrary position in the map. 

	 The commutator $[D_{1+b}, A_{j}] = D_{1+b} A_{j}-A_{j}D_{1+b}$ therefore has the following interpretation: create a path of length $j\!-\!1$, then an edge, or do it in the converse order with a minus sign.
	 	 %By composing these operators, the product $D_{1+b} A_{j+1}$ has the
%effect of adding a new path of length $j$, and an edge of color
%$\{j,j+1\}$ somewhere in the map. Changing the order of the action of
%these operators $A_{j+1} D_{1+b}$ has the effect of adding an edge of
%color $\{j,j+1\}$, and then a new path.
	 The two are almost the same,
except that the second way to proceed does not include the case when the edge is added from
	 the very last corner of the newly created path (note the operator $Y_+$ in~\eqref{eq:EjCombi}), or equivalently when
this creates a path of length~$j$. 
%We conclude that the commutator
%$D_{1+b} A_{j+1}-A_{j+1} D_{1+b}$ has the effect of creating a path
%	 of length $j+1$, i.e. it is equal to $A_{j+2}$. 
	 Thus by~\eqref{eq:commut1} the effect of $A_{j+1}$ is to create a path of length $j$, which concludes the induction. %~\eqref{eq:commut1}.

	 For $b\in \{0,1\}$, the proof of the explicit form~\eqref{eq:EjCombi} of $B_m^{(k)}$ can be done in a similar way (where now $\tfrac{1}{(m-1)!}B^{(k)}_{m}$ and $\Omega^{(k)}_Y$ have the effect of creating a new vertex of colour~$0$ and degree $m$, and a path of length $k$, respectively).
	 See~\cite[Sec. 4.3]{ChapuyDolega2020}.
      \end{proof}

{\bf How to fix the proof for general $b$?} 
We insist that the last sketch of proof does \emph{not} work for a generic $b$. Indeed, it is based on the idea of constructing the same map by adding the same edges in different orders, but in the general case there is no reason \textit{a priori} that different orders give the same contribution to the $b$-weight. This is a \emph{major} problem, and the  whole strategy of~\cite{ChapuyDolega2020} is designed to overcome this difficulty.

A natural idea, to make the same approach work in general, would be to look for some sort of combinatorial operation, such as an involution, which would compare the different possible ways to add two edges (or two $k$-paths) to a given map in different orders, and that would ensure that the overall contributions to the $b$-weight are the same. This turns out to be (very) tricky but possible for the first Lax equation (for $A(s)$), but becomes intractable for the second one, in particular because such an operation would have to be nonlocal (since $k$ is  not a priori bounded). 

Our strategy of proof consists in ``lifting'' these combinatorial heuristics to an algebraic framework of operators, on which involutions and cancellations arguments can be replaced by abstract algebraic relations. This requires in particular to introduce several other sets of variables in addition to $(p_i)_{i\geq 1}, (y_i)_{i\geq 1}$, that heuristically represent several root faces in ``multi-rooted'' objects. We then appropriately promote the operators $D_{1+b}, \Lambda_Y$ to a much larger polynomial ring, on which several remarkable commutation relations can be established, which, in the end, lead us to Theorem~\ref{prop:Lax}. See~\cite[Sec. 4]{ChapuyDolega2020}.


      \subsection{Jack polynomials and the Laplace--Beltrami operator}
\label{subsec:Jack}

     We let $\Symm$ denote the algebra of symmetric functions over the
field $\QQ(b)$ of rational functions in $b$ with rational
coefficients, and we use the standard notation $p_\lambda, m_\lambda, e_\lambda, \dots$ for standard bases~\cite{Macdonald1995}. Since $\Symm = \QQ(b)[p_1,p_2,\dots]$ then clearly
the Laplace--Beltrami operator $D_{1+b}$ given by
\eqref{eq:Laplace-Beltrami} acts on the symmetric function
algebra. Its importance is reflected in the following characterization
of Jack symmetric functions:

\begin{defprop}
  \label{defprop:Jack}
Let $\alpha=1+b$. There is a unique family of symmetric functions
$\{J^{(\alpha)}_\lambda\}$ called \emph{Jack symmetric functions} such that for each partition $\lambda$,
	\begin{itemize}[itemsep=4pt, topsep=4pt,parsep=4pt]
\item $D_{1+b} J^{(\a)}_\lambda = \left(\sum_{\square \in \la}c_{\a}(\square)\right)J^{(\a)}_\lambda$;
\item
$ J_\la^{(\a)} = \hook_{\a}(\la) m_\la + \sum_{\nu < \la}a^{\la}_\nu m_\nu,  \text{ where } a^{\la}_\nu \in \QQ(b).$
\end{itemize}
	Here $c_{\a}(i,j) = \a i-j-b$ is the ``$(\a)$-content'' of the box $\square=(i,j)$
and 
\begin{align*}
  \hook_{\a}(\la) = \prod_{\square
  \in \la}\big(\a a_\la(\square)+\ell_\la(\square)+1 \big),\ \ \hook'_{\a}(\la)=\prod_{\square
  \in \la}\big(\a a_\la(\square)+\ell_\la(\square)+\a \big)
  \end{align*}
are $(\a)$-deformations of
	the hook-length product, with $a_\la(\square)$ and $\ell_\la(\square)$ denoting the ``arm'' and ``leg'' lengths of $\square$ in $\lambda$, \cite{Macdonald1995}.
\end{defprop}
\noindent The following properties are classical results of Stanley~\cite{Stanley1989}:
\begin{itemize}[itemsep=2pt, topsep=2pt,parsep=4pt]
	\item the coefficient
  $[J_\mu^{(\a)}]p_1 J^{(\a)}_\la$ is non-zero only if $\la \subset \mu$,
  \item 
	   $J_\la^{(\a)}(\underline{u}) = \displaystyle \!\prod_{\square \in
		\la\setminus(1,1)} \!(u+c_\a(\square))$, where we recall that  $\underline{u}$ is the specialization $p_i\equiv u$.
    \end{itemize}
Using \cref{defprop:Jack}, these two properties, and a number of algebraic manipulations, one can construct inductively (\cite[Sec.5 ]{ChapuyDolega2020}) a family of differential operators that cancel the function $\tau_b^{(k)}$ defined by \eqref{eq:JackIntro}. The relations defining these operators are equivalent to the Lax equations of Definition~\ref{def:Lax}, and we obtain with some work:
 \begin{theorem}[Main PDE -- Lax form]
  The function $\tau^{(k)}_b$ satisfies the
      equation, for $m\geq 1$:
	 \begin{align}\label{eq:tauLax}
%		\frac{\partial}{\partial q_1} \tau^{(k)}_b = t B^{(k)}_1
%                                                       \tau^{(k)}_b,\qquad C \tau^{(k)}_b = \Omega^{(k)}_Y \tau^{(k)}_b.
		\frac{m!\partial}{\partial q_m} \tau^{(k)}_b = t^m B^{(k)}_m
                                                       \tau^{(k)}_b.
						       %,\qquad C \tau^{(k)}_b = \Omega^{(k)}_Y \tau^{(k)}_b.
	\end{align}
      \end{theorem}
By Theorem~\ref{prop:Lax}, the differential equation~\eqref{eq:tauLax} solved by $\tau^{(k)}_b$ is in fact the same as the combinatorial equation~\eqref{eq:TutteUnconnected} solved by $F_\rho$ (Theorem~\ref{thm:Tutte}, for $\rho$ a coherent MON). It follows easily that these two series are equal, which given Lemma~\ref{lemma:explog}, is precisely the content of \cref{thm:abbreviated}, provided we choose a MON $\rho$ which is both coherent and integral.

 


 % \acknowledgements{}

%% if you use biblatex then this generates the bibliography
%% if you use some other method then remove this and do it your own way
\printbibliography


% \newpage 

% \appendix 
% \section{Differential operators: partial right-paths and commutation
%   relations}
% \label{sec:Operators}

% \subsection{Interpretation of the operator $\Lambda_{Y}$ and proof of the decomposition equation}
% \label{subsec:proofTutte}

% \begin{figure}%
% \center\includegraphics[width=\linewidth]{partialRightPath.pdf}
% 	\caption{A partial right-path as in \cref{def:partialPath}. Black edges belong to a $k$-constellation $\bN$, and red edges form a partial right-path $P_j$. In the case $j=k$, the color $j+1$ should be replaced by $j-1=k-1$ on the illustration.}
% \label{fig:partialRightPath}
% \end{figure}

% To prove the decomposition equation, we will show that each $k$-constellation can be constructed from smaller ones by adding edges one by one, thus working with intermediate objects that do not fully satisfy the constraints defining $k$-constellations.

% \begin{definition}[Partial right-path; see \cref{fig:partialRightPath}]\label{def:partialPath}
% 	Let $\bN$ be a $k$-constellation and $j\in [0..k]$. 
% 	Assume that $\bM_j$ is a map formed by adding a sequence of
%         new edges $P_j = (e_1,e_2, \dots, e_j)$ to $\bN$ (possibly also using some new vertices of color $i$ for $i\in [0..j]$), and assume that $\bM_j$ is rooted at some oriented corner $c_0$ of color $0$, incident to $e_1$, such that:
% 	\begin{itemize}
% 		\item the edge	$e_i$ has color $(i-1,i)$, for $i\in[1..j]$;
% 		\item starting from $c_0$ and following the tour of the face $f_0$ containing it in $\bM_j$, we follow the sequence of edges $e_1,e_2,\dots,e_j$ in this order;
% 		\item For $i<j$, the vertex of color $i$ on $P_j$ satisfies the local constraints of a $k$-constellation in the map $\bM_j$. 
% 		\item the vertex of $\bM_j$ that follows $e_j$  coming from $c_0$ has color $j-1$. 
% 	\end{itemize}
% 	Then  we say that $P_j$ is a \emph{partial right-path of length $j$} for $\bN$.
% \end{definition}

% The following is clear from definitions:
% \begin{lemma}
% 	If $P=(e_1,\dots,e_k)$  is a partial right-path of length $k$ for $\bN$, starting at some corner $c_0$, then $\bN\cup P$ is a $k$-constellation, and $P$ is the right-path of $c_0$ in $\bN \cup P$.
% \end{lemma}

% We will now construct operators that ``build'' partial right-paths, but for this we first have to define what markings we associate to the intermediate objects that are not exactly $k$-constellations.
% \begin{definition}[Markings for partial right-paths]
% 	Let $\bN$ be a constellation and $P_j$ be a partial right-path of length $j$ for $\bN$, of root corner $c_0$ and root vertex $v_0$.
% 	We define the marking $\hat\kappa(\bN\cup  P_j,c_0)$ of the ``intermediate'' constellation $\bN\cup  P_j$ as for usual rooted constellations, except that when measuring the degree of faces, we do not count the root corner $c_0$, and that we do not count the factor $q_i$ corresponding to the root vertex. That is to say:
% $$
% 	 \hat\kappa(\bN\cup  P_j,c_0):=
% 	 %
% 	 y_{\deg (f_0)-1}\prod_{f\in F(\bN\cup  P_j)\setminus\{f_0\}} p_{\deg(f)}\prod_{v\in V_0(M)\setminus\{v_0\}} q_{\deg(v)} \prod_{i=1}^k u_i^{v_i(\bN\cup  P_j)}
% 	 ,
% $$
% where $f_0$ (resp. $v_0$) is the face containing $c_0$
% (resp. the vertex incident to $c_0$) in $\bN\cup
% P_j$. 
% \end{definition}
% \noindent Note that when $c_0$ is the only corner of color $0$ in the face $f_0$, then $\hat\kappa(\bN\cup P_j,c_0) $ involves the variable $y_0$.



% 	The following proposition says that the effect of the operator $\Lambda_{Y}+u_j$ is to extend the length of a partial right-path by one unit.
% \begin{proposition}[Interpretation of $\Lambda_{Y}$]\label{prop:interpretGammaPatched}
% 	Let $\bN$ be a $k$-constellation, $j\in[0..k-1]$, and $P_j=(e_1,\dots,e_j)$ be a partial right-path for $\bN$. 
% Assume that $N\cup P_j$ is connected.  
% 	Let $\rho$ be a coherent MON.
% 	Then
% 	\begin{align*}
% 		(\Lambda_{Y}+u_{j+1})
% 	\hat\kappa(\bN \cup  P_j,c_0)
% 		= \sum_{e_{j+1}} \rho(\bN \cup  P_j \cup \{e_{j+1}\};e_{j+1})
% 	\hat\kappa(\bN\cup P_j \cup \{e_{j+1}\},c_0)
% 	,\end{align*}
% 	where the sum is taken over all possible additions of an edge $e_{j+1}$ (possibly using a new vertex of color $j+1$) such that $(e_1,\dots,e_j,e_{j+1})$ is a partial right-path of length $j+1$ for $\bN$.
% \end{proposition}

% \begin{proof}[Proof of \cref{prop:interpretGammaPatched}]
% 	In order to add the edge $e_{j+1}$ to the partial path $P_j$, we should connect the corner $c_j$ which follows $e_j$ on $P_j$, to some corner of color $(j+1)$, by some new edge  $e_{j+1}$. We can already note that after doing this, the colour constraints of $k$-constellations around the vertex of colour $j$ on $P_j$ will automatically be satisfied, from the last property of \cref{def:partialPath}.
	
% 	We first remark that if $c_{j+1}$ is a corner of color $j+1$ in $\bN$ incident to a vertex $v_{j+1}$, there are two different edges $e_{j+1}, \tilde e_{j+1}$ that can be added joining $c_j$ to $c_{j+1}$, where one is the twist of the other. 
% We distinguish two cases:
% 	\begin{itemize}[itemsep=0pt, topsep=0pt,parsep=0pt, leftmargin=12pt]
% 		\item if $j+1=k$, then both $(e_1,\dots,e_{k-1}, e_k)$ and $(e_1,\dots,e_{k-1}, \tilde{e}_k)$ are (partial) right-paths for $\bN$. Indeed, in this case there are no nontrivial colour constraints to satisfy.
% 		\item if $j+1<k$, then exactly one choice of $e\in \{e_{j+1}\tilde e_{j+1}\} $ is such that $(e_1,\dots,e_{j}, e)$ is a partial right-path for $\bN$. Indeed, since $\bN$ is a $k$-constellation the corner $c_{j+1}$ in $\bN \cup P_j$ is incident to two edges of color $ \{j,j+1\}$ and $\{j+1,j+2\}$, and the last constraint in \cref{def:partialPath} requires that after following $e$ along the path one reaches the edge of color $\{j,j+1\}$, which forces the choice of the twist.  In this proof we will say that this choice of $e$ is the \emph{valid choice}.
% 	\end{itemize}

% 	\smallskip
% 	Then we observe that each vertex of $\bM_j:=\bN\cup\{P_j\}$ satisfies the local constraints of a $k$-constellation, except for the vertex of colour $j$ on $P_j$.
% 	%
% 	This implies that, in the map $\bN\cup P_j$ each non-root face of degree $d$ contains exactly $2d$ corners of label $j+1$ if $j+1<k$, and $d$ corners of label $k$. The same is true for the root face provided we do not count the corner $c_0$ in the degree.
% 	Let $f$ be a face of $\bM_j$ of degree $d$ (or degree $d+1$ if
%         $f$ is the root-face). Orient $f$ arbitrarily and assume that
%         $j+1<k$. Let $u_1,\dots,u_{2d}$ be the list of corners of
%         color $j+1$ in $f$, with respect to the chosen orientation. When following the tour of $f$, the labels of the two corners visited before and after $u_i$ are either $(j-1,j+1)$ or $(j+1,j-1)$, and moreover, corners of the two types alternate. 
% 	For each such corner $u_i$, if we want to create a new edge $e_{j+1}$ from $c_j$ to $u_i$, only one possible twist of that new edge is a valid choice, and moreover, the type of twist which is valid alternates with corners.
% 	This observation being recorded, let us proceed with the proof by distinguishing some cases. There are several ways to create the new edge $e_{j+1}$:
% 	\begin{itemize}[itemsep=0pt, topsep=0pt,parsep=0pt, leftmargin=20pt]
% 		\item[(i)] \textit{we create a new isolated vertex of color $j+1$, linked by an edge to $c_j$.} This does not contribute to the $b$-weight, and the contribution to the marking is $u_{j+1}$. %
% 		\item[(ii)] \textit{we connect $c_j$ to a corner of color $c_{j+1}$ in a non-root face $f$.}
% 			If this chosen face has degree $d$, the degree of the root face will increase by $d$.
% If $j+1<k$, from the observation recorded above, the $2d$ corners of color $(j+1)$ in the chosen face give rise to $2d$ valid choices of edges whose twists alternate, and because $\rho$ is coherent, the total weight of these $2d$ possible additions sum up to
% 			\begin{align}\label{eq:contribution}
% 	(1+b)+(1+b)+\dots+(1+b) = d (1+b).
% 			\end{align}
% 			If $j+1=k$, there are  $d$ corners of color $(j+1)$ in the chosen face, and each of them corresponds to two possible choices of edges $e_{j+1}$ and $\tilde{e}_{j+1}$, which are both valid choices. By Axiom 1(b) of \cref{def:MON}, the sum of contributions to the $b$-weight of adding $e_{j+1}$ or $\tilde{e}_{j+1}$ is $(1+b)$. The total contribution in this case is thus again $d(1+b)$.
				
% 			Therefore both subcases of case $(ii)$ the contribution is $d(1+b)$ where $d$ is the degree of~$f$. We conclude that case (ii) is described by the operator:
% 			$$
% (1+b)\sum_{a,d\geq 1}y_{a+d-1}\frac{d\partial^2}{\partial
%   p_d \partial y_{a-1}} $$


% 		\item[(iii)] \textit{we connect $c_j$ to a corner $c_{j+1}$ of color $j+1$ inside the root face, and we choose the twist of the new edge so that we do not create any new face.} Let $(d+1)$ be the degree of the root face. If $j+1<k$, from the observation recorded above, only half of the $2d$ such edges are valid choices. On the other hand if $j+1=k$, we have $d$ such possible edges and all of them are valid. Hence the number of possible choices is $d$ in both cases. Moreover the degree of the root face does not increase, and the corresponding $b$-weight for each such choice is $b$ by Axiom 1(c) of \cref{def:MON}. Therefore contribution for this case is:
% 	$$b\cdot\sum_{d\geq
% 			0}y_{d}\frac{d\partial}{\partial y_d }.$$

% 		\item[(iv)] \textit{we connect $c_j$ to a corner $c_{j+1}$ of color $j+1$ inside the root face, and we twist the edge so that we create a new face in addition to the root face.}
% 			As before, if $j+1<k$ only half of the $2d$ corners of label $j+1$ in that face are valid choices from the observation above, and the $d$ valid choices alternate around the root face with the $d$ non-root corners of color $0$. Similarly if $j+1=k$, there are $d$ valid choices that alternate around the root face with the $d$ non-root corners of color $0$. 

% 			Therefore given $i,j \geq 1$, if the root face has degree $i+j$, there is exactly one choice of valid corner such that after adding the edge the new root face has degree $i$ (and the newly created face then has degree $j$). Moreover, the corresponding $b$-weight is $1$ (Axiom~1(c)), so the contribution for this case is:
% 			$$\sum_{i,j\geq
%   1}y_{i-1}p_j\frac{\partial}{\partial y_{i+j-1}}.$$
% 	\end{itemize}
% 	By summing contributions of  cases (ii)-(iii)-(iv), we
%         recognize the definition of the operator $\Lambda_{Y}$, so the
%         contribution of the four cases (i)-(ii)-(iii)-(iv) is $\Lambda_{Y}+u_{j+1}$ and the proof is complete.
% \end{proof}



% \begin{remark}We required the MON to be \emph{coherent} so that the second case of the decomposition equation gave rise to the correct weight. One could imagine relaxing further the notion of coherent MON to require only~\eqref{eq:contribution} to hold, rather than the stronger property that edges can be grouped in pairs of weight $(1+b)$ each.
% \end{remark}

% 	\begin{remark}The assumption that $\bN\cup P_j$ is connected is not strictly needed to give an interpretation of the operator $\Lambda_Y+u_i$ as adding an edge, but working with unconnected constellations would require to use a more sophisticated marking taking into account the number of connected components, in the spirit of~\eqref{eq:expansionUnconnected}. We will not need this discussion and prefer to avoid it. We leave to the reader the task of giving a direct interpretation of~\eqref{eq:TutteUnconnected} along these lines.
% \end{remark}


% 	The following proposition is the counterpart of Proposition~\ref{prop:interpretGammaPatched} for the case when the next edge on the partial right-path joins to a new connected component. 
% \begin{proposition}\label{prop:interpretGammaPatchedDisconnect}
% 	Let $\bN$ be a $k$-constellation, $j\in[0..k-1]$, and
%         $P_j=(e_1,\dots,e_j)$ be a partial right-path for $\bN$. 
% 	Assume that $\bN \cup P_j$ is connected. 
% 	Let
%         $\rho$ be a coherent MON.  
% 	Then
% 	\begin{align*}
% 		&\left(\sum_{i,j\geq 1}  y_{j+i-1}
% 		\widetilde{H}_\rho^{[i]} \frac{\partial}{\partial y_{j-1}}\right)
% 	\hat\kappa(\bN \cup  P_j,c_0)t^{|\bN|} \\
% 		&\hspace{2cm}	= \sum_{e_{j+1}, \bN'} \rho(\bN' \cup  P_j \cup \{e_{j+1}\};e_{j+1}) \vec\rho {(\tilde{\bN}'', \tilde{c})}
% 	\hat\kappa(\bN'\cup P_j \cup \{e_{j+1}\},c_0)t^{|\bN'|},
% \end{align*}
% 	where the sum is taken over all $k$-constellations
% 	$\bN'$ such that $P_{j+1}=(e_1,\dots,e_j,e_{j+1})$ is a partial
% 	right-path of length $j+1$ for $\bN'$, and such that removing
%         $e_{j+1}$ from the connected map $\bN'\cup P_{j+1}$ disconnects it into two components such that the component containing $P_j$ is $\bN\cup P_j$.
% 	In the sum, the other connected component is denoted by $\bN''$ and it is rooted at the first corner of colour $0$ following the corner from which $e_{j+1}$ was deleted,
% 	denoted by $c$. We denote by $(\tilde{\bN}'', \tilde{c})$ the dual of the rooted map $(\bN'',c)$.
% \end{proposition}
% \begin{proof}
% 	In order to build a map $\bN'$ as in the statement of the proposition, we should connect the corner $c_j$ which follows $e_j$ on $P_j$, to some corner of color $(j+1)$ in some new connected $k$-constellation $\bN''$, by adding some new edge  $e_{j+1}$ to a corner $c_{j+1}$ of $\bN''$. As in the proof of Proposition~\ref{prop:interpretGammaPatched},  after doing this the colour constraints of $k$-constellations around the vertex of colour $j$ on $P_j$ will automatically be satisfied, from the last property of \cref{def:partialPath}.

% 	Conversely, let $(\bN'',c)$ be a rooted connected $k$-constellation whose
%         root face $f$ has degree $i$. There is a unique
%         way of adding a valid edge from the corner $c_j$ to a
%         corner of colour $(j+1)$ in $f$, such that the first corner of colour $0$
% 	following the edge $e_{j+1}$ is equal to $c$ (indeed, similarly as in the proof of Proposition~\ref{prop:interpretGammaPatched}, valid corners and corners of colour $0$ alternate along faces). Moreover, by Axiom
%         1(a) of Definition~\ref{def:MON}, the contribution to the
%         $b$-weight of this addition is equal to $1$. 

% 	The contribution for the choice of the map $\bN''$, with root
%         face of degree $i$, is given by the generating function
%         $\widetilde{H}_\rho^{[i]}$, where we note that we are
%         computing it with the dual $b$-weight
%         $\vec\rho(\tilde{\bN}'',\tilde{c})$ as in \eqref{eq:dualFunction2}. Moreover, the root face will increase by $i$ when connecting the edge $e_{j+1}$. Therefore the overall contribution for the choice of $\bN'$ and $e_{j+1}$ is equal to
% $$\left(\sum_{i,j\geq 1}  y_{i+j-1}
% 		\widetilde{H}_\rho^{[i]} \frac{\partial}{\partial y_{j-1}}\right)
% 	\hat\kappa(\bN \cup  P_j,c_0)t^{|\bN|}.  ~ ~ ~ ~ ~~ ~ ~ ~ ~ ~ \qedhere
% 	$$
%       \end{proof}

%       We are now ready to prove the decomposition equation.
% \begin{proof}[Proof of \cref{thm:TutteConnected}]
% 	We invert the combinatorial decomposition of the previous
%         section. Any rooted connected (unlabelled) $k$-constellation $\bM$, with a root vertex of degree $m$, can be constructed as follows:
% 	\begin{enumerate}[itemsep=0pt, topsep=0pt,parsep=0pt, leftmargin=20pt]
% 		\item create a new isolated vertex of color $0$, with a marked corner $c_1$; 
% 		\item for $i$ from $1$ to $m$ do:
% 			\begin{enumerate}[itemsep=0pt, topsep=0pt,parsep=0pt, leftmargin=20pt]
% 				\item let $P^{(i)}_0$ be a new (empty) partial-right path of length $0$, rooted at $c_{i}$
% 				\item extend the partial right-path
%                                   $P^{(i)}_0$ into partial right-paths
%                                   $P^{(i)}_1, \dots, P^{(i)}_k$, by
%                                   adding edges one by one (possibly
%                                   adding new vertices, or connecting
%                                   them with rooted connected
%                                   (unlabelled) $k$-constellations along the way);
% 				\item once the right-path $P^{(i)}_k$ has been created, call $c_{i+1}$ the corner that follows $c_i$ around $v_0$, and reroot the current constellation at $c_{i+1}$; 
% 			\end{enumerate}
% 	\end{enumerate}
% 	The contribution of step (1) is simply $y_0$. By \cref{prop:interpretGammaPatched,prop:interpretGammaPatchedDisconnect}, for each $i
%         \in [1..m]$ the contribution of Step 3(b) is given by the
%         product of operators
%         \[\left(\Lambda_{Y}+u_k+\sum_{i,j\geq 1}  y_{j+i-1}
% 		\widetilde{H}_\rho^{[i]}\frac{\partial}{\partial y_{j-1}})\right)\cdots \left(\Lambda_{Y}+u_1+\sum_{i,j\geq 1}  y_{j+i-1}
% 		\widetilde{H}_\rho^{[i]}\frac{\partial}{\partial
%                   y_{j-1}})\right).\]
%             Indeed, note that the dual $b$-weight
%             $\rho(\tilde{\bN}'',\tilde{c})$ appearing in the R.H.S. of
%             the equation given by
%             \cref{prop:interpretGammaPatchedDisconnect} is coherent
%             with the fact that to compute the $b$-weight in the
%             combinatorial decomposition of $\bN'$, the $b$-weight of the smaller component $\bN''$ will be computed from its dual map $\tilde{\bN}''$. 
%             After Step 3(b) the corner $c_i$ is no longer the
%             root corner of the current map, so it has to be counted in
%             the marking $\hat \kappa$. This is taken into account by
%             the operator $Y_+$, so the overall contribution of Steps
%             3(b) and 3(c) is given by the operator
%             \[\Big(Y_+\prod_{l=1}^{k}\big(\Lambda_{Y}+u_l + \sum_{i,j\geq 1} y_{j+i-1}
% 		\widetilde {H}_\rho^{[i]} \frac{\partial}{\partial y_{j-1}}
% 		\big)\Big)^{m}.\]
%         \\
%         At the end of the process, the newly created vertex $v_0$
%         contributes a monomial $q_m$ to the marking, but this
%         contribution is killed by the factor $q_m^{-1}$ in front of
%         the defining equation\eqref{eq:DefHRhoM}. Finally the fact
%         that the size of the map increases by $m$ is taken into
%         account by a factor $t^{m}$. 
% 	Overall, the contribution of steps (1)--(3) is thus equal to
%             \[t^m\Big(Y_+\prod_{l=1}^{k}\big(\Lambda_{Y}+u_l + \sum_{i,j\geq 1} y_{j+i-1}
% 		\widetilde {H}_\rho^{[i]} \frac{\partial}{\partial y_{j-1}}
% 		\big)\Big)^{m}(y_0),\]
% 	which finishes the proof.
%       \end{proof}
      




% \subsection{Commutation relations and Lax pairs}
% \label{subsec:Commutation}



% \subsection{Heuristic: a simple combinatorial proof for $b=0$ or $1$}
% \label{subsec:heuristic}

% It is tempting to prove the commutation relations of \cref{thm:commut1,thm:commut2} by giving them a combinatorial interpretation. This turns out to be possible for $b=0$ or $1$. In this section we quickly sketch this idea because it is the inspiration for the algebraic proof we design in the next sections that works for all $b$.

% \begin{proof}[Sketch of the proof of \cref{thm:commut1} for $b\in \{0,1\}$]
% Similarly as in the proof of the decomposition equation and
% \cref{prop:interpretGammaPatched}, the operator $A_{j+1}:=\Theta_Y
% 	\Lambda_{Y}^{j} \frac{y_0}{1+b}$  can be interpreted as follows. First,
% create a new isolated vertex of color $0$, considered as the root
% vertex and counted by the factor $y_0/(1+b)$. Then create a partial right-path of length $j$ from this vertex, using only existing vertices (operator $\Lambda_{Y}^{j}$). Finally restore the marking of the root face from the $y$ to the $p$ variable (operator $\Theta_Y$). Thus this operator has the effect of creating a root and a partial path of length $j$ from this root, at the level of the $p$ variables.
	
% 	Moreover, it can also be shown with a bit of work that for any $j$ the operator $D_\alpha$ can be interpreted as adding an edge of color $\{j,j+1\}$ at an arbitrary position in the map. Similarly as in the proof of the decomposition equation, if $j+1<k$ only half of the possible edges are valid choices for this construction, while for $k=j+1$ all of them are.

% By composing these operators, the product $D_\alpha A_{j+1}$ has the
% effect of adding a new right path of length $j$, and an edge of color
% $\{j,j+1\}$ somewhere in the map. Changing the order of the action of
% these operators $A_{j+1} D_\alpha$ has the effect of adding an edge of
% color $\{j,j+1\}$, and then a new right path. This is almost the same,
% except that it does not include the case when the edge is added from the very last corner of the newly created right path, or equivalently when this creates a right path of length $j+1$. We conclude that the commutator $D_\alpha A_{j+1}-A_{j+1} D_\alpha$ has the effect of creating a right-path of length $j+1$, i.e. it is equal to $A_{j+2}$. This is precisely the first commutation relation. 
% \end{proof}


% \begin{proof}[Sketch of the proof of \cref{thm:commut2} for $b\in \{0,1\}$]
% 	Similarly as in the proof of \cref{thm:TutteConnected}, the operator $B^{(k)}_{m}=(m-1)! \Theta_Y \big(Y_+\prod_{i=1}^{k}(\Lambda_{Y}+u_i)\big)^m y_0$ can be interpreted as creating a new vertex of colour~$0$ and degree $m$, with an ordering of the edges incident to it, at the level of $p$ variables.
% 	Moreover it can be shown that $\Omega^{(k)}_Y$ has the effect
%         of adding a right-path (of length $k$, possibly using new
%         vertices along the way) to a $k$-constellation, while
%         $B^{(k)}_1$ has the effect of adding a new vertex of color $0$
%         and a right-path starting from it. Therefore, the operators
%         $\Omega^{(k)}_Y B^{(k)}_{m}$ and
%         $B^{(k)}_{m}\Omega^{(k)}_Y $ both have the effect of adding
%         a new vertex of color $0$ and degree $m$, and a new right-path
%         in the map, except that the second one does not include the
%         case when the new right path is incident to the new vertex. This corresponds precisely to creating a new vertex of degree $m+1$ with an ordering of its edges, i.e. to $B^{(k)}_{m+1}$, which gives the second commutation relation.
%       \end{proof}


% The sketch of the proof we just gave can be made fully rigorous in the
% case $b=0$ or $b=1$ with a bit of work (what remains to be done is the
% proof of the fact that the operators $D_\alpha$ and $\Omega^{(k)}_Y$
% can be interpreted as we claimed, with appropriate marking
% conventions). However, these proofs do \emph{not} work for general
% $b$. Indeed, they are based on the idea of constructing the same map
% by adding the same edges in different orders, but in the general case there is no reason \textit{a priori} that different orders give the same contribution to the $b$-weight. Our whole strategy is designed to overcome this difficulty, see \cref{rem:promotion} below.

% \subsection{Proof of the first commutation relations (\cref{thm:commut1})}
% \label{subsec:proofCommut1}

% The idea of the proof of \cref{thm:commut1} is the following: we
% ``promote'' the operator $D_\alpha$ to an operator (noted
% $D_\alpha+D_\alpha'$) acting on a larger space $\PPP_Y$ such that
% $\Theta_Y(D_\alpha+D_\alpha') = D_\alpha \Theta_Y$. This promoted
% operator commutes with $\Lambda_{Y}$ and its commutator with
% $\Lambda_Y$ is given by $Y_+\Lambda_Y$, see
% \cref{lemma:relations}. This enables us to perform simple algebraic
% manipulations leading to the proof of \cref{thm:commut1} by projecting
% the operators acting on $\PPP_Y$ on the subspace
% $\PPP$. \cref{rem:promotion} describes the (combinatorial) origin of this proof.

% We let $\PPP_Y$ be the set of polynomials in the variables $y_i$ and $p_j$ that are at most linear in the variables $y_i$, that is
% \begin{align}
% %
% \PPP_{Y} &:= \Span_{\mathbb{Q}(b)}\{p_\lambda, y_i p_\lambda\}_{ i \in \mathbb{N}, \lambda \in \Y}.
% %
% \end{align}
% Note that $\PPP \subset \PPP_{Y}$ and that differential operators in the variables $p_i$, such as $D_\alpha$, naturally act on $\PPP_Y$. 
% We now define the operator $D_\alpha'$ on $\PPP_Y$ by
% \begin{align*}
% D_\alpha' := 1/2 \left(   (1+b)\sum_{i,j\geq 1} 2y_{i+j} ij \frac{\partial^2 }{\partial y_i \partial p_j }+\sum_{i,j\geq 1} 2i y_{i}p_j \frac{\partial }{ \partial y_{i+j} }+b\sum_{i,j\geq1} i(i-1)y_i\frac{\partial }{\partial y_i} \right).
% \end{align*}


% \begin{lemma}\label{lemma:relations}
% 	We have the following commutation relations, as operators on $\PPP_Y$.
% 	\begin{subequations}
% 	\begin{align}
% 		[D_\alpha+D_\alpha',\Lambda_{Y}]&=0 \label{eq:rela1}\\
% 		[D_\alpha+D_\alpha',Y_+]&=Y_+\Lambda_{Y} \label{eq:rela2}\\
% 		\Theta_Y (D_\alpha+D_\alpha')&= D_\alpha\Theta_Y. \label{eq:rela3}
% 	\end{align}
% 	\end{subequations}
% \end{lemma}
% \begin{proof}
% The proof of these equations presents no difficulty since all
% operators have finite order. We refer the interested reader to \cref{sec:AppendixComputations}.
% \end{proof}

% \begin{proof}[Proof of \cref{thm:commut1}]
% 	The first two relations of \cref{lemma:relations} imply that
%         $[D_\alpha+D_\alpha',Y_+\Lambda_{Y}^j]=Y_+\Lambda_{Y}^{j+1}$
%         by induction on $j\geq 0$. Applying $\Theta_Y$ to this
%         identity we get:
% $$\Theta_Y  (D_\alpha+D_\alpha')Y_+\Lambda_{Y}^j-\Theta_Y Y_+\Lambda_{Y}^j (D_\alpha+D_\alpha') =\Theta_Y Y_+\Lambda_{Y}^{j+1}.$$
% Using the third relation of the lemma we obtain
% $$D_\alpha \Theta_Y Y_+\Lambda_{Y}^j-\Theta_Y Y_+\Lambda_{Y}^j (D_\alpha+D_\alpha') =\Theta_Y Y_+\Lambda_{Y}^{j+1} .$$
% 	Now we multiply by $y_0$ on the right, and notice that
%         $D_\alpha'y_0$ annihilates the space $\PPP$. 
% 	Therefore, as operators on $\PPP$ we have
% $$D_\alpha \Theta_Y Y_+\Lambda_{Y}^j y_0-\Theta_Y Y_+\Lambda_{Y}^j D_\alpha y_0=\Theta_Y Y_+\Lambda_{Y}^{j+1} y_0.$$ 
% Using that $D_\alpha y_0=y_0D_\alpha$ this shows that we have the
% following equality between operators on $\PPP$:
% $$
% 	\left[ D_\alpha,\Theta_Y Y_+\Lambda_{Y}^j y_0\right] =\Theta_Y Y_+\Lambda_{Y}^{j+1} y_0.
% $$
% Since for $j=0$ we have $\Theta_Y Y_+\Lambda_{Y}^{j} y_0= \Theta_Y Y_+
% y_0 = p_1$, we obtain \eqref{eq:commut1} by induction on $j$, which
% finishes the proof.
% \end{proof}

% \begin{remark}[Origin of this proof]\label{rem:promotion}
% %
% %
% %
% %
% %
% %
% Let us quickly explain the origin of this proof and of the operator $D_\alpha'$.
% 	The  idea of the combinatorial proof of \cref{subsec:heuristic} is that a given map can be obtained from a smaller one by adding the missing edges in several different orders. It fails in the context of $b$-weights because these different orders may give different contributions. To overcome this, it is natural to look for an ``exchange lemma'' that would say that in fact, the contributions are the same. More precisely, we would need to say that the operation of adding an edge $e$
% to a partial right-path in a rooted map $\bM$, and of adding an edge
% $f$ of a given colour not incident to this path, ``commute'' with respect to the $b$-weight. For example, for any map $\bM$, one could look for an involution $(e,f)\mapsto (e',f')$ on the set of such pairs of edges that preserves the rooted marking of the final map  and such that $\rho(\bM\cup \{e,f\},e,f) =\rho(\bM\cup \{e',f'\},f',e')$. 

% If such a proof exists, it should be represented algebraically by a
% simple commutation relation between the operator $\Lambda_{Y}$ and the
% operator that adds an edge of a given colour somewhere in the map. This operator needs to be a ``promoted'' version of $D_\alpha$ acting on the space $\PPP_Y$, that needs to take into account the case when the edge $f$ is incident to the root face. This is precisely what the operator $D_\alpha+D_\alpha'$ does.

% 	In fact, such a combinatorial proof can be given and we found it before the algebraic proof given here. We were able to make it work by using the coherent MON $\rho_{SYM}$, see \cref{rem:rhoSYM}.	
% 	However writing all the details turns out to be tedious and we
%         decided to give only the algebraic proof, leaving this remark for the interested reader.
% \end{remark}

% \subsection{Proof of the second commutation relations (\cref{thm:commut2})}
% \label{subsec:proofCommut2}

% In order to prove \cref{thm:commut1}, we promoted our operators from
% $\PPP$ to the larger space $\PPP_Y$, where we were able to control
% commutation relations.
%  In order to prove \cref{thm:commut2} we follow a similar approach, but we need to use the larger space $\PPP_{\tilde{Y},\tilde{Z}}$ defined below.
%  The rest of this section is dedicated to this proof.

% %
% %
% %
% %
% %
% %
% %
% %
% %
% %
% %
% %
% %
% %
% We introduce three new families of indeterminates $\bm{y'} = (y_i')_{i\geq 0}$ ,$\bm{z}=(z_i)_{i\geq 0}$,
% $\bm{z'} = (z_i')_{i\geq 0}$. We let $\PPP_{\tilde{Y},\tilde{Z}}$ be the space of polymomials in $\bm{y},\bm{y'},\bm{z},\bm{z'},\bm{p}$ which are at most linear in each of the families $\bm{y},\bm{y'},\bm{z},\bm{z'}$, and that do not involve simultaneously prime and non-prime variables in these families. Namely:
% %
% %
% %
% %
% %
% \begin{align}
% \PPP_{\tilde{Y},\tilde{Z}} &:= \Span_{\mathbb{Q}(b)}\{y_i z_j p_\lambda, y'_i z'_j p_\lambda, y_i p_\lambda, z_j p_\lambda, y'_i p_\lambda, z'_j p_\lambda, p_\lambda\}_{i,j \in \mathbb{N}, \lambda \in \Y}.
% \end{align}
% Clearly
% \[ \PPP \subset \PPP_{Y} \subset \PPP_{\tilde{Y},\tilde{Z}}.\]


% \begin{remark}[Origin of the space $\PPP_{\tilde{Y},\tilde{Z}}$]\label{rem:promotion2}
% 	In the spirit of \cref{rem:promotion}, in order to make the
%         heuristic proof of \cref{subsec:heuristic} work for the second
%         commutation relation, one would need an exchange lemma that
%         enables one to add two different right-paths to the same map,
%         with different roots, in two different orders, in such a way
%         that the contributions to the $b$-weights of both additions
%         are the same. But since the construction of right-paths only
%         applies to rooted objects, the proof of this lemma, which
%         would have to be inductive and work with partial right-paths,
%         would need  to keep track of \emph{two} root face degrees. It
%         is thus natural to use new variables $z_j$ to mark the size of
%         this second root face. However, one should not forget to
%         consider the case where, at some point of the construction,
%         both roots lie in the \emph{same} face of the map, thus
% 	splitting it into two intervals (say of length $i$ and $j$). 
% 	For this case, we use the variables $y'_i z'_j$, hence
%         the need of working with the big space
%         $\mathcal{P}_{\tilde{Y},\tilde{Z}}$. One needs to promote the
%         various operators we consider to this bigger setting, and
%         understand their commutators. For example, the operators
%         $\Lambda_{\tilde{Y}}$ and $\Lambda_{\tilde{Z}}$ defined below
%         are the promoted versions of the operator $\Lambda_{Y}$ (and
%         its $\mathbf{z}$-analogue $\Lambda_Z$), and they have the
%         effect of extending the first and second partial right path by
%         one unit, respectively. The key ``commutation relations''
%         between these operators are presented in \cref{lemma:relations3}. 
% \end{remark}


% We first define variants of the operator $\Lambda_{Y}$ for other alphabets. Since $\La_{Y}$ is acting also on the bigger space $\PPP_{\tilde{Y},\tilde{Z}}$ we can define operators $\La_{Y'},\La_{Z}$ and $\La_{Z'}$ acting on $\PPP_{\tilde{Y},\tilde{Z}}$ by analogy, that is $\La_A$ is obtained from the formula for $\La_{Y}$ by replacing each occurrence of $y_i$ and $\frac{\partial}{\partial y_i}$ by $a_i$ and $\frac{\partial}{\partial a_i}$ for each symbol $a\in\{z,y',z'\}$ of capital symbol $A\in\{Z,Y',Z'\}$. Using the same analogy, we define the operators $\Theta_{Y'},\Theta_{Z}, \Theta_{Z'}$,  $Y'_+,Z_+,Z'_+$, and $\Omega^{(k)}_Z$.
% Next, we define 
% \begin{align*}
% 	\La_{Z,Z'}^{Y,Y'} :=   (1\splus{}b)\ssum_{i,j,k\geq 1} \frac{y'_{i\splus{}j\sminus{}1} z'_{k\sminus{}1}\cdot\partial^2 }{\partial y_{i\splus{}k\sminus{}1} \partial z_{j\sminus{}1} }
% \splus{}\ssum_{i,j,k\geq 1}\frac{y_{i\splus{}j\sminus{}1} z_{k\sminus{}1}\cdot\partial^2 }{\partial y'_{i\splus{}k\sminus{}1} \partial z'_{j\sminus{}1} }\splus{}b\ssum_{i,j,k\geq 1} \frac{y'_{i\splus{}j\sminus{}1} z'_{k\sminus{}1}\cdot\partial^2 }{\partial y'_{i\splus{}k\sminus{}1} \partial z'_{j\sminus{}1} },
% \end{align*}
% and its version $\La_{Y,Y'}^{Z,Z'}$ with appropriately exchanged variables:
% \begin{align*}
% \La_{Y,Y'}^{Z,Z'} :=  (1\splus{}b)\ssum_{i,j,k\geq 1} \frac{z'_{i\splus{}j\sminus{}1} y'_{k\sminus{}1}\cdot\partial^2 }{\partial z_{i\splus{}k\sminus{}1} \partial y_{j\sminus{}1} }
% \splus{}\ssum_{i,j,k\geq 1}\frac{z_{i\splus{}j\sminus{}1} y_{k\sminus{}1}\cdot\partial^2 }{\partial z'_{i\splus{}k\sminus{}1} \partial y'_{j\sminus{}1} }\splus{}b\ssum_{i,j,k\geq 1} \frac{z'_{i\splus{}j\sminus{}1} y'_{k\sminus{}1}\cdot\partial^2 }{\partial z'_{i\splus{}k\sminus{}1} \partial y'_{j\sminus{}1} }.
% \end{align*}
% We also define
% \begin{align*}
% \Theta_{\tilde{Z}} &:=  \sum_{i\geq 0}p_i\frac{\partial}{\partial z_i} + \sum_{i,j\geq 0}y_{i+j}\frac{\partial^2}{\partial y'_i z'_j},
% \end{align*}
% with the convention that $p_0 = 1$, and 
% %
% $\Theta_{\tilde{Y}}$ by analogy.
% Finally, we define
% \begin{align*}
% 	\La_{\tilde{Y}} &:=  \La_{Y} + \La_{Y'} + \La_{Y,Y'}^{Z,Z'}\ , \ \ 
% \tilde{Y}_+ := Y_+ + Y'_+,\\
% 	\La_{\tilde{Z}} &:=  \La_{Z} + \La_{Z'} + \La_{Z,Z'}^{Y,Y'}\ , \ \ 
% \tilde{Z}_+ := Z_+ + Z'_+.
% \end{align*}
% %


% The following lemma is the analogue of \cref{lemma:relations} and it easily implies \cref{thm:commut2}.
% \begin{lemma}\label{lemma:relations2}
% 	We have the following relations between operators acting on $\PPP_Y$.
% 	\begin{align*}
% 		\left[\Omega^{(k)}_Z + \square^{(k)}, Y_+\prod_{i=1}^{k}(\Lambda_{Y}+u_i)\right]&=\left(Y_+\prod_{i=1}^{k}(\Lambda_{Y}+u_i)\right)^2, \\
% 		\Theta_Y (\Omega^{(k)}_Z + \square^{(k)}) &= \Omega^{(k)}_Z\Theta_{Y},\\
% 		y_0\cdot\PPP &\subset \ker\square^{(k)},
% 	\end{align*}
% 	where 
% 	\begin{align*}
% 		\square^{(k)} &= \Theta_{\tilde{Z}}\tilde{Z}_+\prod_{1 \leq i \leq k}(\La_{\tilde{Z}}+u_i)\La_{\tilde{Z}}z_0 - \Theta_{Z}Z_+\prod_{1 \leq i \leq k}(\La_{Z}+u_i)\La_{Z}z_0 \\
% 		&= \Theta_{\tilde{Z}}\tilde{Z}_+\prod_{1 \leq i \leq k}(\La_{\tilde{Z}}+u_i)\La_{\tilde{Z}}z_0 - \Omega^{(k)}_Z.
% 	\end{align*}
% \end{lemma}

% \begin{proof}[Proof of \cref{thm:commut2}]
% %
% %
% The first relation of \cref{lemma:relations2} and a direct induction imply that for all $m \geq 1$ one has
% \[ [\Omega^{(k)}_Z + \square^{(k)}, \big(Y_+\prod_{i=1}^{k}(\Lambda_{Y}+u_i)\big)^m]=m \big(Y_+\prod_{i=1}^{k}(\Lambda_{Y}+u_i)\big)^{m+1}.\]
% %
% %
% %
% %
% %
% Acting by $y_0$ on the right and by $\Theta_Y$ on the left, we obtain
% the following identity between operators on $\PPP$:
% 	\begin{align}\label{eq:interm11}
% \Theta_Y[\Omega^{(k)}_Z + \square^{(k)}, \big(Y_+\prod_{i=1}^{k}(\Lambda_{Y}+u_i)\big)^m]y_0=\Theta_Ym \big(Y_+\prod_{i=1}^{k}(\Lambda_{Y}+u_i)\big)^{m+1}y_0.
% \end{align}
% %
% %
% %
% %
% %

% 	Now, we know by \cref{lemma:relations2} that $y_0\cdot\PPP
%         \subset \ker\square^{(k)}$. Moreover the operator $[\Omega^{(k)}_Z,y_0] =
%         0$ obviously annihilates $\PPP$. Thus we get the following identity between operators on $\PPP$:
% \begin{align*} &\Theta_Y
%   \big(Y_+\prod_{i=1}^{k}(\Lambda_{Y}+u_i)\big)^m(\Omega^{(k)}_Z +
%   \square^{(k)})y_0 = \Theta_Y
%   \big(Y_+\prod_{i=1}^{k}(\Lambda_{Y}+u_i)\big)^m\Omega^{(k)}_Z y_0
%                       =\\
%   &=
%   \Theta_Y \big(Y_+\prod_{i=1}^{k}(\Lambda_{Y}+u_i)\big)^m
%   y_0\Omega^{(k)}_Z=
%   \Theta_Y \big(Y_+\prod_{i=1}^{k}(\Lambda_{Y}+u_i)\big)^m
%   y_0\Omega^{(k)}_Y.
%   \end{align*}
% Moreover, the second relation in \cref{lemma:relations2} gives 
% \[\Theta_Y(\Omega^{(k)}_Z + \square^{(k)}) \big(Y_+\prod_{i=1}^{k}(\Lambda_{Y}+u_i)\big)^m y_0 = \Omega^{(k)}_Z\Theta_Y \big(Y_+\prod_{i=1}^{k}(\Lambda_{Y}+u_i)\big)^m y_0.\]
% Thus
% \[ \Theta_Y[\Omega^{(k)}_Z + \square^{(k)}, \big(Y_+\prod_{i=1}^{k}(\Lambda_{Y}+u_i)\big)^m]y_0 = [\Omega^{(k)}_Z,\Theta_Y \big(Y_+\prod_{i=1}^{k}(\Lambda_{Y}+u_i)\big)^m y_0],\]
% 	which together with~\eqref{eq:interm11} implies the second relation of~\eqref{eq:commut2} and concludes the proof.
% \end{proof}

% We prove \cref{lemma:relations2} using an equality between operators acting on $\PPP_{\tilde{Y},\tilde{Z}}$ (\cref{lemma:relations3}) which we then project to $\PPP_Y$. In order to do this, we first need \cref{lemma:relations4}. 
% We let 
% \[ \Delta := (1+b)\sum_{i,j \geq 0}z'_i y'_j \frac{\partial^2}{\partial y_i\partial z_j} + \sum_{i,j \geq 0}z_i y_j \frac{\partial^2}{\partial y'_i\partial z'_j}+b\sum_{i,j \geq 0}z'_i y'_j \frac{\partial^2}{\partial y'_i\partial z'_j}.\]

% \begin{lemma}\label{lemma:relations4}
% 	We have the following relations between operators acting on $\PPP_{\tilde{Y},\tilde{Z}}$.
% 	\begin{subequations}
% 	\begin{align}
% 		\La_{\tilde{Z}}\Delta &= \Delta \La_{\tilde{Y}}, \label{eq:rel1} \\ 
% 	%
% 		 (\La_{\tilde{Z}}+\Delta)\La_{\tilde{Y}}&=\La_{\tilde{Y}}(\La_{\tilde{Z}}+\Delta)
% 		=(\La_{\tilde{Y}}+\Delta)\La_{\tilde{Z}}=\La_{\tilde{Z}}(\La_{\tilde{Y}}+\Delta)
% 		\label{eq:rel2bis}\\
% 		[\La_{\ZZ},\YY_+] &= \YY_+ \Delta, \mbox{ i.e. } \La_{\ZZ}\YY_+ = \YY_+ (\La_{\ZZ} + \Delta) ,\label{eq:rel3} \\
% 				[\La_{\YY},\ZZ_+] &= \ZZ_+ \Delta, \mbox{ i.e. } \La_{\YY}\ZZ_+ = \ZZ_+ (\La_{\YY} + \Delta) ,\label{eq:rel3bis} \\
% 		\Theta_{\tilde{Z}}\La_{\tilde{Y}} &=
%                                                     \La_{Y}\Theta_{\tilde{Z}}  \mbox{ and }\Theta_{\tilde{Y}}\La_{\tilde{Z}} = \La_{Z}\Theta_{\tilde{Y}}\label{eq:rel4}\\
%           		\Theta_{\tilde{Z}}\tilde{Y}_+ &= Y_+\Theta_{\tilde{Z}} \mbox{ and }\Theta_{\tilde{Y}}\tilde{Z}_+= Z_+\Theta_{\tilde{Y}}\label{eq:rel5}.%
% 		%
% 		%
% 	\end{align}
% 	\end{subequations}
% \end{lemma}

% \begin{proof}
% The proof of these equations presents no conceptual difficulty since all
% operators have finite order. We refer the interested reader to \cref{sec:AppendixComputations}.
% \end{proof}

% \begin{lemma}\label{lemma:relations3}
% 	For $n,m\geq 0$, we have the following equality between operators acting on $\PPP_{\tilde{Y},\tilde{Z}}$:
% 	\begin{align}\label{eq:crazySymmetrizedRelation}
% 		[\tilde{Z}_+\La_{\tilde{Z}}^{m+1},\tilde{Y}_+\La_{\tilde{Y}}^{n}] + [\tilde{Z}_+\La_{\tilde{Z}}^{n+1},\tilde{Y}_+\La_{\tilde{Y}}^{m}] &=\tilde{Y}_+\La_{\tilde{Y}}^n\tilde{Z}_+\La_{\tilde{Z}}^m\Delta + \tilde{Y}_+\La_{\tilde{Y}}^m\tilde{Z}_+\La_{\tilde{Z}}^n\Delta.
% 	\end{align}
% \end{lemma}

% \begin{proof}
% Without loss of generality we can assume that $m\geq n$, that is $m = n+i$ for $i \geq 0$.
% We first claim that it suffices to prove the formula
% \begin{align}\label{eq:explicitCommutator}
% 	[\tilde{Z}_+\La_{\tilde{Z}}^m,\tilde{Y}_+\La_{\tilde{Y}}^{n}] = \sum_{1\leq j \leq i}\YY_+\La_{\YY}^{n+i-j}\ZZ_+\La_{\tilde{Z}}^{n+j-1}\Delta = [\tilde{Y}_+\La_{\tilde{Y}}^m,\tilde{Z}_+\La_{\tilde{Z}}^{n}].
% \end{align}
% 	Indeed, assuming~\eqref{eq:explicitCommutator}, the L.H.S. of~\eqref{eq:crazySymmetrizedRelation} is equal to
% \begin{align*} 
% %
% 	&	[\tilde{Z}_+\La_{\tilde{Z}}^{m+1},\tilde{Y}_+\La_{\tilde{Y}}^{n}]-[\tilde{Z}_+\La_{\tilde{Z}}^{m},\tilde{Y}_+\La_{\tilde{Y}}^{n+1}] \\
% %
% 	&=\sum_{1\leq j \leq i+1}\YY_+\La_{\YY}^{n+i+1-j}\ZZ_+\La_{\tilde{Z}}^{n+j-1}\Delta - \sum_{1\leq j \leq i-1}\YY_+\La_{\YY}^{n+i-j}\ZZ_+\La_{\tilde{Z}}^{n+j}\Delta 
% \\ 
% 	&=\tilde{Y}_+\La_{\tilde{Y}}^n\tilde{Z}_+\La_{\tilde{Z}}^m\Delta +
% \tilde{Y}_+\La_{\tilde{Y}}^m\tilde{Z}_+\La_{\tilde{Z}}^n\Delta,
% \end{align*}
% 	since all  terms in the first sum cancel with the second sum, except $j\in\{1,i+1\}$. This is the desired equality.
	
	
% 	We now prove~\eqref{eq:explicitCommutator} by a repeated use of the relations of \cref{lemma:relations4}.
% 	We rewrite
%         \begin{multline}
%           \label{eq:takietam}
% 	[\tilde{Z}_+\La_{\tilde{Z}}^m,\tilde{Y}_+\La_{\tilde{Y}}^{n}] = \tilde{Z}_+\La_{\tilde{Z}}^{n+i}\tilde{Y}_+\La_{\tilde{Y}}^{n}-\tilde{Y}_+\La_{\tilde{Y}}^{n}\tilde{Z}_+\La_{\tilde{Z}}^{n+i} \\
% %
% 	=
% \tilde{Z}_+\tilde{Y}_+\big((\La_{\tilde{Z}}+\Delta)^{i}-\La_{\tilde{Z}}^i\big)(\La_{\tilde{Z}}+\Delta)^n\La_{\tilde{Y}}^{n},
% \end{multline}
% 	by applying first the relations \eqref{eq:rel3}--\eqref{eq:rel3bis} to move the operators $\YY_+$ and $\ZZ_+$ to the left, and then rearranging with \eqref{eq:rel2bis}.
% %
% %
% 	%

% We now expand $(\La_{\tilde{Z}}+\Delta)^{i}$ according to the position of the leftmost $\Delta$, and we get:
% \begin{align*} 
% \tilde{Z}_+\tilde{Y}_+\big((\La_{\tilde{Z}}+\Delta)^{i}-\La_{\tilde{Z}}^i\big)(\La_{\tilde{Z}}+\Delta)^n\La_{\tilde{Y}}^{n}
% 	&=\tilde{Z}_+\tilde{Y}_+ \sum_{1\leq j \leq i}\La_{\tilde{Z}}^{j-1}\Delta
% (\La_{\tilde{Z}}+\Delta)^{i-j}(\La_{\tilde{Z}}+\Delta)^n\La_{\tilde{Y}}^{n}
%  \\
% %
% %
% 	&=\tilde{Z}_+\tilde{Y}_+ \sum_{1\leq j \leq i}(\La_{\tilde{Y}}+\Delta)^{n+i-j}\La_{\tilde{Z}}^{n+j-1}
% \Delta,
% \end{align*}
% 	where for the second equality we first used the relation~\eqref{eq:rel1} to move the isolated operator $\Delta$ to the right, and then rearrange with~\eqref{eq:rel2bis}. Using the relation \eqref{eq:rel3bis} we can move the operator $\ZZ_+$ inside the sum, and we obtain the first equality in~\eqref{eq:explicitCommutator}. 
	
% 	If we expand $(\La_{\tilde{Z}}+\Delta)^{i}$
%         in~\eqref{eq:takietam} according to the position of the rightmost $\Delta$, we get
% 	\begin{align*} 
% \tilde{Z}_+\tilde{Y}_+ \big((\La_{\tilde{Z}}+\Delta)^{i}-\La_{\tilde{Z}}^i\big)(\La_{\tilde{Z}}+\Delta)^n\La_{\tilde{Y}}^{n}
% %
% 		&=\tilde{Z}_+\tilde{Y}_+ \sum_{1\leq j \leq i}(\La_{\tilde{Z}}+\Delta)^{j-1}\Delta
% \La_{\tilde{Z}}^{i-j}\La_{\tilde{Z}}^n(\La_{\tilde{Y}}+\Delta)^{n} \\
% %
% %
% %
% 		&=\tilde{Z}_+\tilde{Y}_+ \sum_{1\leq j \leq i}(\La_{\tilde{Z}}+\Delta)^{n+j-1}\La_{\tilde{Y}}^{n+i-j}\Delta,   
% 	\end{align*}
% using the same relations as before. Applying \eqref{eq:rel3} to move
% the operator $\YY_+$ yields the second equality in~\eqref{eq:explicitCommutator}.
% \end{proof}

% \begin{proof}[Proof of~\cref{lemma:relations2}]
% We have three statements to prove:

% \noindent $\bullet$
% We first prove that $y_0\cdot\PPP \subset \ker\square^{(k)}$.
% To see this, we replace in the formula
% \[ \square^{(k)} = \Theta_{\tilde{Z}}\tilde{Z}_+\prod_{1 \leq i \leq k}(\La_{\tilde{Z}}+u_i)\La_{\tilde{Z}}z_0 - \Theta_{Z}Z_+\prod_{1 \leq i \leq k}(\La_{Z}+u_i)\La_{Z}z_0 \]
%  the operator  $\La_{\tilde{Z}}$ by its definition $\La_{Z} + \La_{Z'} + \La_{Z,Z'}^{Y,Y'}$ and we expand the first product. We notice that the monomials in the expansion that involve only the operator $\La_Z$ cancel with the second product, therefore all remaining monomials involve one of the operators  $\La_{Z'}$ or $\La_{Z,Z'}^{Y,Y'}$ at least once. Therefore each term in the expansion involves either a derivative with respect to a prime variable, or a derivative $\frac{\partial}{\partial y_k}$ with $k\geq 1$, and the statement follows.

% 	\medskip

% \noindent $\bullet$
% We now prove that
% 	$[\Omega^{(k)}_Z + \square^{(k)}, \big(Y_+\prod_{i=1}^{k}(\Lambda_{Y}+u_i)\big)]=\big(Y_+\prod_{i=1}^{k}(\Lambda_{Y}+u_i)\big)^2.$ 
% Since
% 	\begin{align}\label{eq:Omega+square}
% 	\Omega^{(k)}_Z + \square^{(k)} = \Theta_{\tilde{Z}}\tilde{Z}_+\prod_{1 \leq i \leq k}(\La_{\tilde{Z}}+u_i)\La_{\tilde{Z}}z_0,   	\end{align}
% %
% %
% 	we can rewrite the desired identity as
% 	\begin{align}\label{eq:quadratic}
% 		[\Theta_{\tilde{Z}}\tilde{Z}_+P\big(\La_{\tilde{Z}}\big)\cdot \La_{\tilde{Z}},Y_+P\big(\La_{Y}\big)] &= \bigg(Y_+P\big(\La_{Y}\big)\bigg)^2,
% 	\end{align}
% 	where $P(x):=\prod_{1 \leq i \leq k}(x+u_i)x$.
% 	To prove this quadratic identity on polynomials, it is sufficient to prove the corresponding symmetrized bilinear identity\footnote{It would of course be sufficient to prove the non-symmetrized version of this bilinear identity, namely that the first terms on each side of~\eqref{eq:commutatorYZ} are equal, but this is not true!}
% \begin{multline}
% \label{eq:commutatorYZ}
% [\Theta_{\tilde{Z}}\tilde{Z}_+\La_{\tilde{Z}}^{m+1},Y_+\La_{Y}^n] + [\Theta_{\tilde{Z}}\tilde{Z}_+\La_{\tilde{Z}}^{n+1},Y_+\La_{Y}^m]
% = Y_+\La_{Y}^m Y_+\La_{Y}^n+Y_+\La_{Y}^n Y_+\La_{Y}^m,
% \end{multline}
% for $m,n\geq 0$.
% 	Indeed, assuming~\eqref{eq:commutatorYZ} and  writing $P(x)=\sum_{0\leq m \leq k} a_m x^m$, the L.H.S. of~\eqref{eq:quadratic} rewrites
% %
% %
% \begin{align*}
% %
% %
% 	&	\sum_{0 \leq m < n\leq k} a_m a_n\Big([\Theta_{\tilde{Z}}\tilde{Z}_+\La_{\tilde{Z}}^{m+1},Y_+\La_{Y}^n] + [\Theta_{\tilde{Z}}\tilde{Z}_+\La_{\tilde{Z}}^{n+1},Y_+\La_{Y}^m] \Big) 
% + \sum_{0 \leq m \leq k} a_m^2[\Theta_{\tilde{Z}}\tilde{Z}_+\La_{\tilde{Z}}^{m+1},Y_+\La_{Y}^m] \\
% 	&= \sum_{0 \leq m < n\leq k} a_m a_n\Big(Y_+\La_{Y}^m Y_+\La_{Y}^n+Y_+\La_{Y}^n Y_+\La_{Y}^m\Big) 
% + \sum_{0 \leq m \leq k} a_m^2 Y_+\La_{Y}^m Y_+\La_{Y}^m = \bigg(Y_+P\big(\La_{Y}\big)\bigg)^2.
% \end{align*}
% %
% %
% 	Now, by acting with the operators $\Theta_{\tilde{Z}}$ and
%         $z_0$, respectively on the left and right of the relation of
%         \cref{lemma:relations3} we have the following identity between
%         operators acting on $\PPP_Y$:
% \begin{equation}
% \label{eq:commutatorYZagain}
% \Theta_{\tilde{Z}}\Bigg([\tilde{Z}_+\La_{\tilde{Z}}^{m+1},\tilde{Y}_+\La_{\tilde{Y}}^{n}] + [\tilde{Z}_+\La_{\tilde{Z}}^{n+1},\tilde{Y}_+\La_{\tilde{Y}}^{m}]\Bigg)z_0 =\Theta_{\tilde{Z}}\Bigg(\tilde{Y}_+\La_{\tilde{Y}}^n\tilde{Z}_+\La_{\tilde{Z}}^m\Delta + \tilde{Y}_+\La_{\tilde{Y}}^m\tilde{Z}_+\La_{\tilde{Z}}^n\Delta\Bigg)z_0.
% \end{equation}
	
% 	It thus suffices to prove that the left (right, resp.) hand side of \eqref{eq:commutatorYZ} and \eqref{eq:commutatorYZagain} coincide. 
% We start with the left hand side. First, we claim that
% 	\begin{align}\label{eq:halfRelation}
% 		\Theta_{\tilde{Z}}[\tilde{Z}_+\La_{\tilde{Z}}^{m+1},\tilde{Y}_+\La_{\tilde{Y}}^{n}]z_0 = [\Theta_{\tilde{Z}}\tilde{Z}_+\La_{\tilde{Z}}^{m+1}z_0,Y_+\La_{Y}^n].
% 	\end{align}
% Indeed, 
% \[
% \Theta_{\tilde{Z}}[\tilde{Z}_+\La_{\tilde{Z}}^{m+1},\tilde{Y}_+\La_{\tilde{Y}}^{n}]z_0 = \Theta_{\tilde{Z}}\tilde{Z}_+\La_{\tilde{Z}}^{m+1}\tilde{Y}_+\La_{\tilde{Y}}^{n}z_0 - \Theta_{\tilde{Z}}\tilde{Y}_+\La_{\tilde{Y}}^{n}\tilde{Z}_+\La_{\tilde{Z}}^{m+1}z_0.\]
% Using the fact that $[z_0,\La_{Y}] = 0$ annihilates $\PPP_Y$, and the
% fact that $\La_{Y'}$ and $\La_{Y,Y'}^{Z,Z'}$ annihilate $z_0\PPP_Y$ we
% substitute $\La_{\tilde{Y}} = \La_{Y}+\La_{Y'}+\La_{Y,Y'}^{Z,Z'}$ and obtain
% \[ \Theta_{\tilde{Z}}\tilde{Z}_+\La_{\tilde{Z}}^{m+1}\tilde{Y}_+\La_{\tilde{Y}}^{n}z_0 = \Theta_{\tilde{Z}}\tilde{Z}_+\La_{\tilde{Z}}^{m+1}z_0 Y_+\La_{Y}^{n}.\]
% Similarly, using the relations~\eqref{eq:rel4}-\eqref{eq:rel5}, we obtain
% \[ \Theta_{\tilde{Z}}\tilde{Y}_+\La_{\tilde{Y}}^{n}\tilde{Z}_+\La_{\tilde{Z}}^{m+1}z_0 = Y_+\La_{Y}^{n}\Theta_{\tilde{Z}}\tilde{Z}_+\La_{\tilde{Z}}^{m+1}z_0,\]
% which together with the previously proved equation, implies~\eqref{eq:halfRelation}. 
% Since the same equation with $m$ and $n$ exchanged also holds, this proves that the left hand sides of \eqref{eq:commutatorYZ} and \eqref{eq:commutatorYZagain} coincide. 
	
% 	We now turn to the right-hand sides of \eqref{eq:commutatorYZ} and \eqref{eq:commutatorYZagain}. First, we have
% \[ \Theta_{\tilde{Z}}\tilde{Y}_+\La_{\tilde{Y}}^m\tilde{Z}_+\La_{\tilde{Z}}^n\Delta z_0 = Y_+\La_{Y}^m \Theta_{\tilde{Z}}\tilde{Z}_+\La_{\tilde{Z}}^n\Delta z_0\]
% by relations~\eqref{eq:rel4}-\eqref{eq:rel5}. Moreover
% \begin{multline*} 
% 	\Theta_{\tilde{Z}}\tilde{Z}_+\La_{\tilde{Z}}^n\Delta z_0 y_i
%         p_\lambda = \Theta_{\tilde{Z}}\tilde{Z}_+\La_{\tilde{Z}}^n y_0'z'_i p_\lambda=	\Theta_{\tilde{Z}}y_0'\tilde{Z}_+\La_{\tilde{Z}}^n z'_i p_\lambda=\Theta_{\tilde{Z}}y_0'Z'_+\La_{Z'}^n z'_i p_\lambda=
%         Y_+ \La_{Y}^n y_i p_\lambda
%       \end{multline*}
% by direct inspection of the definitions of operators. This implies
% that the action of
% $\Theta_{\tilde{Z}}\tilde{Y}_+\La_{\tilde{Y}}^m\tilde{Z}_+\La_{\tilde{Z}}^n\Delta
% z_0$ and $Y_+ \La_{Y}^m Y_+ \La_{Y}^n$ on $\PPP_Y$ coincides.
% Therefore the right hand sides of \eqref{eq:commutatorYZ} and \eqref{eq:commutatorYZagain} coincide, which finally implies that \eqref{eq:commutatorYZ} holds true. This concludes the proof of the desired identity.


% \medskip

% \noindent $\bullet$
% 	It only remains to prove that $\Theta_Y (\Omega^{(k)}_Z + \square^{(k)}) = \Omega^{(k)}_Z\Theta_{Y}$. From~\eqref{eq:Omega+square} and from the relations~\eqref{eq:rel4}-\eqref{eq:rel5}, we directly obtain $\Theta_{\tilde Y} (\Omega^{(k)}_Z + \square^{(k)}) = \Omega^{(k)}_Z\Theta_{\tilde{Y}}$. But $\Theta_{\tilde{Y}}$  and $\Theta_{Y}$ have the same action on $\PPP_Y$, therefore we get that $\Theta_{Y} (\Omega^{(k)}_Z + \square^{(k)}) = \Omega^{(k)}_Z\Theta_{Y}$, as operators on $\PPP_Y$, which is the desired relation.
% 	\end{proof}



% \section{$b$-deformation of the tau-function}
% \label{sec:Jack}

% %
% %
% %
% %
% %
% %
% %
% %


% In this section we study the $b$-deformed tau-function $\tau^{(k)}_b$, defined in~\eqref{eq:JackIntro} using Jack symmetric functions. We show that it is annihilated by the operators defined in the previous section, which makes the connection with the generating function of coverings $F_\rho$ and prove our main result, \cref{thm:mainInSection5}.

% \subsection{Jack symmetric functions}

% \subsubsection{Partitions and symmetric functions}
% %
% %
% %
% %
% The group
% $\Sym{\infty}$ of permutations of $\mathbb{N}_{\geq 1}$ with a finite support
% acts naturally on the set of 
% sequences of nonnegative integers with finite support $\A = \bigoplus_{i = 1}^{\infty}\N$, and
% partitions represents orbits of this action. We can rephrase this
% observation as follows. Let $\Symm_n :=
% \QQ[x_1,\dots,x_n]^{\Sym{n}}$ be the algebra of symmetric polynomials,
% that is polynomials in $x_1,\dots,x_n$ invariant by the natural action
% of $\Sym{n}$ permuting their variables. Let $\Symm :=
% \underleftarrow{\Symm_n}$ be the projective limit with respect to the
% natural morphism $\Symm_{n+1} \ni f(x_1,\dots,x_n,x_{n+1}) \mapsto
% f(x_1,\dots,x_n,0)$. The algebra of symmetric functions $\Symm$ has a natural homogenous basis
% indexed by partitions and obtained by symmetrizing monomials:
% \[m_\la = \sum_{\alpha \in \Sym{\infty} \la} \xx^\alpha,\]
% where $\Sym{\infty} \la$ is the orbit of the partition $\la$ by the
% action of the permutation group $\Sym{\infty}$ on $\A$, and
% $\xx^\alpha$ is the monomial $\xx^\alpha = \prod_i
% x_i^{\alpha_i}$. In particular $\Symm = \bigoplus \Symm^n$ has a natural gradation by
% degree, and $\Symm^n$ is a finite-dimensional vector space,
% whose dimension is given by the number of partitions of size
% $n$.

% There is another base of $\Symm^n$ of a great importance in this
% paper, which is called \emph{power-sum} basis, and is given by 
%     \[ p_\lambda = \prod_{i=1}^{\ell(\lambda)} p_i; \quad p_i = \sum_j x_j^i
%       \text{ for } i>0.\]
% An immediate consequence of this fact is that $\Symm$ is a polynomial
% algebra, $\Symm = \QQ[p_1,p_2,\dots]$.

% \subsubsection{Laplace-Beltrami operator and Jack symmetric function}

% In order to define Jack symmetric functions, which are the main
% characters of this section, we need to introduce some simple
% combinatorial statistics of partitions.

% We let $\PPP_n$ denote the set of partitions of size $n$. There is an important poset structure on $\PPP_n$
% given by the \emph{dominance order:}
% \[ \lambda \leq \mu \iff \sum_{i\leq j}\lambda_i \leq \sum_{i\leq
%     j}\mu_i \text{ for any positive integer } j.\]
% To any partition $\lambda \in \PPP_n$ we can associate
% a conjugate partition $\la^t = (\la^t_1,\dots,\la^t_{\ell'})$, where
% $\ell' = \la_1$, and for any $1 \leq i \leq \ell'$
% \[ \la^t_i = \#\{j: \la_j \geq i\}.\]
% The concept of conjugate partition is very natural from a geometric
% point of view. Indeed we can represent a partition $\la$ by drawing its
% \emph{Young diagram}, which consists of the set 
% \[ \la = \{(i, j):1 \leq i \leq \la_j, 1 \leq j \leq \ell(\la) \}\]
% and then conjugating $\la$ corresponds to reflecting its Young diagram
% through the line $x=y$.  For any
%  box $\square := (i,j) \in \la$ from Young diagram we define its
%  \emph{arm-length} by $a(\square) := \la_j-i$ and its
%  \emph{leg-length} by $\ell(\square) := \la_i^t-j$. These definitions follow~\cite[Chapter I]{Macdonald1995}.

% Let $\alpha = 1+b$ be an indeterminate. There are
% several natural statistics on the set of partitions (or,
% equivalently, Young diagrams) that we need:
% \begin{align}
% %
% %
% \label{eq:HookProduct}
% \hook_\a(\la) &:= \prod_{\square \in \la}\left(\a\ a(\square) + \ell(\square) +1 \right),\\
% \label{eq:HookProduct2}
% \hook'_\a(\la) &:= \prod_{\square \in \la}\left(\a\ a(\square) +
%                  \ell(\square) +\a \right),\\
% \label{eq:NumericalFactor}
% z_\la &:= \prod_{i \geq 1}i^{m_i(\la)}m_i(\la)!,
% \end{align}
% where $m_i(\la)$ denotes the number of parts of $\la$ equal to $i$
% (therefore $n!z_\lambda^{-1}$ is the number of permutations from the
% conjugacy class of type $\lambda$). We
% also recall that for a box $\square = (x,y) \in \la$ its
% content is equal to $x-y = (x-1) - (y-1)$. We define its
% $\a$-deformation by
% \[c_\a(\square) := \a(x-1)-(y-1).\]

% Let $\Symm_\alpha$ denote the algebra of symmetric functions over the
% field $\QQ(\alpha)$ of rational functions in $\alpha$ with rational
% coefficients. Since $\Symm_\alpha = \QQ(\a)[p_1,p_2,\dots]$ then clearly
% the Laplace-Beltrami operator $D_\a$ given by
% \eqref{eq:Laplace-Beltrami} acts on the symmetric function
% algebra. Its importance is reflected in the following result.

% \begin{defprop}
%   \label{defprop:Jack}
% There is a unique family of symmetric functions
% $\{J^{\alpha}_\lambda\}$ such that for each partition $\lambda$,
% \begin{itemize}
% \item $D_\alpha J^{\alpha}_\lambda = \left(\sum_{\square \in \la}c_\a(\square)\right)J^{\alpha}_\lambda$;
% \item
% $ J_\la^\a = \hook_\a(\la) m_\la + \sum_{\nu < \la}a^{\la}_\nu m_\nu,  \text{ where } a^{\la}_\nu \in \QQ(\a).$
%   \end{itemize}
%   We call them \emph{Jack symmetric functions}.
% \end{defprop}

% \begin{remark}
%   Jack symmetric functions
%   are usually defined by three conditions: orthogonality,
% 	normalization, and triangularity (which is the second property in our definition). However,
%   \cref{defprop:Jack} is the core of the proof that the classical definition
%   makes sense. Therefore we are
%   going to treat \cref{defprop:Jack} as a definition of Jack symmetric functions in
%   this paper and we refer to \cite{Stanley1989,Macdonald1995} for completeness.
% \end{remark}

% We can endow $\Lambda_\alpha$ with a scalar product by defining it on
% the basis of power-sums
% \begin{equation}
%   \label{eq:product}
%   \langle p_\mu,p_\nu\rangle_\a = \a^{\ell(\mu)}z_\mu\delta_{\mu,\nu},
% \end{equation}
% where $\delta_{\mu,\nu}$ is the Kronecker delta. It turns out that Jack
% symmetric functions are also orthogonal with the following squared norm:
% \[\langle J_\la^\a,J_\la^\a\rangle_\a = \hook_\a(\la)
%   \hook'_\a(\la) =: j_\la.\]
% Note that this is a one-parameter deformation of the factor
% $(\frac{f_\la}{n!})^2$ appearing in the definition~\eqref{eq:SchurIntro}
% of the classical
% tau-function. Indeed, for  $\alpha = 1$, $\hook_1(\la) = \hook'_1(\la)$
% coincides with the classical hook-length appearing in the hook-length formula for $f_\lambda$ (see e.g.~\cite{Stanley:EC2}), thus \[ \frac{n!^2}{f_\la^2} = j_\la.\]

% For any linear operator $D \in \End(\Symm_\alpha)$ we can
% define its adjoint $D^\perp$ with respect to
% $\langle\cdot,\cdot\rangle_\alpha$ so that
% \[\langle Df,g\rangle_\alpha = \langle
% f,D^\perp g\rangle_\alpha\] for all symmetric functions $f,g \in
% \Symm_\alpha$.
% For instance
% \begin{equation}  
%   \left(p_j/\a\right)^\perp = j\partial/\partial p_j,
% \end{equation}
% which is a direct consequence of \eqref{eq:product}.

% From now on, we will use the notation $J_\la^{(\a)}(\pp)$ to indicate
% that we are treating Jack polynomials as polynomials in the ``power-sum'' variables
% $p_1,p_2,\dots$, considered as indeterminates.

% \medskip

% We are now going to show that the operators $G_j := j!\partial/\partial p_j$
% are determined by a similar recursion as the operators $A_j$ from
% \eqref{thm:commut1}:

% \begin{lemma}\label{lem:commutSimple}
% Define the differential operators $(G_j)_{j\geq 1}$ on $\PPP$ by:
% \begin{align}
%   \label{eq:GjCombi}
%  G_{j}:= j!\partial/\partial p_j \ \ j\geq0.
% \end{align}
% Then these operators satisfy the recurrence formula
% \begin{align}\label{eq:commutSimple}
% 	G_1 &= \partial/\partial p_1 \ \ \ ,\ \ G_{j+1} = [G_j,A_2^\perp], \mbox { , for } j\geq 1.
% \end{align}
% \end{lemma}

% \begin{proof}
%   The proof is made by induction on $j$ and it is an easy
%   computation. Note that
%   \[ A_2^\perp = \left(\Theta_Y Y_+\Lambda_{Y} \frac{y_0}{\alpha}\right)^\perp = \left(\sum_{i
%       \geq 1} p_{i+1}\cdot \left(p_{i}/\a\right)^\perp\right)^\perp
% %
% %
% %
% 	= \sum_{i
%         \geq 1} \left(p_{i+1}/\a\right)^\perp\cdot p_{i}.\]
%     Since
%     \[j![    \partial/\partial p_j, \left(p_{i+1}/\a\right)^\perp\cdot p_{i}] =
%     j!\delta_{i,j}\left(p_{i+1}/\a\right)^\perp\]
%   we obtain 
%   \[ G_{j+1} = j! \left(p_{j+1}/\a\right)^\perp = j! \sum_{i \geq 1}[
%     \partial/\partial p_j, \left(p_{i+1}/\a\right)^\perp\cdot
%     p_{i}] = [G_j,A_2^\perp],\]
%   and we conclude the proof.
% \end{proof}

% Stanley obtained in his seminal paper \cite{Stanley1989} some results concerning Jack symmetric functions, which are of special
% interest for us. These results can be seen as $\a$--deformations of
% a classical product formula and a special case of
% Pieri rule for Schur polynomials. 
% Moreover, for two partitions $\la,\mu \in \PPP$ we write $\la
% \nearrow \mu$ if $|\mu|-|\la| = 1$ and the Young diagram  of $\la$ is
% contained in the one of $\mu$.


% \begin{theorem}[\cite{Stanley1989}]
%   For any $\lambda \in \PPP_n$ one has
%   \begin{align}
%     \label{eq:JackEvaluationProdForm}
%      J_\la^{(\a)}(\underline{u}) &= \prod_{\square \in
%                          \la\setminus(1,1)}(u+c_\a(\square)),\\
%     \label{eq:JackPieriRule}
%     p_1 J_\la^{(\a)}(\pp) &= \sum_{\la \nearrow \mu}c_{\la \nearrow \mu}
%                        J_\mu^{(\a)}(\pp),
%     \end{align}
% where $c_{\la \nearrow \mu} \in \N[\a]$ is a (explicit) polynomial in
% $\a$ with nonnegative integer coefficients.

% \end{theorem}

% \begin{corollary}
%   \label{cor:ActionOfEOnJack}
%   For any $i \geq 1$ the following identity holds true
%   \[ A_i J_\la^{(\a)}(\pp) = \sum_{\la \nearrow \mu}c_\a(\mu\setminus\la)^{i-1}\frac{c_{\la \nearrow \mu}}{\a}
%                        J_\mu^{(\a)}(\pp).\]
%                    \end{corollary}

%                    \begin{proof}
%                      We use induction on $i$. For
%                      $i=1$ one has $A_1 = p_1/\a$, so this is simply
%                      \eqref{eq:JackPieriRule}.
%                      We recall that
% 			   \begin{align}\label{eq:jackEigenvector}
% 				   D_\a  J_\la^{(\a)}(\pp) = \left(\sum_{\square \in \la} c_{\a}(\square)\right)  J_\la^{(\a)}(\pp).
% 			   \end{align}
%                      Thus
%                      \begin{multline*}
%                        A_{i+1} J_\la^{(\a)}(\pp) =
%                          [D_\a,A_i]J_\la^{(\a)}(\pp)  = \sum_{\la \nearrow \mu}\left(\sum_{\square \in\mu}c_\a(\square)- \sum_{\square \in\la}c_\a(\square)\right)c_\a(\mu\setminus\la)^{i-1}\frac{c_{\la \nearrow \mu}}{\a}
%                          J_\mu^{(\a)}(\pp)\\
%                          =\sum_{\la \nearrow
%                            \mu}c_\a(\mu\setminus\la)^i \frac{c_{\la \nearrow \mu}}{\a}
%                        J_\mu^{(\a)}(\pp). \ \ \  \qedhere
%                        \end{multline*}
%                      \end{proof}

% \subsection{The $b$-deformation of the tau-function}

% In this section we prove our main theorem.
% In the proof, the differential operators defined in previous sections with respect to the variables $\pp$ will also be used with respect to the variables $\qq$, and for this we introduce the following more precise notation. We denote by $A_i(\pp)$, $B^{(k)}_i(\pp)$, $G_i(\pp)$, respectively,
%  the operators defined by \eqref{eq:EjCombi},
% \eqref{eq:FmCombi} and \eqref{eq:GjCombi}. We
% denote by $A_i(\qq)$, $B^{(k)}_i(\qq)$, $G_i(\qq$), respectively,
% the operators obtained from $A_i(\pp)$ $B^{(k)}_i(\pp)$, $G_i(\pp)$
%  by replacing each occurence of the indeterminate $p_i$ in
% their definition by the indeterminate $q_i$ for each $i>0$.
% Moreover we recall that, everywhere, $\a = 1+b$.

% Define $\tau^{(k)}_b(t;\pp,\qq,u_1,\dots,u_k) \in
% \QQ(b)[\pp,\qq,u_1,\dots,u_k][[t]]$ by
% \begin{align}
%   \label{eq:MultiJackDef}
%   \tau^{(k)}_b(t;\pp,\qq,u_1,\dots,u_k) := \sum_{n \geq 0}t^n\sum_{\la
%     \vdash n}\frac{J_\la^{(1+b)}(\pp)J_\la^{(1+b)}(\qq)J_\la^{(1+b)}(\underline{u_1})\cdots J_\la^{(1+b)}(\underline{u_k})}{j_\la^{(1+b)}},
% \end{align}
% with the convention that the constant term in $t$ is equal to $1$.



%         \begin{lemma}
%       \label{lem:MultiJack1}
%       The generating series $\tau^{(k)}_b$ satisfies the following
%       equation:
%       \begin{equation}
%         \label{eq:VirasoroJack1}
%         G_1(\qq) \tau^{(k)}_b(t;\pp,\qq,u_1,\dots,u_k) =
%           tB^{(k)}_1(\pp) \tau^{(k)}_b(t;\pp,\qq,u_1,\dots,u_k).
%           \end{equation}
%       \end{lemma}

% \begin{proof}
%         We fix an integer $n \geq 0$ and we look at the coefficient of $t^{n+1}$ in the %
% %
% L.H.S.
%         of \cref{eq:VirasoroJack1}, which is given by the formula:
%         \begin{multline*}
%           \sum_{\la
%     \vdash
%     n+1}\frac{J_\la^{(1+b)}(\pp)\JJ_\la^{(1+b)}(u_1,\dots,u_k)}{j_\la^{(1+b)}}G_1(\qq)
% J_\la^{(1+b)}(\qq) = \\
%   \sum_{\la
%     \vdash
%     n+1}\frac{J_\la^{(1+b)}(\pp)\JJ_\la^{(1+b)}(u_1,\dots,u_k)}{j_\la^{(1+b)}}\frac{\partial}{\partial
%     q_1}   J_\la^{(1+b)}(\qq),
% \end{multline*}
% where
% \[\JJ_\la^{(1+b)}(u_1,\dots,u_k) := J_\la^{(1+b)}(\underline{u_1})\cdots J_\la^{(1+b)}(\underline{u_k}).\]
% Applying the Pieri rule \eqref{eq:JackPieriRule} we obtain the following
% expression
% \[ \sum_{\la
%     \vdash
%     n+1} J_\la^{(1+b)}(\pp)\JJ_\la^{(1+b)}(u_1,\dots,u_k)\sum_{\mu \nearrow \la}\frac{c_{\mu \nearrow \la}}{1+b}\frac{J_\mu^{(1+b)}(\qq)}{j_\mu^{(1+b)}}.\]
% It is straightforward from \eqref{eq:JackEvaluationProdForm} that for
% any $\mu \nearrow \la$ we have
% 	\begin{align}\label{eq:contentForm}
% 		J_\la^{(1+b)}(\underline{u}) =
% \big(u+c_{\a}(\la\setminus\mu)\big)\cdot
% J_\mu^{(1+b)}(\underline{u}),
% 	\end{align}
% which gives the following identity:
% \[ \JJ_\la^{(1+b)}(u_1,\dots,u_k) = \big(\sum_{1 \leq i \leq
%     k+1}e_{k+1-i}(u_1,\dots,u_k) c_{\a}(\la\setminus\mu)^{i-1}\big)
%   \JJ_\mu^{(1+b)}(u_1,\dots,u_k).\]
% Plugging it into the expression of the L.H.S. of \cref{eq:VirasoroJack1}
% and changing the order of summation we obtain the following formula:
% \[ \sum_{\mu
%     \vdash
%     n} \frac{J_\mu^{(1+b)}(\qq)\JJ_\mu^{(1+b)}(u_1,\dots,u_k)}{j_\mu^{(1+b)}}\big(\sum_{1 \leq i \leq
%     k+1}e_{k+1-i}(u_1,\dots,u_k) \sum_{\mu \nearrow
%     \la}c_{\a}(\la\setminus\mu)^{i-1}\frac{c_{\mu \nearrow \la}}{1+b} J_\la^{(1+b)}(\pp)\big).\]
% Finally, \cref{cor:ActionOfEOnJack} implies that this expression is
% equal to
% \[ \sum_{\mu
%     \vdash
%     n}
%   \frac{J_\mu^{(1+b)}(\qq)\JJ_\mu^{(1+b)}(u_1,\dots,u_k)}{j_\mu^{(1+b)}}B^{(k)}_1(\pp)J_\mu^{(1+b)}
%   = [t^{n+1}]\RHS,\]
% which leads to the desired identity:
% \[ G_1(\qq)\tau^{(k)}_b(t;\pp,\qq,u_1,\dots,u_k) =
%   tB^{(k)}_{1}(\pp) \tau^{(k)}_b(t;\pp,\qq,u_1,\dots,u_k). \qedhere\]
% \end{proof}


% \begin{lemma}
%       \label{lem:MultiJack2}
%       The generating series $\tau^{(k)}_b$ satisfies the following
%       equation:
%       \begin{equation}
%         \label{eq:VirasoroJack2}
%         A_2^\perp(\qq) \tau^{(k)}_b(t;\pp,\qq,u_1,\dots,u_k) =
%           t\Omega_Y^{(k)}(\pp) \tau^{(k)}_b(t;\pp,\qq,u_1,\dots,u_k).
%           \end{equation}
%       \end{lemma}

% \begin{proof}
% The proof is very similar to the proof of the previous lemma. We fix an integer $n \geq 0$ and we look at the coefficient of $t^{n+1}$ in the L.H.S.
%         of \cref{eq:VirasoroJack2}:
%         \begin{multline*}
%           \sum_{\la
%     \vdash
%     n+1}\frac{J_\la^{(1+b)}(\pp)\JJ_\la^{(1+b)}(u_1,\dots,u_k)}{j_\la^{(1+b)}} A_2^\perp(\qq)
%   J_\la^{(1+b)}(\qq) \\
%   = \sum_{\la
%     \vdash
%     n+1}J_\la^{(1+b)}(\pp)\JJ_\la^{(1+b)}(u_1,\dots,u_k)
%   \sum_{\mu \nearrow \la} c_\a(\la\setminus\mu)\frac{c_{\mu \nearrow \la}}{1+b}\frac{J_\mu^{(1+b)}(\qq)}{j_\mu^{(1+b)}},
% \end{multline*}
% by \cref{cor:ActionOfEOnJack}. Applying the same substitutions as in the
% proof of \cref{lem:MultiJack2} we transform the coefficient of
% $t^{n+1}$ on the L.H.S. of \cref{eq:VirasoroJack2} to the following form:
% \[ \sum_{\mu
%     \vdash
%     n} \frac{J_\mu^{(1+b)}(\qq)\JJ_\mu^{(1+b)}(u_1,\dots,u_k)}{j_\mu^{(1+b)}}\big(\sum_{1 \leq i \leq
%     k+1}e_{k+1-i}(u_1,\dots,u_k) \sum_{\mu \nearrow
%     \la}c_{\a}(\la\setminus\mu)^{i}\frac{c_{\mu \nearrow \la}}{1+b} J_\la^{(1+b)}(\pp)\big).\]
% Finally, \cref{cor:ActionOfEOnJack} gives that this expression is
% equal to
% \[ \sum_{\mu
%     \vdash
%     n}
%   \frac{J_\mu^{(1+b)}(\qq)\JJ_\mu^{(1+b)}(u_1,\dots,u_k)}{j_\mu^{(1+b)}}\Omega_Y^{(k)}(\pp)
%   J_\mu^{(1+b)}(\pp)
%   = [t^{n+1}]\RHS,\]
% which finishes the proof.
% \end{proof}

% These two lemmas composed together give us the following equations, 
% which are a keystone of this paper.

%     \begin{lemma}
%       \label{lem:MultiJack}
%       For any $m \geq 1$ the generating series $\tau^{(k)}_b$ satisfies the following
%       equation:
%       \begin{equation}
%         \label{eq:VirasoroJack}
%         G_m(\qq) \tau^{(k)}_b(t;\pp,\qq,u_1,\dots,u_k) =
%           t^mB_m^{(k)}(\pp) \tau^{(k)}_b(t;\pp,\qq,u_1,\dots,u_k).
%           \end{equation}
%       \end{lemma}

%       \begin{proof}
%         We use induction on $m$. For $m=1$ the
%         statement is precisely 
%         \cref{lem:MultiJack1}. We fix a nonnegative integer $l >1$ and
%         suppose that \cref{eq:VirasoroJack}
%         holds true for any $m<l$. Then 
%       \[  G_l(\qq) \tau^{(k)}_b(t;\pp,\qq,u_1,\dots,u_k)
%         = [G_{l-1}(\qq), A_2^\perp(\qq)]\tau^{(k)}_b(t;\pp,\qq,u_1,\dots,u_k),\]
%       which is equal to
%       \begin{align*}
%         \big(&G_{l-1}(\qq)\cdot\Omega_Y^{(k)}(\pp) t-
%         A_2^\perp(\qq) \cdot B^{(k)}_{l-1}(\pp)
%                                                 t^{l-1}\big)\tau^{(k)}_b(t;\pp,\qq,u_1,\dots,u_k) 
%         = \big(\Omega_Y^{(k)}(\pp)\cdot G_{l-1}(\qq) t-\\
%      -   &B^{(k)}_{l-1}(\pp)\cdot A_2^\perp(\qq) 
%                                                 t^{l-1}\big)\tau^{(k)}_b(t;\pp,\qq,u_1,\dots,u_k)
%                                                   = [\Omega_Y^{(k)}(\pp),
%         B^{(k)}_{l-1}(\pp)]t^l\tau^{(k)}_b(t;\pp,\qq,u_1,\dots,u_k)
%         \end{align*}
%       by \cref{lem:MultiJack2}, induction hypothesis, and the fact
%       that operators $A_2^\perp(\qq), G_{l-1}(\qq)$ commute with $\Omega_Y^{(k)}(\pp), B^{(k)}_{l-1}(\pp)$. This
%       finishes the proof since
%       \[ [\Omega_Y^{(k)}(\pp),
%         B^{(k)}_{l-1}(\pp)] = B^{(k)}_{l}(\pp). \qedhere\]
%     \end{proof}
   

% \subsection{Proof of the main results}

% We are now ready to make the connection between $k$-constellations and the function $\tau^{(k)}_b$.
% For $m\geq 1$ we introduce the functions
% $$
% U_m := (1+b) m \frac{\partial}{\partial q_m} \log \tau^{(k)}_b \ \ 
% , \ \ V_m =(1+b) m \frac{\partial}{\partial p_m} \log \tau^{(k)}_b.
% $$
% Recall that $\pi$, defined in Section~\ref{sec:MonTutte} is
%   the operator exchanging the sets of variables
%   $\pp\leftrightarrow\qq$ and $u_i\leftrightarrow u_{k+1-i}$ for $1
%   \leq i \leq k$. 
% \begin{proposition}\label{prop:mixDiffEqs}
% 	For $m\geq 1$ one has:
% 	\begin{align*}
% 		U_m =t^m  \cdot
% 		\Theta_Y \Big(Y_+\prod_{l=1}^{k}\big(\Lambda_{Y}+u_l + \sum_{i,j\geq 1}V_i  y_{j+i-1}
% 		 \frac{\partial}{\partial y_{j-1}}
% 		\big)\Big)^{m}  (y_0),
% 	\end{align*}
% 	and moreover $V_m = \pi U_m$. 
% \end{proposition}
% \begin{proof}
% 	From the previous lemma and from the definition~\eqref{eq:FmCombi} of $B_m^{(k)}$, we have for $m\geq 1$,
% 	\begin{align}\label{eq:patch1}
% 	m\frac{\partial}{\partial q_m} \tau^{(k)}_b =
% 	\Theta_Y \big(Y_+\prod_{l=1}^{k}(\Lambda_{Y}+u_l)\big)^m \frac{y_0}{1+b} \tau^{(k)}_b.
% 	\end{align}
% 	Now, for any series $A(\yy,\pp)$ depending on variables $\yy$ and $\pp$ (and possibly other parameters), we have by definition of the operator $\Lambda_Y$:
% 	\begin{align*}%
% 		(\Lambda_{Y}+u_l) A(\yy,\pp) \tau^{(k)}_b =
% 		\Big((\Lambda_{Y}+u_l+ 
% 		\sum_{i,j\geq 1}  V_i y_{j+i-1}
% 		\frac{\partial}{\partial y_{j-1}} )A(\yy,\pp) \Big) \tau^{(k)}_b,
% 	\end{align*}
% 	where we used that $(1+b) i\frac{\partial}{\partial p_{i}}\tau^{(k)}_b = V_i \tau^{(k)}_b$.
% 	Applying this identity repeated times to~\eqref{eq:patch1}, and using also $m\frac{\partial}{\partial q_m} \tau^{(k)}_b=\frac{1}{1+b} U_m \tau^{(k)}_b$ , we get
% 	\begin{align*}
% 		\frac{1}{1+b} U_m =
% 		\Theta_Y \big(Y_+\prod_{l=1}^{k}
% 		\big(\Lambda_{Y}+u_l+ 
% 		\sum_{i,j\geq 1}  V_i y_{j+i-1}
% 		\frac{\partial}{\partial y_{j-1}} \big)
% 		\Big)^m \left(\frac{y_0}{1+b}\right),
% 	\end{align*}
% 	which is the desired identity upon multiplying by $(1+b)$.

% 	The fact that $V_m = \pi U_m$ is clear since $\tau^{(k)}_b = \pi \tau^{(k)}_b$.
% \end{proof}

% From Corollary~\ref{cor:TutteThetaConnected} we immediately deduce
% \begin{corollary}\label{cor:equalInSection5}
% Let $\rho$ be a coherent MON. 
% Then for any $m\geq 1$, one has $U_m = H^{[m]}$, where $H^{[m]}$ is the generating function of constellations defined by~\eqref{eq:DefHRhoM}.
% \end{corollary}
% Recall that $\sum_{m \geq 1} m q_m \frac{\partial}{\partial q_m}$ and $ t \frac{\partial}{\partial t}$ act similarly on $\tau^{(k)}_b$, therefore we obtain in particular
% \begin{theorem}\label{thm:mainInSection5}
% Let $\rho$ be a coherent MON. Then one has
% $$
% 	(1+b) t \frac{\partial}{\partial t} \ln \tau^{(k)}_b = \Theta_Y \vec H_\rho.
% $$
% In particular, if $\rho$ is integral then we have
% 	\begin{align}\label{eq:mainInSection5}
% (1+b) t \frac{\partial}{\partial t} \ln \tau^{(k)}_b =
% 	\sum_{n \geq 1}\sum_{(\bM,c)} t^{n} b^{\nu_\rho(\bM,c)} \kappa(\bM),
% 	\end{align}
% 	where the second sum is taken over rooted connected $k$-constellations $(\bM,c)$ of size $n$.
% \end{theorem}
% Note that, up to the correspondence between constellations and
% generalized branched coverings, the last statement is precisely our main result stated in the introduction (Theorem~\ref{thm:abbreviated}).



%   \section{Weighted Hurwitz numbers and projective limits}
% \label{sec:infinite}

% In previous sections we have considered $k$-constellations for an arbitrary fixed $k$, which correspond to coverings with $k+2$ ramification points. However in the literature concerning the orientable case, much interest has been given to cases where the number of ramification points is unbounded, which is the case for (weighted) Hurwitz numbers.
% In this section we explain how to obtain this case as a limit of the previous one by allowing $k$ to become, in some sense, infinite.
% %

% \begin{quote}
% 	\centering \textit{In this section, a coherent MON $\rho$ is fixed.}
% \end{quote}


% \subsection{Infinite constellations and projective limits}

% We let $\mathcal{C}^{(k)}$ be the set of $k$-constellations, and $\mathcal{C}^{(k)}_n$ the subset formed by the ones of size $n$. 
% We equip these objects with the \emph{modified marking} 
% $$
% 	\hat \kappa(\bM) := \prod_{f\in F(\bM)} p_{\deg(f)}\prod_{v\in V_0(\bM)} q_{\deg(v)} \prod_{i=1}^k u_i^{n-v_i(\bM)},
% $$
% which differs from the marking~\eqref{eq:marking} used in the previous
% section only by the exponent of $u_i$ which counts "non-vertices"
% instead of vertices. We let $\hat F_k$ denote the corresponding
% ``modified'' generating function, given by the substitution:
% \begin{align}\label{eq:hatFk}
% 	\hat F^{(k)}(t,\mathbf{p},\mathbf{q},u_1,\dots,u_k) :=F_\rho(t
%   \cdot u_1\cdots u_k,\mathbf{p},\mathbf{q},u_1^{-1},\dots,u_k^{-1}).
% \end{align}
% We define the modified marking in the same way for rooted objects.
% In the rest of \cref{sec:infinite} we will work with modified markings. To avoid confusion, all the new generating functions we define in this section will be also denoted with a hat symbol`$~\widehat{~}~$'.

% \begin{remark}
% 	In previous sections, the parameter $k$ was fixed, and it could be deduced from the notation for the function $F_\rho$ only by its number of arguments. Since in this section it is crucial to let $k$ vary, we indicate it explicitly in the notation $\hat F^{(k)}$. On the other hand, now that we have proved that our generating functions do not depend on the choice of a coherent MON, we drop the dependency in $\rho$ from the notation.
% 	The two sides of Equation~\eqref{eq:hatFk} reflect these differences of notation.
% \end{remark}

% We observe that the addition of a leaf of colour $(k+1)$ to every vertex of colour $k$ gives an inclusion 
% \begin{align}\label{eq:inclusion}
% 	\mathcal{C}_n^{(k)} \hookrightarrow \mathcal{C}_n^{(k+1)}.
% \end{align}
% This inclusion preserves the modified marking.
% The same inclusion holds at the level of rooted objects and it preserves the rooted modified marking.

% Moreover, one deduces from~\eqref{eq:TutteUnconnected}
% %
% %
% %
%   %
%   %
%   %
%   %
%   that
% \begin{align}\label{eq:projectiveCondition}
% 	\hat F^{(k)}(t,\mathbf{p},\mathbf{q},u_1,\dots,u_k) =\hat F^{(k+1)}(t,\mathbf{p},\mathbf{q},u_1,\dots,u_k,0).
% \end{align}
% This implies that it is possible to define the $b$-weights $\tilde\rho(\bM)$, $\vec\rho(\bM,c)$, and $b^{\nu_\rho(\bM,c)}$ in a way that respects the inclusion~\eqref{eq:inclusion} and which are compatible with the generating function expansions proved in previous sections.

% These inclusions enable us to define the projective limit 
% $$\mathcal{C}_n^{(\infty)}:=\underleftarrow{\mathcal{C}_n^{(k)}}.$$ 
% Elements of $\mathcal{C}^{(\infty)} := \bigcup \mathcal{C}_n^{(\infty)}$ can be viewed as ``constellations
% with infinitely many colors whose all except finitely many vertices
% are leaves'', and for short will be called \emph{infinite constellations} in what follows. They are equipped with a well-defined (modified) marking and $b$-weight\footnote{We used~\eqref{eq:projectiveCondition} to argue that the $b$-weight could be defined coherently respecting the inclusion~\eqref{eq:inclusion}, so the quantity $\tilde{\rho}(\bM)$ is \textit{a priori} defined abstractly. However, if one is only interested in some family of infinite constellations corresponding to a finite underlying value of $k$, one can use the $b$-weights explicitly defined in Section~\ref{sec:MonTutte} via the combinatorial decomposition of $k$-constellations. This is the case for example for the $b$-Hurwitz numbers $H^\ell(\lambda,\mu)(b)$ defined in Section~\ref{subsec:bHurwitz}, for a fixed value of $\ell$.}. Their generating function
% 	$$
% 	\hat F^{(\infty)}(t,\mathbf{p},\mathbf{q},u_1,\dots ):=
% 	1+\sum_{n \geq 1}\sum_{\bM \in \mathcal{C}^{(\infty)}_n} \frac{t^{n}}{2^{n-cc(\bM)}n!} \frac{\tilde{\rho}(\bM)}{(1+b)^{cc(\bM)}} \hat \kappa (\bM)
% 	$$
% is given by the projective limit
% $$
% \hat F^{(\infty)}(t,\mathbf{p},\mathbf{q},u_1,\dots ) = \underleftarrow{\hat F^{(k)}}(t,\mathbf{p},\mathbf{q},u_1,\dots,u_k).
% $$



% \begin{remark}[Infinite constellations are generalized branched coverings]
% 	The interpretation of infinite constellations in terms of
%         generalized branched coverings is straightforward. Elements of
%         $\mathcal{C}^{(\infty)}_n$ correspond to generalized branched
%         coverings of the sphere $\mathcal{S}\rightarrow \Sphere_+$ of
%         degree~$n$, whose number of non-trivial ramification points is
% 	(finite but) not bounded \textit{a priori}. The first ramification point (allowed to be trivial) is numbered $-1$ and corresponds to faces of the constellation. The other (nontrivial) ramification points are numbered with some (non necessarily consecutive) nonnegative integers, corresponding to the integers $i$ such that $n-v_i(\bM)$ is nonzero in the constellation.
% \end{remark}

% We denote the set of \emph{rooted} infinite constellations of size $n$ by
% $\mathcal{C}^{(\infty)}_{n,\bullet}$. We have the following theorem.
% \begin{theorem}[Main results reformulated for infinite constellations]\label{thm:infinite}
% 	The generating function $\hat F^{(\infty)} \equiv \hat F^{(\infty)} (t;\pp,\qq,u_1,\dots)$ of infinite constellations satisfies the equation
% \begin{equation}\label{eq:TutteThetaInfinite}
% 	\frac{t\partial}{\partial t} \hat F^{(\infty)} = \Theta_Y \sum_{m \geq 1}t^m \cdot q_m \cdot
% 	\left(Y_+  \prod_{i=1}^\infty (1+u_i \Lambda_{Y})\right)^m  \frac{y_0}{1+b} \hat F^{(\infty)}.
% \end{equation}
% It is equal to the Jack polynomial expansion
% \begin{equation}\label{eq:jackInfinite}
% 	\hat F^{(\infty)}(t,\mathbf{p},\mathbf{q},u_1,\dots ) = \sum_{n \geq 0}t^n
% 	\sum_{\la
% 	\vdash n}\frac{J_\la^{(1+b)}(\pp)J_\la^{(1+b)}(\qq) \prod_{i\geq 1} u_i^{|\lambda|}J_\la^{(1+b)}(\underline{1/u_i})  }{j_\la^{(1+b)}}.
% \end{equation}
% 	Moreover $(1+b) \frac{ t \partial} {\partial t} \log \hat  F^{(\infty)}$ has coefficients which are polynomials in $b$ with non negative integer coefficients, explicitly given by
% \begin{equation}\label{eq:jackInfiniteConnected}
% (1+b) \frac{ t \partial} {\partial t} \log
% 	\hat F^{(\infty)} = 
% 	\sum_{n \geq 1}\sum_{(\bM,c) \in \mathcal{C}^{(\infty)}_{n,\bullet}} t^{n} \hat \kappa(\bM) b^{\nu_\rho(\bM,c)}.
% \end{equation}
% \end{theorem}
% \begin{proof}
% 	This is a direct consequence of the results of previous sections.
% \end{proof}


% We recall that $J_\la^{(1+b)}$ is a polynomial in the power-sum variables of homogeneous graded degree $|\lambda|$, normalized in such a way that the coefficient of $p_1^n$ is $1$.  Therefore $u_i^{|\lambda|}J_\la^{(1+b)}(\underline{1/u_i})$ is a polynomial in $u_i$ with constant term $1$. Thus, despite the infinite product in its definition, the expression~\eqref{eq:jackInfinite} is a well defined formal power series in $t$ and the $u_i$.
% In fact, from~\eqref{eq:contentForm}, we have the explicit formula 
% $u^{|\la|}J_\la^{(1+b)}(\underline{1/u}) = \prod_{\square \in \la}\big(1+u c_{\a}(\square)\big)$, where the product is taken over all boxes of $\la$, and we can write
% \begin{align}\label{eq:contentProductForm}
% \hat F^{(\infty)}(t,\mathbf{p},\mathbf{q},u_1,\dots ) = \sum_{n \geq 0}t^n
% 	\sum_{\la
% 	\vdash n}\frac{J_\la^{(1+b)}(\pp)J_\la^{(1+b)}(\qq) \prod_{i\geq 1}  \prod_{\square \in \la}\big(1+u_i c_{\a}(\square)\big)  }{j_\la^{(1+b)}}.
% \end{align}



% We also observe that the case of finite $k$ considered in previous
% sections can be recovered from the infinite case by considering the
% case where $u_i$ is equal to zero for all $i>k$. 

% \subsection{Weighted $b$-Hurwitz numbers}

% We now introduce $b$-weighted analogues of the weighted Hurwitz numbers of~\cite{Guay-PaquetHarnad2017}. In order to avoid the terminology ``$b$-weighted weighted Hurwitz numbers'', we will use ``weighted $b$-Hurwitz numbers'' instead.

% We note that the parameters $u_i$ appear in the equations of \cref{thm:infinite} only through the generating function $z \mapsto \prod_{i=1}^\infty (1+u_i z)$. Therefore it can be interesting to use the coefficients of this power series as parameters, rather than the $u_i$ themselves. Let 
% $$
% G(z) = 1+\sum_{k\geq 1} g_k z^k
% $$
% be a formal power series, where the $g_k$ are indeterminates. 
% Define the generating function $\hat F^G \equiv \hat F^G(t,\mathbf{p},\mathbf{q},g_1,\dots)$ by the following variant of Equation~\eqref{eq:TutteThetaInfinite}
% \begin{equation}\label{eq:TutteThetaInfiniteG}
% 	\frac{t\partial}{\partial t} \hat F^{G}  = \Theta_Y \sum_{m \geq 1}t^m \cdot q_m \cdot
% 	\Big(Y_+  G\big( \Lambda_{Y}\big)\Big)^m  \frac{y_0}{1+b} \hat F^{G}.
% \end{equation}

% When the variables $(g_k)_{k\geq 1}$ and $(u_i)_{i\geq 1}$ are related by the equation $G(z)=\prod_{i=1}^\infty (1+z u_i)$, then we have
% \begin{align}\label{eq:F=F}
% 	\hat F^G(t,\mathbf{p},\mathbf{q},g_1,\dots)=\hat F^{(\infty)}(t,\mathbf{p},\mathbf{q},u_1,\dots).
% \end{align}
% Therefore in this case $\hat F^G$ can be seen as the generating function of infinite constellations, with weights $u_i$ related to $g_k$ by $$g_k =  e_k (u_1,\dots)$$ where $e_k$ denote the elementary symmetric functions. 
% Call an infinite constellation $\bM$ of size~$n$ \emph{normal} if the sequence 
% $$\mathfrak{v}(\bM):=(n-v_i(\bM))_{i\geq 1}$$ is nondecreasing, i.e. if it is an integer partition. Then $\hat F^{(\infty)}$ can be viewed as the generating function of normal infinite constellations with their usual marking in the $\mathbf{p}$-$\mathbf{q}$-$t$ variables, and a marking  $m_{\mathfrak{v}(\bM)}(u_1,u_2,\dots)$ giving the contribution of vertices of color $i\geq 1$, where $m_{\cdot}$ denotes monomial symmetric functions (see e.g.~\cite{Stanley:EC2}). Indeed, the summation hidden in the definition of monomial symmetric functions consists in summing over all possible reorderings of the sequence $\mathfrak{v}(\bM)$, accounting also for constellations which are not normal.  This observation enables us to give an interpretation of $\hat F^{G}$ in terms of the indeterminates $g_k$ with no reference to underlying variables $u_i$, as follows.
% %
% \begin{theorem}\label{thm:weighted}
% 	Let $G(z)=1+\sum_{k\geq 1}g_k z^k$. Then the series $\hat F^G$ is the generating function of normal infinite constellations with their $b$-weight, and with a marking $p_i$ (resp $q_i$) per face (resp. vertex of colour $0$) of degree $i$, and a marking $f_{\mathfrak{v}(M)}(g_1,g_2,\dots)$ where  for an integer partition $\iota$, $f_{\iota}$ is the polynomial that expresses the monomial symmetric function of index $\iota$ in terms of the elementary symmetric functions. 
% 	At the level of rooted connected objects, we have
% 	$$
% (1+b)\frac{t\partial}{\partial t} \log	\hat F^G =
% 		\sum_{n \geq 1}\sum_{(\bM,c) \in \mathcal{C}^{(\infty)}_{n,\bullet}\atop \bM~\mbox{\tiny normal}} 
% 		t^{n} b^{\nu_\rho(\bM,c)} \prod_{f\in F(\bM)} p_{\deg(f)}\prod_{v\in V_0(\bM)} q_{\deg(v)}\ f_{\mathfrak{v}(\bM)}(g_1,\dots). 
% 	$$
% \end{theorem}

% The polynomials $f_\iota$ can be constructed as follows. For an integer partition $\mu$ of size $n$, we have $e_\mu = \sum_{\iota \vdash n} R_{\mu,\iota} m_\iota$ where the square matrix $R$ has its rows and columns indexed by integer partitions of size $n$, and $R_{\mu,\iota}$ is equal to the number of $0$-$1$ matrices with row-sum (resp. column-sum) equal to $\mu$ (resp. $ \iota$), see~\cite{Stanley:EC2}. Then  
% $f_\iota(g_1,\dots)  = \sum_{\mu \vdash n} (R^{-1})_{\iota,\mu} \prod_{i=1}^{\ell(\mu)} g_{\mu_i}$, where $R^{-1}$ denotes the inverse matrix of $R$. Since $R$ is triangular for the dominance order with a diagonal of $1$, the matrix $R^{-1}$ has integer coefficients, that we will not try to make explicit here.


% \begin{remark}\label{rem:genus}
% 	An extra
%         parameter\footnote{The parameter $\hbar$ is noted $\beta$
%           in~\cite{AlexandrovChapuyEynardHarnad2020,Okounkov2000a} but
%           we prefer to avoid the confusion with $\beta$-ensembles.}
%         $\hbar$ can be added to the series $\hat F^G$ (or $\hat
%         F_{\infty}$) via the scaling $g_k \mapsto \hbar ^k g_k$ (or
%         $u_i\mapsto \hbar u_i$). The exponent $d$ of $\hbar$ is directly related to the Euler
% 	characteristic of the underlying constellation (or covering) via $\chi = \ell(\lambda)+\ell(\mu)-d$, where $\lambda$ and $\mu$ represent respectively the degrees of faces and of vertices of colour $0$ (or the profiles of the first two ramification points). In particular, the $\hbar$-expansion of $\log \hat F^G$ is a topological expansion, with powers $\hbar^{\ell(\lambda)+\ell(\mu)-\chi}$ appearing in the coefficient of $p_\lambda q_\mu$ for each $\lambda, \mu \vdash n$. We will not need this viewpoint here in general, but we will introduce the variable $\hbar$ in the special cases of the next sections. 
% \end{remark}


% \begin{remark}
% 	In the case $b=0$, the interpretation of the series $\hat F^G$ that we gave, in terms of normal constellations, is \emph{not} the standard one. Indeed, in that case it is possible to give an interpretation in terms of factorizations of permutations using \emph{transpositions} (which at the level of coverings correspond to simple branchpoints), with a weighting system that can be made explicit in terms of the $g_k$, which gives rise to the so-called \emph{weighted Hurwitz numbers}, see~\cite{Guay-PaquetHarnad2017}.
% %
% 	%
% %
% %
% %
% %
% %
% %
% %
% %
% %
% %
% %
% 	The connection between the two interpretations goes through the group algebra of the symmetric group and the Jucys-Murphy elements, which are specific to $b=0$. 	We leave as an open problem to give a similar interpretation of the series $\hat F^G$ in full generality. In the next section, we will address, however, the case of $b$-weighted analogues of \emph{classical} Hurwitz numbers.

% \end{remark}

% We observe that the function $\hat F^G$ has the following expression
% \begin{align}\label{eq:GProductForm}
% \hat F^{G}(t,\mathbf{p},\mathbf{q},g_1,\dots ) = \sum_{n \geq 0}t^n
% 	\sum_{\la
% 	\vdash n}\frac{J_\la^{(1+b)}(\pp)J_\la^{(1+b)}(\qq) \prod_{\square \in \la} G(c_{\a}(\square))  }{j_\la^{(1+b)}}.
% \end{align}
% Indeed, when $g_k=e_k(u_1,\dots)$ this is a consequence of \eqref{eq:contentProductForm} and~\eqref{eq:F=F}. But the fact that elementary symmetric functions are a basis of the space of symmetric functions implies that this is true with the $g_k$ being independent indeterminates.


% \subsection{$b$-Hurwitz numbers, $G(z)=\exp(\hbar z)$.}
% \label{subsec:bHurwitz}

% \begin{definition}
% 	For $n, \ell \geq 1$ and $\lambda, \mu$ two partitions of $n$, the (connected) $b$-Hurwitz number $H^\ell(\lambda,\mu)(b)$ is defined as the
% 	polynomial
% 	$$
% 	H^\ell(\lambda,\mu)(b):= \sum_{(\bM,c)} b^{\nu_\rho(\bM,c)},
% 	$$
% 	where the sum is taken over rooted connected $\ell$-constellations $\bM$ of size $n$ with full profile 
% 	\begin{align}\label{eq:profileHurwitz}
% 		(\lambda, \mu, \underbrace{[2,1^{n-2}], \dots, [2,1^{n-2}]}_{\ell \mbox{ times}}).
% 	\end{align}
% 	Equivalently, this sum is taken over rooted generalized branched coverings  of degree $n$ of the sphere  by a connected surface, with  $\ell+2$  numbered ramification points, the first two with profiles $\lambda$, $\mu$, and all the other ones being simple.
% \end{definition}

% Since each rooted connected constellation of size $n$ has $(n-1)$
% non-root corners, we note that
% $\frac{(n-1)!}{n!}H^\ell(\lambda,\mu)(0) = \frac{1}{n}H^\ell(\lambda,\mu)(0)$ is nothing but the usual (orientable) double Hurwitz number of parameters $\lambda, \mu, \ell$. For $\mu=[1^n]$, we recover single Hurwitz numbers. Also note that $H^\ell(\lambda,\mu)(b)-H^\ell(\lambda,\mu)(0)$ gives the contribution of coverings from non-orientable surfaces. In both cases, the Euler characteristic of the underlying surface can be recovered by~\eqref{eq:EulerCharacteristic}, namely	$$
% 	\chi = \ell(\mu)+\ell(\lambda)-\ell.
% 	$$


% We now form the corresponding generating function for non-necessarily connected objects,
% $$
% \hat F_{\mathrm{Hurwitz}}\equiv \hat F_{\mathrm{Hurwitz}}
% (t,\mathbf{p},\mathbf{q}, \hbar) := 1 + \sum_{n\geq 1} \frac{t^n}{2^{n}n!} \sum_{\lambda, \mu \vdash n\atop \ell \geq 0 } \sum_{\bM}\frac{\tilde\rho(\bM)}{2^{-cc(\bM)}(1+b)^{cc(\bM)}} \frac{\hbar^\ell}{\ell!} p_\lambda q_\mu,
% $$
% where $\hbar$ is a new indeterminate and where the last sum is taken over \emph{labelled}, connected or not, $\ell$-constellations $\bM$ of size $n$ with full profile~\eqref{eq:profileHurwitz}. 
% We have
% \begin{theorem}\label{thm:FHurwitz}
% 	The generating function $\hat F_{\mathrm{Hurwitz}}$ is equal to the function $\hat F^G$ with $G(z) = \exp_\hbar (z):= \exp \hbar z$, \textit{i.e.}
% 	\begin{align}\label{eq:bHurwitzGen}
% 	\hat F_{\mathrm{Hurwitz}}  = \hat F^{\exp_\hbar}.
% 	\end{align}
% 	The function $(1+b)\frac{t \partial}{\partial t} \log \hat F_{\mathrm{Hurwitz}}$ is the generating function of rooted connected coverings, explicitly given by
% 	\begin{align}\label{eq:bHurwitzGenConn}
% 	(1+b)\frac{t \partial}{\partial t} \log \hat F_{\mathrm{Hurwitz}}
% 		=\sum_{n\geq 1} t^n \sum_{\lambda, \mu \vdash n\atop \ell \geq 0 } H^\ell(\lambda,\mu)(b) \frac{\hbar^\ell}{\ell!} p_\lambda q_\mu
% 		.
% 	\end{align}


% \end{theorem}
% \begin{theorem}[$b$-Cut and Join equation]\label{thm:cutAndJoin}
% The series $\hat F_{\mathrm{Hurwitz}} \equiv \hat F_{\mathrm{Hurwitz}} (t,\mathbf{p},\mathbf{q}, \hbar)$ satisfies the equation
% 	\begin{align}\label{eq:cutAndJoin}
% 	\frac{\partial}{\partial \hbar}\hat F_{\mathrm{Hurwitz}} = D_\alpha \hat F_{\mathrm{Hurwitz}},
% 	\end{align}
% 	where $\alpha=1+b$ and $D_\alpha$ is the Laplace-Beltrami operator~\eqref{eq:Laplace-Beltrami}.
% \end{theorem}
% We observe that the function $\hat F^{\exp_\hbar}$ is well defined as a formal power series, since the $\hbar$-grading makes the substitution $g_k \mapsto \frac{\hbar^{k}}{k!}$ well defined.
% \begin{proof}[Proof of \cref{thm:FHurwitz}]
% 	We will obtain the generating function of Hurwitz numbers as a suitable limit of $\hat F^{(k)}$ for $k\rightarrow \infty$. This idea is inspired from the orientable case where a similar argument is classical~\cite{BousquetSchaeffer2000,GouldenJackson2008}.

% 	Let us study the expansion of $\hat F^{(k)}\equiv \hat F^{(k)} (t,\mathbf{p}, \mathbf{q},u_1,\dots,u_k)$ for $u_1=\dots=u_k = \frac{\hbar}{k}$, by grouping monomials $u_1^{n_1}\dots u_k^{n_k}$ according to the number $p$ of nonzero exponents.
% 	It will be convenient to use (only) in this proof the
%         following notation: $[\dots]$ denotes the coefficient extraction with respect to the $u_i$-variables only, as if the variables $t, \hbar, p_i, q_j$ were constants.
% 	We have,
% 	\begin{align}
% 	\left. \hat F^{(k)} \right|_{u_i = \frac{\hbar}{k} }
% 		&= \sum_{\ell \geq 0} \frac{\hbar^\ell}{k^\ell} \sum_{p\geq 0}  \sum_{n_{1}+\dots+n_{k}=\ell, n_{i} \geq 0 \atop |\{i,n_i > 0\}|=p}
% 	[u_{1}^{n_1}\dots u_k^{n_k}] \hat F^{(k)}
% 		\nonumber
% 		\\ 
% 		&=\sum_{\ell \geq 0} \frac{\hbar^\ell}{k^\ell}
% 		\sum_{0\leq p \leq \ell}  \sum_{n_{1}+\dots+n_{p}=\ell\atop n_{i} > 0 }
% 		{k \choose p} [u_{1}^{n_1}\dots u_p^{n_p}] \hat F^{(k)},
% 		\label{eq:int23}
% 	\end{align}
% where the second inequality uses the fact that coefficients of $\hat F^{(k)}$ are symmetric functions in the $u_i$.
% 	We now consider the limit of~\eqref{eq:int23} when $k$ goes to infinity.
% 	First remark that for fixed $\ell$ and $p\leq \ell$, we have when $k$ goes to infinity:
% $$
% 	\frac{1}{k^\ell} {k \choose p} \longrightarrow \mathbf{1}_{p=\ell} \frac{1}{\ell!}.
% $$
% 	Therefore, for each fixed coefficient in $\hbar$, when $k$ goes to infinity, only the term $p=\ell$ contributes to~\eqref{eq:int23} at the first order. Further, for $p=\ell$ only the term $n_1=\dots=n_\ell=1$ is possible in~\eqref{eq:int23}. We thus get the (coefficientwise) limit
% $$
% \lim_k \left. \hat F^{(k)} \right|_{u_i = \frac{\hbar}{k} }
% 		= \sum_{\ell\geq 0}  
% 	\frac{\hbar^\ell}{\ell!} [u_{1}^{1}\dots u_\ell^{1}] \hat F^{(\infty)},
% $$
% which, by definition, is equal to $\hat F_{\mathrm{Hurwitz}}$.


% Now, when $k$ goes to infinity, we also have the limit: 
% $$\prod_{i=1}^k (1+ z u_i) \big|_{u_i = \frac{\hbar}{k}}  = \left(1+\frac{z \hbar}{k}\right)^k\longrightarrow \exp (\hbar z).$$
% 	Because the series $\hat F^{(k)}$ satisfies the decomposition equation~\eqref{eq:TutteThetaInfiniteG} with $G(z)=\prod_{i=1}^k (1+u_i z)$, and because all coefficients (in $t$) of the series $\hat F^{(k)}$ are symmetric polynomials in the $u_i$, this implies that the series $\lim_k \left. \hat F^{(k)} \right|_{u_i = \frac{\hbar}{k} }$ satisfies the decomposition equation~\eqref{eq:TutteThetaInfiniteG} with $G(z)=\exp(\hbar z)$. Therefore $\lim_k \left. \hat F^{(k)} \right|_{u_i = \frac{\hbar}{k} } = \hat F^{\exp_\hbar}$ and the proof is finished.
% \end{proof}
% \begin{proof}[Proof of \cref{thm:cutAndJoin}]
% %
% 	This is a direct consequence of~\eqref{eq:GProductForm} with $G=\exp_\hbar$ and of the fact~\eqref{eq:jackEigenvector} that $J_\la^\alpha(\mathbf{p})$ is an eigenvector of $D_\alpha$ with eigenvalue $\sum_{\square \in \la} c_\alpha(\square)$.
%       \end{proof}

%       We conclude this subsection by proving piecewise
%       polynomiality of the $b$-weighted double Hurwitz numbers. 
%       In the case $b=0$ our result  is slightly weaker than the 
%       result of Goulden, Jackson and Vakil~\cite{GouldenJacksonVakil2005}
%      which states that the classical double Hurwitz number 
%       $\frac{1}{|\la|}H^\ell(\lambda,\mu)(0)$ is a piecewise polynomial.
%       The extra factor of $|\la|$ comes from the fact that we need to average over the choice of the root to define the $b$-weight, and we do not know if the stronger result holds for general $b$.
%       \begin{theorem}[Piecewise polynomiality]
%         Let us fix $\ell, m, n$ and consider $H^\ell(\lambda,\mu)(b)$
%         as a function on partitions $\lambda,\mu$ whose number of
%         parts is equal to $m$ and $n$ respectively. Then
%         $H^\ell(\lambda,\mu)(b)$ is a polynomial in $b$, whose
%         coefficients are piecewise polynomial functions in $\lambda_1,\dots,\la_m,\mu_1,\dots,\mu_n$.
%       \end{theorem}

% 	    \begin{proof}
% Our proof is inspired by the combinatorial proof of Goulden, Jackson, and Vakil for the case $b=0$, but instead of working directly at the level of combinatorial objects it works inductively at the level of generating functions, using the decomposition equation.


%         We will prove the result by induction on $\ell$.
%         Let $\hat{H}^{[d]}_\rho$ and $\hat{\widetilde{H}}^{[d]}_\rho$
%         denote the projective limit (over $\ell$) of $H^{[d]}_\rho(\frac{1}{u_1\cdots
%                                    u_\ell};\pp,\qq,u_1^{-1},\dots,u_\ell^{-1})$
%                                  and $\widetilde
%             {H}_\rho^{[d]}(\frac{1}{u_1\cdots u_\ell};\pp,\qq,
%             u_1^{-1},\dots,u_\ell^{-1})$, respectively. We define
%             $\hat{\vec H}_\rho$ similarly. Note that for any function
% 		    $H \in\{ \hat{H}^{[d]}_\rho, \hat{\widetilde{H}}^{[d]}_\rho,
% 		    \hat{\vec H}_\rho\}$ and for any sequence $i_1<\dots <i_\ell$
%             one has $[u_{i_1}\cdots u_{i_\ell}]H = [u_{1}\cdots
%             u_{\ell}]H = [e_i(\uu)]H$ since $H$ is symmetric in
%             $\uu$. In particular
%         \begin{align*}
%           H^\ell(\lambda,\mu)(b) &= [u_1\cdots u_\ell]\Theta_Y \hat{\vec
%           H}_\rho= \sum_{d \geq
%             1}[u_1\cdots
%           u_\ell]q_d\cdot \hat{H}_\rho^{[d]}.
%             \end{align*}
%         Let us define $h_\ell: \N^{m+n}\to \N[b]$ as the coefficient
% 		    %
% %
% %
% 		    $$h_\ell(\lambda_1,\dots,\lambda_m,\mu_1,\dots,\mu_{n-1},d) = 
% 		    [p_{\lambda_1}\dots p_{\lambda_m} q_{\mu_1}\dots q_{\mu_{n-1}}] [u_1\cdots u_\ell] \hat{H}_\rho^{[d]}.$$ 
%         We will prove by induction on $\ell$ that for any $n$ and $m$,  $h_\ell: \N^{m+n}\to
%         \N[b]$ is polynomial in $b$ with coefficients which are
%         piecewise polynomials, which implies the result we want. 

%         \cref{cor:TutteThetaConnected} implies that
% 		    \[\hat{H}_\rho^{[d]} = t^d p_d +\text{(monomials involving
% 		    variables $\uu$)},\]
%         therefore 
% 		    $h_0(\lambda_1,\dots,\lambda_m,\mu_1,\dots,\mu_{n-1},d)$ is equal to $1$ if $d=\lambda_i\geq 1$ for some $i$ and all other variables are equal to zero, and it is equal to zero otherwise. This is a piecewise
%       polynomial function. %
%       %
%       %
%       %

% Moreover \cref{cor:TutteThetaConnected} implies that
% \begin{equation}
%   \label{eq:pomocnicze}
%   [u_1\cdots u_\ell]\hat{H}_\rho^{[d]}=
% 		\Theta_Y [u_1\cdots
%                 u_\ell]\Big(Y_+\prod_{i}\big(1+u_i(\Lambda^{[1]}_{Y}+\Lambda^{[2]}_{Y}+\Lambda^{[3]}_{Y}
%                 + \R)\big)\Big)^{d}  (y_0),
%                 \end{equation}
%                 where
% \begin{align*}
% 	\Lambda^{[1]}_{Y} &=        (1+b)\sum_{i,j\geq 1}
%                      y_{j+i-1}\frac{i\partial}{\partial
% 		     p_{i}}\frac{\partial}{\partial y_{j-1}},                       &\Lambda^{[2]}_{Y} =        \sum_{i,j\geq
%                                           1}
%                                           y_{j-1}p_i\frac{\partial}{\partial
%                                           y_{i+j-1}},\\
%                                            \Lambda^{[3]}_{Y} &=
%                                                                b\sum_{i\geq
%                                                                1}
%                                                                y_{i}\frac{i\partial}{\partial
%                                                                y_{i}},
% 							      &\R = \sum_{i,j\geq 1} y_{j+i-1}\hat{\widetilde
%             {H}}_\rho^{[i]}\frac{\partial}{\partial y_{j-1}}.
%                                                                \end{align*}
% 		    For $\ell'\geq 0$ let $\R_{\ell'}:=[u_1\cdots u_{\ell'}]\R  = [e_{\ell'}(\uu)]\R$. Expanding the
%             R.H.S. of~\eqref{eq:pomocnicze} we can express it as the
%             following linear combination:
%             \begin{align*}
%               \Theta_Y\sum_{s \geq 1}\sum_{I_1,\dots,I_d \subset
%               [1..\ell]}\Bigg[\prod_{i \in
%               \left(\bigcup_{i=1}^d I_i\right)^c}u_i\Bigg]\prod_{i=1}^dY_+(\Lambda^{[1]}_{Y}+\Lambda^{[2]}_{Y}+\Lambda^{[3]}_{Y}+\R)^{|I_i|}(y_0),
%             \end{align*}
%             where we sum over all pairwise disjoint (possibly empty)
%             subsets of $[1..\ell]$, whose union is non-empty.
%             Expanding the product we find a linear combination of quantities of the form
%             \begin{equation}
%               \label{eq:kombinacja}
%               \Theta_YY_+^{k_1}w_1\cdots Y_+^{k_s}w_s(y_0),
%               \end{equation}
%             for some $1 \leq s \leq \ell$, and positive integers $k_1,\dots,k_s$
%             whose sum is equal to $d$, where
%             \begin{itemize}
%               \item $w_k$ is a non-empty word in
%                 $\Lambda^{[1]}_{Y},\Lambda^{[2]}_{Y},\Lambda^{[3]}_{Y},\R_{0},\dots,\R_{\ell-1}$,
%                 \item the total length of words satisfies
%                   $\ell(w_1)+\cdots+\ell(w_s) \leq \ell$,
%                   \item the total sum of indices of the variables
%                     $\R_{0},\dots,\R_{\ell-1}$ appearing in $w_1,\dots,w_s$ is equal to $\ell-\sum_{i=1}^s\ell(w_i)$.
%                   \end{itemize}
%                   For a fixed sequence of words $w_1,\dots,w_s$, the
%                   element of the form~\eqref{eq:kombinacja} appears in the
%                   R.H.S. of~\eqref{eq:pomocnicze} with 
%                   coefficient
%                   $\binom{\ell}{\ell(w_1),\dots,\ell(w_s)}$, which does not depend on
%                   $k_1,\dots,k_s$.
% 		  Therefore, since the number of choices for $s$ and  $(w_i)_{1\leq i \leq s}$ is finite, it is enough to show that for each such choice, the quantity		  
% 		  \begin{align}\label{eq:fixedWord}
% 			  [p_{\lambda_1} p_{\lambda_2}\dots p_{\lambda_m} q_{\mu_1}q_{\mu_2}\dots q_{\mu_{n-1}}] \sum_{k_1+\dots+k_s=d} \Theta_YY_+^{k_1}w_1\cdots Y_+^{k_s}w_s(y_0)
% 		  \end{align}
% 		  is a piecewise polynomial in $(\lambda_1,\dots,\lambda_m,\mu_1,\dots, \mu_{n-1},d)$.

% To see this, we notice that to compute the wanted coefficient  in~\eqref{eq:fixedWord}, one needs to sum over all ``trajectories'' of the monomials in $p_i,q_j, y_k$ appearing from right to left along the product of operators. Such a trajectory consists in a tuple  of monomials
% $$(p_{\lambda^{(0)}}q_{\mu^{(0)}}y_{i_{0}},p_{\lambda^{(1)}}q_{\mu^{(1)}}y_{i_{1}}, \dots, p_{\lambda^{(2s)}}q_{\mu^{(2s)}}y_{i_{2s}}),$$
% where $p_{\lambda^{(2r)}}q_{\mu^{(2r)}}y_{i_{2r}}$ (resp. $p_{\lambda^{(2r-1)}}q_{\mu^{(2r-1)}}y_{i_{2r-1}}$) is the monomial appearing to the right (resp. left) of the operator $w_r$ for $1\leq r \leq s$, with 
% $p_{\lambda^{(2s)}}q_{\mu^{(2s)}}y_{i_{2s}}=y_0$ and
% %
% $p_{\lambda^{(0)}}q_{\mu^{(0)}}p_{i_{0}}=p_{\lambda_1} p_{\lambda_2}\dots p_{\lambda_m} q_{\mu_1}q_{\mu_2}\dots q_{\mu_{n-1}}.
% $
% %

% Note that the nature of the operators $\Lambda^{[i]}_{Y}$ and $\R_{i}$ implies that the possible values of the indices of successive monomials appearing along this tuple are constrained by a finite number of linear equalities and inequalities. In other words, the set of valid trajectories is parametrized by tuples of integers
% \begin{align}\label{eq:tuples}
% (\lambda^{(i)}_j, 1\leq j \leq \ell(\lambda^{(i)});
% \mu^{(i)}_j, 1\leq j \leq \ell(\mu^{(i)});
%  i_j;
% 	k_j)_{1\leq j \leq s}
% \end{align}
% subject to linear constraints, {\textit i.e.} by integer points in a polytope. Moreover, note that for fixed $\ell, n, m$, the maximum number of parts appearing in any monomial is bounded, so this polytope is finite-dimensional. We include the equality
% $$
% k_1+\dots +k_s =d,
% $$
% which involves the parameter $d$, in the linear constraints defining this polytope.

% Finally, each $r \in [1..s]$ such that $w_r$ is equal to $\R_{\ell'}$ for some $\ell'\in [0..\ell-1]$ corresponds to the fact that one passes from the monomial $p_{\lambda^{(2r)}}q_{\mu^{(2r)}}y_{i_{2r}}$ to $p_{\lambda^{(2r-1)}}q_{\mu^{(2r-1)}}y_{i_{2r-1}}$ by using the operator 
% $$\R_{\ell'}=[u_1\cdots u_{\ell'}]  y_{j+i-1}\hat{\widetilde
%             {H}}_\rho^{[i]}\frac{\partial}{\partial y_{j-1}}. 
% $$
% Therefore, assuming that the trajectory is valid, the coefficient 
% \begin{align}\label{eq:piecewisePolyStep}
% 	[u_1\cdots u_{\ell'}][p_{\lambda^{(2r-1)}\setminus \lambda^{(2r)}}] [q_{\mu^{(2r-1)}\setminus \mu^{(2r)}}] \hat{\widetilde {H}}_\rho^{[i_{2r-1}-i_{2r}]} 
% \end{align}
% is collected along the way. By the induction hypothesis, and since
% $\ell'<\ell$, the quantity~\eqref{eq:piecewisePolyStep} is a
% polynomial, whose coefficients are piecewise polynomials in all parameters $\lambda^{(2r-1)}_j, \mu^{(2r-1)}_j, i_{2r-1}, \lambda^{(2r)}_j, \mu^{(2r)}_j, i_{2r}$ involved.

% In conclusion, we have proved the following fact: the coefficients of
% the quantity~\eqref{eq:fixedWord} are the sums over integer
% trajectories of the form~\eqref{eq:tuples}, constrained to live in a
% finite-dimensional polytope, of products of quantities of the
% form~\eqref{eq:piecewisePolyStep}, which are piecewise polynomials in
% the coordinates. Therefore the coefficients of~\eqref{eq:fixedWord} are piecewise polynomials, which is what we wanted to prove.
% \end{proof}



%       \begin{remark}
% 	      In the case $b=0$, piecewise polynomiality-type
%         results were an important motivation to look for a hidden geometric explanation
%         in the spirit of the
%         ELSV-formula~\cite{EkedahlLandoShapiroVainshtein2001}. Our
%         result gives one more motivation to explore the underlying, yet to be found,
%         geometric structure describing the $b$-deformation.
%         \end{remark}


% \subsection{Dessins d'enfants, the $b$-conjecture, and $\beta$-ensembles}
% \label{subsec:betaEnsembles}

% The case $k=1$ of our main results, or equivalently the case $G(z)=(1+u_1z)$ of $b$-weighted Hurwitz numbers, corresponds to coverings with three ramification points, or combinatorially, to $1$-constellations. 
% In the orientable case, $1$-constellations are called \emph{dessins d'enfants}, \emph{bipartite maps}, or \emph{hypermaps}. See~\cite{LandoZvonkin2004}.

% Bipartite maps on non-orientable surfaces have been considered before. They  can be encoded combinatorially by matchings, or equivalently by products in the double coset algebra of the Gelfand pair $(\mathfrak{S}_{2n}, H_n)$, see~\cite{HanlonStanleyStembridge1992, GouldenJackson1996}.
% In~\cite{GouldenJackson1996} the following formal power series is introduced:
% \begin{align}\label{eq:bFunction}
%   B(t;\pp,\qq,\rr) :&=
% 	\sum_{n \geq 0}t^n\sum_{\la
% 	\vdash n}\frac{1}{j_\la^{(\alpha)}} J_\la^{(\alpha)}(\pp)J_\la^{(\alpha)}(\qq) J_\la^{(\alpha)}(\rr),
%    \end{align}
% where $\rr=(r_i)_{i\geq 1}$ is an infinite family of variables. Of course, this function becomes equal to our function $\tau^{(1)}_b(t;\pp,\qq,u_1)$ under the specialization $\rr=\underline{u_1}$.
% In the same paper~\cite{GouldenJackson1996}, Goulden and Jackson
% state the $b$-conjecture\footnote{In the same paper Goulden and
%   Jackson also state the closely related \emph{Matching-Jack
%     conjecture}, which deals with non-necessarily connected bipartite
%   maps. Our statements for non-necessarily connected
%   $1$-constellations are related to this conjecture in the same way as
%   our statements for rooted connected $1$-constellations are related
%   to the $b$-conjecture. We choose to focus the discussion on the
%   latter. }, namely that one can write
% \begin{align}\label{eq:bConjecture}
% 	(1+b)\frac{t \partial}{\partial t} B(t;\pp,\qq,\rr) :&=
%                                                                \sum_{n
%                                                                \geq 1}\sum_{\lambda, \mu, \iota \vdash n} \sum_{(\bM,c)} t^n  p_\lambda q_\mu r_\iota b^{s(\bM,c)},
% \end{align}
% where the last sum is taken over rooted $1$-constellations (rooted bipartite maps in the language of~\cite{GouldenJackson1996}) on a connected surface (orientable or not), of size $n$ and of full profile $(\lambda, \mu, \iota)$, and where $s(\bM,c)$ is an (unspecified) integer parameter attached to $(\bM,c)$ which is zero if and only if the surface is orientable.
% As they show, this statement is true for $b\in\{0,1\}$,  which is
% proved using the connection between Schur (or zonal, respectively)
% polynomials and representation theory of $\mathfrak{S}_n$ (or of the
% Gelfand pair $(\mathfrak{S}_{2n},
% H_n)$). See~\cite{Macdonald1995,HanlonStanleyStembridge1992,
%   GouldenJackson1996}. For general values of $b$, no suitable connection to
% representation theory is known, and this conjecture is still open due
% to the lack of tools to attack it. However, it is interesting to
% remark that both the $b$-conjecture and our main result (say, in the
% formulation of \cref{thm:infinite}) involve \emph{three}
% infinite families of parameters, respectively $\{\pp,\qq,\rr\}$ and
% $\{\pp,\qq,(u_i)_{i\geq 1}\}$. In fact, the special cases $b=0,1$ of our result can be used to prove the
% $b$-conjecture in the corresponding cases. Therefore, we suspect that these two incomparable results are in fact of equal strength, and that our result could be used to prove the full $b$-conjecture. 

% Polynomiality over the \emph{rationals} in the $b$- and Matching-Jack conjectures was proved in \cite{DolegaFeray2016,DolegaFeray2017}, but integrality and positivity of coefficients had been proved until now only in a few very special cases. The
% case $k=1$ of our main result coincides with the $b$-conjecture in
% the case where $\rr=\underline{u_1}$, if we identify the unknown
% parameter $s(\bM,c)$ with our parameter $\nu_\rho(\bM,c)$.  In other
% words, the $b$-conjecture is now proved when one keeps \emph{two} full
% sets of indeterminate variables ($\pp$ and $\qq$). This is by far the best progress towards it and, in particular,
% it covers all the special cases proved in the literature so far, as we now quickly explain. 

% The most general case proven so far was the case of the simultaneous specialisations $\rr=\underline{u_1}$ and $q_i=\mathbf{1}_{i=2}$ (thus keeping \emph{one} full set of indeterminate variables). This was done by Lacroix~\cite{LaCroix2009}.
% This case corresponds to $1$-constellations in which all vertices of colour $0$ have degree $2$, which are in bijection (by lifting these vertices into single edges)  with general, uncoloured, maps on surfaces (the only remaining vertices have colour~$1$ and can be thought of as uncoloured, they are counted with a modified weight $u_1$; vertices of colour $0$ have become edges, with weight $t$; faces of degree $i$ are counted with a weigth $p_i$).
% Lacroix used a connection between the function $B$ under this specialization and the $\beta$-ensembles of random matrix theory, together with a connection due to Okounkov~\cite{Okounkov1997} between $\beta$-ensembles and Jack polynomials -- for $1+b = 2/\beta$. The paper~\cite{Okounkov1997} enables one to work with certain \emph{linear} combinations of Jack polynomials, which is what~\cite{LaCroix2009} uses.
% In our work, we crucially need to consider \emph{bilinear} sums instead (with Jack polynomials in two set of variables $\pp$ and $\qq$) which makes it inaccessible by this method.
% Similarly, the equations of~\cite{AdlervanMoerbeke2001} in the context of $\beta$-ensembles deal with a function of \emph{one} infinite family of variables.
% It is reasonable to expect that our work could be related to multi-matrix analogues of the $\beta$-ensembles.

% Some special cases of the $b$-conjecture had been established for
% bipartite maps in some other very restricted cases, for example for
% bipartite maps with a unique face and of genus at most $2$,
% see~\cite{Dolega2017a}. See also~\cite{KanunnikovVassilieva2016,KanunnikovPromyslovVassilieva2018} for
% other partial results, concerning mostly the coefficients of $B$
% itself (as in the Matching-Jack conjecture). All these cases are fully covered by the case $k=1$ of our result by taking specializations or extracting coefficients.



% \subsection{$b$-Monotone Hurwitz numbers and $\beta$-HCIZ integral}
% \label{subsec:HCIZ}

% In the case $b=0$, the particular choice of function $G(z)=Z(z):=\frac{1}{1-\hbar z}$ is known to generate the monotone Hurwitz numbers, see~\cite{GouldenGuayPaquetNovak2014}. That is, $\hat F^{Z}\Big|_{b=0}$ has an explicit interpretation as a generating function of factorizations of a product of two permutations (whose cycles lengths are marked by variables $p_i$ and $q_i$) into a product of transpositions whose maximal transposed elements are weakly increasing.


% It is also known~\cite{GouldenGuayPaquetNovak2014} that this same function is a $1/N$-expansion of the Harish-Chandra-Itzykson-Zuber (HCIZ) integral, 
% $$I_N (t) = \int _{U(N)} e^{t N \mathop{Tr} (A_N U B_N U^{-1})} dU,$$
% where $dU$ is the Haar measure over the unitary group $U(N)$.
% Here the variables $p_i$ and $q_j$ are the power-sum symmetric functions in the eigenvalues of the diagonal matrices $A_N$ and $B_N$, and $\hbar$ plays the role of $-\frac{1}{N}$ in a double expansion in $t$ and $\frac{1}{N}$ of $I_N(t)$. See~\cite{GouldenGuayPaquetNovak2014} again for the precise meaning of this statement.

% The function $\hat F^{Z}$ provides a natural $b$-deformation of the generating function of monotone Hurwitz numbers. This deformation satisfies the equation: 
% \begin{equation}
% 	\frac{t\partial}{\partial t} \hat F^{Z}  = \Theta_Y \sum_{m \geq 1}t^m \cdot q_m \cdot
% 	\Big(Y_+  \frac{1}{1-\hbar \Lambda_{Y}}\Big)^m  \frac{y_0}{1+b} \hat F^{Z},
% \end{equation}
% and our main theorem implies in particular that  $(1+b)\frac{t\partial}{\partial t} \log \hat F^Z$ has coefficients which are polynomials in $b$ with an explicit combinatorial interpretation.

% Moreover, deformations $I^{\beta}_N(t)$ of the HCIZ integral obtained by deforming the Laplace-Beltrami operator have been considered before, at least since Brézin and Hikami~\cite{BrezinHikami2003} (there are many other references, see e.g.~\cite{BergereEynard2009}). It would be interesting to study if they admit $1/N$ expansions, and if they are related to the $b$-deformed monotone Hurwitz numbers which we introduce here. In other words, can the following diagram be made commutative?
% $$
% \begin{array}{rcccl}
% 	&I_N(t) & \stackrel{\frac{1}{N}-\mbox{expansion, \cite{GouldenGuayPaquetNovak2014}}}{\xrightarrow{\hspace*{4cm}}} & \hat F^{Z}(t,\hbar) \Big|_{b=0}& \\
% 	\stackrel{\mbox{$\beta$-deformation}}{\mbox{e.g.\ \cite{BrezinHikami2003}}}	&\xdownarrow{6mm} & &\xdownarrow{6mm}& 	\stackrel{\mbox{$b$-deformation}}{\mbox{(this paper)}} \\
% 	&I^{\beta}_N(t) & \stackrel{\frac{1}{N}-\mbox{expansion ?} }{\xrightarrow{\hspace*{4cm}}} & \hat F^{Z}(t,\hbar)&
% \end{array}
% $$
% This question is beyond the scope of our paper and is left as an open problem.

% %
% %
% %
% %
% %

% 	\medskip
% 	\noindent {\bf Acknowledgements.} We thank Houcine Ben Dali for pointing a mistake in the interpretation of the decomposition equation in the first version of this paper.
	

% \newpage
% \appendix

% %
% \newcommand{\dop}[1]{\includegraphics{diag#1.pdf}}
% %
% \newcommand{\sdop}[1]{\includegraphics[scale=0.7]{diag#1.pdf}}
% \newcommand{\bdop}[1]{\includegraphics[scale=1.3]{diag#1.pdf}}
% %
% %
% %
% %

% \section{Relations between operators of finite order}
% \label{sec:AppendixComputations}



% In this appendix, for completeness, we prove 
% \cref{lemma:relations} and
% \cref{lemma:relations4}.
% All the equalities of operators appearing there can be proved by hand, either by computing commutators or by checking the action on a basis. These computations are lengthy but present no difficulty.
% Here we present them in a diagrammatic way, which has the advantage of grouping terms together in a way which makes them easier to check -- however each reader may prefer to rely on their own technique to group terms, depending on their taste. The diagrams are especially useful to record the calculations to prove~\eqref{eq:rela1} and~\eqref{eq:rel2bis}. Other relations of the lemmas are quickly checked by any mean.
% %

% \medskip
% {\bf Diagrammatic conventions.} Operators are represented by tree-like diagrams, read from top to bottom. Each top or bottom vertex of the diagram represents a variable. Variables from the families $\pp, \yy, \yy', \zz,  \zz'$ are represented by white, black, grey circles  and black, grey squares, respectively ($\circ,\bullet, \bdop{GreyCircle},\bdop{BlackSquare},\bdop{GreySquare}$). Inner nodes of the diagram have the effect of merging or splitting variables in all possible ways, conserving the sum of their indices. For example, the diagrams  \dop{1} and \dop{2}  represent respectively the operators 
% $$\sum_{i,j}  y_{i+j} \frac{\partial^2}{\partial y_j \partial p_i}
% \mbox{ and }
% \sum_{i,j}  p_{i+j} \frac{\partial^2}{\partial y_j \partial p_i}.
% $$
% It is implicit that all sums are taken over combinatorially meaningful values of the parameters, \textit{i.e.} indices of variables $\pp$ are positive while indices of other variables are nonnegative.
% A fat edge in the diagram means that the contribution is weighted by the index (say $j$) of the variable appearing at the top of this edge, for example \dop{3} represents  
% $\displaystyle\sum_{i,j} j y_{i+j} \frac{\partial^2}{\partial y_j \partial p_i}.$
% Similarly, we use a fat dotted edge to weight the contribution by a factor of $j(j-1)$.

% {\bf Composition.}
% Composition of operators corresponds to concatenation of diagrams from top to bottom, respecting the type of variable at the gluing points.
% For example the composition \dop{5}$\circ$\dop{4} is equal to \dop{6}+\dop{7}+ \dop{4}~\dop{5}+\dop{41}. Note that since we work on monomials which are at most linear in $\yy$, the second and third term are the null operator on the spaces we consider.

% {\bf Example.}
% The  operator \dop{6} has two top vertices of types $\yy, \pp$ and two bottom vertices of types $\yy,\pp$, it 
% is equal to\footnote{Indeed  for a given choice of $i,k,i',k'$ such that $i+k=i'+k'$, if one assigns variables $p_i,y_k$ and $p_{i'},y_{k'}$ to top and bottom vertices according to their type, 
% it is possible to distribute indices at each node if and only if $i'<k$ (as one sees by considering the topmost inner node) and in this case there is a unique way to do it. Moreover the two fat edges have respective weights $k-i'$ and $i$, from left to right.}
% $\displaystyle
% \sum_{i+k=i'+k'} \bI{i'<k} i (k-i')  y_{k'} p_{i'} \frac{\partial^2}{\partial y_k \partial p_i} .
% $



% %
% %
% %
% %
% %

% %



% \begin{proof}[Proof of~\eqref{eq:rela1}]
% 	We compute the commutator $[D_\alpha+D_\alpha',\Lambda_Y]$ by collecting its coefficients as a polynomial in $\alpha$ and $b$, viewed as independent variables. We have
% $$
% 	D_\alpha+D_\alpha'=\sdop{26} \ , \ 
% 	\Lambda_Y =\sdop{27}. 
% $$
% We need to evaluate the coefficients of $\alpha^2, \alpha b,  b^2, \alpha, b$, and $1$.
% We start with the coefficient of $\alpha$, which is equal to\footnote{In equations below we expand commutators from left to right, e.g.\ $[a+b,c+d]=[a,c]+[a,d]+[b,c]+[b,d]$ in this order. We make an exception to this rule when, as in~\eqref{eq:commalphab} below, it is convenient to group terms of the R.H.S. in blocks according to the nature of incoming and outgoing variables -- in that case, the left-to-right order of expansion is preserved inside each block. Also, note that unconnected diagrams such as \sdop{4}~\sdop{40} do not appear in commutators, since the two ways to obtain them cancel.}
% 	\begin{align}
% 		\sdop{29}. \label{eq:commIalpha1}
% 	\end{align}
% The third and last terms cancel, and by checking the types of top of bottom vertices, we see that what remains is a linear combination of operators of the form $\bI{i'+k'=i+k} p_{i'}y_{k'}\frac{\partial^2}{\partial y_k \partial p_i}$, 
% 	with coefficients that can be read from the diagrams and are equal to\footnote{In the following equality we assume the equation $i'+k'=i+k$ which is implicit from context. We will do similar assumptions in other computations without recalling it.} 
% 	\begin{align*}
% 		&i(k\sminus{}k')\bI{i'>i}
% \splus{}i(k\sminus{}i')\bI{i'<k}
% \sminus{}ik
% \sminus{}i(i\sminus{}i')\bI{i'<i}
% \splus{}ik'
% 		\sminus{}i(k\sminus{}i')\bI{i'<k}\\=&
% 		i(k\sminus{}k'\splus{}i\sminus{}i')\bI{i'>i}
% 		\splus{}i(k'\sminus{}k\sminus{}i\splus{}i')
% 			=0,
% 	\end{align*}
% 	where we used $(i\sminus{}i')\bI{i'<i}=(i\sminus{}i')(1\sminus{}\bI{i'>i})$.
% Similarly the coefficients $\alpha^2$, $\alpha b$, $b$, and $1$, are respectively equal to 
% \begin{align*}
% 	\sdop{28}, \sdop{30},\\
% 	\sdop{31}, \sdop{32}.
% \end{align*}
% These are respectively linear combinations of operators of the form
% $$
% 	\frac{ y_{i+j+k}\partial^3}{\partial y_k \partial p_i \partial p_j} \ , \
% 	\frac{ y_{i+k}\partial^2}{\partial y_k \partial p_i} \ , \
% 	\frac{ y_{k}p_i\partial}{\partial y_{i+k}} \ , \
% 	\frac{ y_k p_i p_j\partial}{\partial y_{i+j+k}}.$$
% 	The coefficients can be read on the diagrams and are equal\footnote{Note that for the contribution of the second diagram in the coefficient of $\alpha^2$, we have symmetrized in $(i,j)$, namely we substituted $ij (i+k) \rightarrow ij\frac{ (i+k) + (j+k)}{2}$, since we know that the sum we take over $i$ and $j$ is symmetric. We do the same for the very last diagram appearing in the coefficient of $1$.} to, respectively
% 	\begin{align*}
% 		\sminus{}ij\tfrac{i\splus{}j}{2} \splus{} ij\tfrac{ (i\splus{}k) \splus{} (j\splus{}k)}{2}\sminus{}ijk = 0 
% 		\ &, \  
% 		k^2 i \sminus{}ik(i\splus{}k)\sminus{}\tfrac{i(i\sminus{}1)i}{2}\splus{}\tfrac{i (i\splus{}k)(i\splus{}k\sminus{}1)}{2}\sminus{}\tfrac{ik(k\sminus{}1)}{2}=0,
% 		\\
% 		k(k\splus{}i)\sminus{}k^2\splus{}\tfrac{i(i\sminus{}1)}{2} \splus{}\tfrac{k(k\sminus{}1)}{2}\sminus{}
% 		\tfrac{(k\splus{}i)(k\splus{}i\sminus{}1)}{2}=0
% 		\ &, \
% 		\tfrac{1}{2}(i\splus{}j) \splus{} k \sminus{}  \tfrac{(i\splus{}k)\splus{}(j\splus{}k)}{2}=0.
% 	\end{align*}
% 	This concludes the proof that $[D_\alpha+D_\alpha',\Lambda_Y]=0$. 
% \end{proof}

% Before completing the proof of \cref{lemma:relations}, we now prove the main commutation relation of \cref{lemma:relations4}.
% \begin{proof}[Proof of~\eqref{eq:rel2bis}]
% Assuming other relations it is enough to prove the first equality, namely
% 	$[\La_{\tilde{Z}}+\Delta,\La_{\tilde{Y}}]=0$.
% %
% 	We have
% 	\begin{align*}
% 		\Lambda_{\tilde{Z}}+\Delta&=\sdop{18}, \\
% 		\Lambda_{\tilde{Y}}&=\sdop{19}.
% 	\end{align*}
% 	We compute the commutator $[\La_{\tilde{Z}}+\Delta,\La_{\tilde{Y}}]$ as in the proof of~\eqref{eq:rel1}, by collecting its coefficients as a polynomial in $\alpha$ and $b$. Thus we need to evaluate the coefficients of $\alpha^2, \alpha b,  b^2, \alpha, b$, and $1$.
% 	The coefficient of $\alpha^2$ is equal to
% 	\begin{align}
% 		\sdop{20}, \label{eq:commalpha2}
% 	\end{align}
% 	which is a linear combination of operators of the form $\bI{j'+k'=i+j+k}\frac{y_{j'}z_{k'}\partial^3}{\partial p_{i} \partial y_j \partial z_k}$, with coefficients that can be read from the diagrams\footnote{For example, the contribution of the leftmost diagram is computed as follows. Call $p_i,y_j,z_k$ and $y_{j'} z_{k'}$ the bottom and top variables, respectively. The fat edge gives a factor of $i$. Moreoever, for the indices of variables to be distributed at the inner nodes coherently, one sees that the index $i+k$ of the inner node inherited from the concatenation of the two smaller diagrams, has to be larger than the index $j'$ of the bottom $\yy'$-node. It is easy to see that this is the only constraint, hence a total contribution of $i\bI{j'<i+k}$ for that diagram. Other diagrams are treated similarly.}:
% 	\begin{align}\label{eq:luckyAgain}
% i (-\bI{j'<i+k}+\bI{j'<k}+\bI{j'> k}-\bI{j'> i+k}+\bI{j'=k}-\bI{j'=i+k})
% 	= i(1-1) = 0.
% 	\end{align}
% 	Similarly, the coefficient of $\alpha b$ is equal to
% 	\begin{align}
% 		\sdop{21},\!\! \label{eq:commalphab}
% 	\end{align}
% 	Note that the first term of~\eqref{eq:commalphab} is the same as~\eqref{eq:commalpha2} with permuted variables, so it is equal to zero. The second term of~\eqref{eq:commalphab} is a linear combination of operators $\bI{j'+k'=j+k}\frac{y_{j'}z_{k'}\partial^2}{\partial y_{j} \partial z_k}$, with coefficient%
% 	\footnote{In the second term of~\eqref{eq:commalphab}, the coefficient of the third diagram is computed as follows. Call $y_j,z_k$ and $y_{j'}, z_{k'}$ the top and bottom variables, respectively. Then the index (say $u$) of the ``square'' inner node inherited from the concatenation of the two smaller diagrams has to satisfy $u<j$ (from the top part of the diagram) and $u>j'$ (for the bottom part). This requires that $j>j'$, and in this case there are $j\sminus{}j'\sminus{}1$ possible choices for $u$. Similar constraints hold for the ``circle'' inner node but thay are equivalent to the previous ones provided that $j'+k'=j+k$. Therefore the contribution of this diagram is $(j\sminus{}j'\sminus{}1)\bI{j>j'}$. Other cases of the same sort appear in the computations and are treated similarly.}
% 	\begin{align}
% 		&	j\bI{j'>k}
% 	\sminus{}j'\bI{j'>k}
% 		\sminus{}(j\sminus{}j'\sminus{}1)\bI{j>j'}
% 	\splus{}j\bI{j'=k}
% 	\sminus{}j'\bI{j'=k} 
% 	\sminus{}\bI{j'<j}
% 	\sminus{}k\bI{j'<k}
% 	\splus{}k'\bI{j'<k}\label{eq:lucky}
% 		\\&\hspace{10cm} \nonumber
% 		\splus{}(k\sminus{}k'\sminus{}1)\bI{j'>j}
% 		\splus{}\bI{j'>j}   \\
% 		&=
% 		(j\sminus{}j')\bI{j'\geq k}
% 		\splus{}(k'\sminus{}k)(1\sminus{}\bI{j'\geq k})
% 		\splus{}(j'\sminus{}j)\bI{j>j'}
% 		\splus{}(k\sminus{}k')(1\sminus{}\bI{j\geq j'}) \nonumber
% 		\\
% 		&=0\cdot \bI{j'\geq k}
% 		\splus{}0\cdot \bI{j\geq j'}
% 		\splus{}k'\sminus{}k\splus{}k\sminus{}k' =0, \nonumber
% 	\end{align}
% 	Similarly, the coefficient of $b^2$ is equal to
% 	\begin{align}
% 		\sdop{22}, \label{eq:commb2}	
% 	\end{align}
% 	which is a linear combination of operators $\bI{j'\splus{}k'=j\splus{}k}\frac{y'_{j'}z'_{k'}\partial^2}{\partial y'_{j} \partial z'_k}$, with coefficient
% 	\begin{align*}
% 		&k'\bI{j'<k}
% 	\sminus{}k\bI{j'<k}
% 	\splus{}j\bI{j'>k}
% 	\sminus{}j'\bI{j'>k}
% 		\splus{}(k'\sminus{}k\sminus{}1)\bI{j'<j}
% 		\sminus{}(j'\sminus{}j\sminus{}1)\bI{j'>j}
% 	\splus{}j\bI{j'=k}
% 	\sminus{}j'\bI{j'=k}
% 		\\&\hspace{12cm} \nonumber
% 	\splus{}\bI{j'<k}
% 	\sminus{}\bI{j'<j},
% 	\end{align*}
% 	which is zero by the same computation we performed on~\eqref{eq:lucky} (up to exchanging prime and non-primes and reversing inequalities).

% The coefficient of $\alpha$ is 
% 	\begin{align}
% 		\sdop{25},  \label{eq:commalpha1}
% 	\end{align}
% which is a linear combination of operators of the form 
% $\bI{j'\splus{}k'=j\splus{}k}\frac{y_{j'}z_{k'}\partial^2}{\partial y_{j} \partial z_k}$,
% $\bI{j'\splus{}k'=j\splus{}k}\frac{y'_{j'}z'_{k'}\partial^2}{\partial y'_{j} \partial z'_k}$,
% $\bI{i'\splus{}j'\splus{}k'=j\splus{}k}\frac{p_iy'_{j'}z'_{k'}\partial^2}{\partial y_{j} \partial z_k}$,
% $\bI{j'\splus{}k'=i\splus{}j\splus{}k}\frac{y_{j'}z_{k'}\partial^3}{\partial p_i\partial y'_{j} \partial z'_k}$.
% The first and third  have respective coefficients
% $$
% (j\sminus{}j')\bI{j'<j}\sminus{}
% (j\sminus{}j'\sminus{}1)\bI{j'<j}
% \sminus{}\bI{j<j}
% \sminus{}(k\sminus{}k')\bI{k'<k}
% \splus{}(k\sminus{}k'\sminus{}1)\bI{k'<k}
% \splus{}\bI{k'<k}=0,
% $$
% $$
% \bI{j'<k}
% -\bI{j'<k-i'}
% +\bI{j'>k}
% -\bI{j'>k-i'}
% +\bI{j'=k}
% -\bI{j'=k-i'}=1-1=0,
% $$
% while the second is the same as the first up to permutation of colours and the fourth is analogous to the third.


% The coefficient of $b$ is
% 	\begin{align}
% 		\sdop{23}. \label{eq:commb1}	
% 	\end{align}
% The first term is a linear combination of operators $\bI{i'\splus{}j'\splus{}k'=j\splus{}k}\frac{p_{i'}y'_{j'}z'_{k'}\partial^2}{\partial y'_{j} \partial z'_k}$, with coefficient
% $$
% \bI{j'>k}-\bI{j'>k-i'}+\bI{j'=k}-\bI{j'=k-i'}+\bI{j'<k}-\bI{j'<k-i'}=1-1=0.
% $$
% (in fact this computation is the same as~\eqref{eq:luckyAgain} with diagrams upside-down), while the second term is again similar to the second term of~\eqref{eq:commalphab} and is zero by the same computation we performed on~\eqref{eq:lucky}.
% Finally, the coefficient of $1$ is
% 	\begin{align}
% 		\sdop{24}, \label{eq:comm1}	
% 	\end{align}
% 	which is the same as the first term of \eqref{eq:commb1} up to an exchange of variables, so is equal to zero too. This concludes the proof that $[\La_{\tilde{Z}}+\Delta,\La_{\tilde{Y}}]=0$.
% \end{proof}

% The remaining computations are much shorter and can easily be done without any diagrams. We include them here for completeness. We introduce further notations: increment of the index of the variable is represented by an arrow, for example $Y_+=\dop{11}$.

% \begin{proof}[Proof of~\eqref{eq:rela2}]
% 	We directly compute the commutator $[D_\alpha\splus{} D_\alpha',Y_+]$
% \begin{align*}
% 	&\big[\dop{10}, \dop{11}\big]=\dop{12}\\
% 	&=\alpha  \sum_{i,k} [i(k\splus1)\sminus ik  ] \frac{y_{i+k+1}\partial^2}{\partial p_i y_k}
% 	 + \sum_{i,k} [ (k\splus1)\sminus k ] \frac{p_i y_{k+1}\partial}{\partial y_{i+k}}
% 	 + \frac{b}{2}\sum_{k} [(k\splus1)k \sminus k (k\sminus1)  ] \frac{y_{k+1}\partial}{\partial y_{k}}\\
% 	 &=Y_+ \Lambda_Y. \qedhere
% \end{align*}
% \end{proof}

% \begin{proof}[Proof of~\eqref{eq:rela3}]
% We want to prove that $[D_\alpha, \Theta_Y]=\Theta_Y D'_\alpha$. We have
% 	\begin{flalign*}
% 		[D_\alpha, \Theta_Y] = [\dop{14},\dop{13}]=\dop{15},\\
% %
% %
% 		\ \Theta_Y D'_\alpha = \dop{16}.
% 	\end{flalign*}
% 	To conclude it is enough to notice that \dop{17}, \textit{i.e.} $\sum_{i,j} i \frac{p_{i+j}\partial^2}{\partial p_i \partial p_j} = \frac{1}{2}\sum_{i,j} (i+j) \frac{p_{i+j}\partial^2}{\partial p_i \partial p_j}$.
% \end{proof}

% \begin{proof}[Proof of~\eqref{eq:rel1}]
% 	We compute the operators $\La_{\tilde{Z}}\Delta$ and $\Delta \La_{\tilde{Y}}$, which are equal respectively to
% 	\begin{align*}
% 		\sdop{33}
% 	\end{align*}
% 	and
% 	\begin{align*}
% 		\sdop{34}.
% 	\end{align*}
% We conclude by noticing that the two quantities differ only by the colour of inner vertices and reordering of terms, more precisely both are equal to	
% 	\begin{align*}
% 		\sdop{35}. 
% 	\end{align*}
% \end{proof}

% \begin{proof}[Proof of~\eqref{eq:rel3} and~\eqref{eq:rel3bis}]
% 	It is enough to prove the first one, namely $[\La_{\ZZ},\YY_+] = \YY_+ \Delta$. The commutator $[\La_{\ZZ},\YY_+]$ is equal to
% 		\begin{align*}
% 			&	\sdop{36}\\
% 			&=\sum_{i'\splus{}k'=i\splus{}k\splus{}1}
% 			\Big(\alpha  (\bI{k'<i\splus{}1}\sminus{}\bI{k'<i})  \frac{y'_{i'}z'_{k'}\partial^2}{\partial y_i z_k}
% 	 \splus{}  (\bI{k'<i\splus{}1}\sminus{}\bI{k'<i})  \frac{y_{i'}z_{k'}\partial^2}{\partial y'_i z'_k}
% 		\splus{}  b  (\bI{k'<i\splus{}1}\sminus{}\bI{k'<i})  \frac{y'_{i'}z'_{k'}\partial^2}{\partial y'_i z'_k}
% 			\Big)	\\
% 	 &
% 			=\sum_{i,k}
% 			\alpha  \frac{y'_{k+1}z'_{i}\partial^2}{\partial y_i z_k}
% 	 \splus{}    \frac{y_{k+1}z_{i}\partial^2}{\partial y'_i z'_k}
% 		\splus{}  b    \frac{y'_{k+1}z'_{i}\partial^2}{\partial y'_i z'_k}
% 			=\YY_+ \Delta,
% 	\end{align*}
% 	where for the last equality we used that $\bI{k'<i\splus{}1}\sminus{}\bI{k'<i} =\bI{k'=i}$ and that $k'=i$ and  
% $i'\splus{}k'=i\splus{}k\splus{}1$ imply that $i'=k+1$.
% \end{proof}


% \begin{proof}[Proof of~\eqref{eq:rel4}] It is enough to prove the first equality $\Theta_{\tilde{Z}}\La_{\tilde{Y}} = \La_{Y}\Theta_{\tilde{Z}}$, \textit{i.e.} $\Theta_{\tilde{Z}}(\La_{\tilde{Y}}-\La_Y) = [\La_{Y},\Theta_{\tilde{Z}}]$.
% Now $[\La_{Y},\Theta_{\tilde{Z}}]$ and $\Theta_{\tilde{Z}}(\La_{\tilde{Y}}-\La_Y) $ are respectively equal to	
% 	\begin{align*}
% 		\sdop{37}
% 	\end{align*}
% 	and
% 	\begin{align*}
% 		\sdop{38}.
% 	\end{align*}
% 	The difference of these two quantities is a linear combinations of operators 
% 	$\frac{y_{i+j+k}\partial^3}{\partial p_i \partial y'_j \partial z'_k}$,
% 	$\bI{i'+j'=j+k}\frac{p_{i'}y_{j'}\partial^2}{\partial y'_j \partial z'_k}$,
% 	$\frac{y_{i+j}\partial^2}{\partial y'_i \partial z'_j }$,
% 	$\frac{y_{i+j}\partial^2}{\partial y_i \partial z_j }$.
% The coefficients of these operators in this difference are respectively equal to
% 	\begin{align*}
% 		\alpha i  - \alpha i =0 \ , \  
% 		1 - \bI{i'\leq j} -\bI{i'>j}=0 \ ,\  
% 		b(i+j) -bi -bj =0 \ ,\  
% 		\alpha j - \alpha j=0.
% 	\end{align*}


% %


% \end{proof}

% \begin{proof}[Proof of~\eqref{eq:rel5}]
% 	It is enough to prove the first equality
% $\Theta_{\tilde{Z}}\tilde{Y}_+ = Y_+\Theta_{\tilde{Z}}$, which is straightforward
% 	since both are equal to
% 	$	\sum_{i,j}p_i y_{j+1}\frac{\partial^2}{\partial
%           y_j\partial z_i} +
%         \sum_{i,j}y_{i+j+1}\frac{\partial^2}{\partial y'_i z'_j}$ with
%         the convention that $p_0=1$. To be consistent with the rest we still provide the diagrammatic interpretation:
% 	\begin{align*}
% 		\sdop{39}.
% 	\end{align*}
% \end{proof}


% %\bibliographystyle{amsalpha}

% %\bibliography{biblio2015}


% \section{Deleted stuff from the long version}



% Let $\mathcal{S}$ be a \emph{surface}, that is to say a compact, two dimensional, real
% manifold. By the classification theorem a connected surface $\mathcal{S}$ is
% uniquely determined by its Euler characteristic 
% $\chi_{\mathcal{S}} \leq 2$ 
%  (or, equivalently, its \emph{genus} $g_{\mathcal{S}} \in \frac{1}{2}\N$  given by $\chi_\mathcal{S} = 2-2 g_{\mathcal{S}}$) together with 
% the information whether $\mathcal{S}$ is orientable or not. 
% For two surfaces $\mathcal{S}_1, \mathcal{S}_2$ we call a continuous map $$f: \mathcal{S}_1 \to \mathcal{S}_2$$ a
% \emph{branched covering} (also known as \emph{ramified covering} or \emph{branched
%   cover}) 
%   %
%   of $\mathcal{S}_2$ by $\mathcal{S}_1$ 
%   if every point $s \in
% \mathcal{S}_2$ has an open neighborhood $U\ni s$ such that $f^{-1}(U)$ is a union of disjoint open sets $V_1\dots,V_\ell$, for some $\ell\geq 1$, such that 
%  on each $V_i$ the map $f$ is topologically equivalent to the complex map
%  $z \to z^{p_i}$ for some positive integer $p_i$ (with $s$ corresponding to $0\in \mathbb{C}$).
%  %
%  %
% For each $s\in \mathcal{S}_2$, we can reorder the multiset
% $\{p_1,\dots,p_\ell\}$ to form a \emph{partition} $\la = (\la_1,\dots,\la_\ell)$ of length $\ell$, that is a sequence of integers such that
% $\la_1\geq \cdots \geq \la_\ell >0$. 
% %
% This partition, denoted by $\lambda(s)$, is called the \emph{ramification profile over $s$}, or \emph{profile over $s$} for short.
% When the surfaces are connected, the size
% $n=\lambda_1+\dots+\lambda_\ell$ of the partition $\lambda(s)$ does
% not depend on the point $s$, and it is called the \emph{degree} of the covering. In particular, it is equal to the number of preimages of a generic point of $\mathcal{S}_2$.
% %
% The integer
% $\ell$ which is \emph{the length} of the partition $\la(s)$
% %
% %
% is called the \emph{multiplicity} of $s$. 

% %
% %
% %
% %
% %
% %
% %
% %
% %
% %
% %
% %
% %
% %
% %
% %
% %
% %
% There are finitely
%  many points $s_1\dots,s_k \in \mathcal{S}_2$ of multiplicity
%  smaller than the degree -- they are called \emph{critical values}, or
%  \emph{ramification points}. 
%  The multiset $\{\la(s_1),\dots,\la(s_k)\}$ of
%  profiles over all ramification points is called \emph{the full profile}
%  of the covering $f$.
% %
%  Sometimes the ramification points will be numbered from $1$ to $k$, in which case the full profile will be defined as the ordered $k$-tuple $(\la(s_1),\dots,\la(s_k))$. 
% %
%  We will (classically) allow the partitions $\la(s_i)$ to be equal to $[1^n]$, \textit{i.e.} we allow  ``trivial ramification points''.
%  %
%  We say that two branched coverings $f_1: \mathcal{S}_1 \to
%  \mathcal{S}, f_2: \mathcal{S}_1 \to
%  \mathcal{S}$ are \emph{equivalent} if there exists a homeomorphism $\phi: \mathcal{S}_1 \to
%  \mathcal{S}_2$ such that $f_2 = f_1\circ\phi$. When ramification points of
%  $f_1$ and $f_2$ are numbered we additionally require $\phi$ to
%  preserve this numbering.


% %
%  %
%  %

% \subsection{Generalized branched coverings.}
% When $f:\mathcal{S}_1 \to \mathcal{S}_2$ is a branched covering and
% $\mathcal{S}_2$ is orientable, then necessarily $\mathcal{S}_1$ is
% orientable. This is the case in particular when $\mathcal{S}_2$ is the sphere $\Sphere$. We will now
% generalize the definition of branched coverings to allow arbitrary
% surfaces as the covering space of the sphere.


% Let $\mathcal{S}$ be a surface. We let
% $\pi_{\mathcal{S}}:\widetilde{\mathcal{S}} \rightarrow \mathcal{S}$ be
% the orientation double cover of $\mathcal{S}$, and we let $\sigma:
% \widetilde{\mathcal{S}} \to \widetilde{\mathcal{S}}$ be the
% corresponding involution of $\widetilde{\mathcal{S}}$, so that
% %
% %
% $\mathcal{S} \equiv \widetilde{\mathcal{S}}/\sigma$.
% We also define $\Sphere_+$ and $\Sphere_-$ as the upper and the
%   lower hemisphere, respectively, (with the equator $\partial \Sphere_+ = \partial
%   \Sphere_-$ as common boundary) and the natural projection $p:\Sphere \to
%   \Sphere_+$ identifying both hemispheres. 
%   \begin{definition}\label{def:generalizedBC}
% Let $f: \mathcal{S} \to \Sphere_+$ be a continuous map, which
% (restricting its domain) is a covering of
% $\Sphere_+\setminus\partial\Sphere_+$. We say that $f$ is a
% \emph{generalized branched covering} of the sphere if there exists a
% branched covering of the sphere
% $\tilde{f}:\widetilde{\mathcal{S}}\to \Sphere$ such that
% 	  \begin{align}\label{eq:compatibility}
% 		  f \circ \pi_{\mathcal{S}} = p\circ\tilde{f}.
% 	  \end{align}
% 	We say that two generalized branched
% coverings $f: \mathcal{S}\to \Sphere_+$ and $f':\mathcal{S}'\to \Sphere_+$ are
%         \emph{equivalent}  if the branched
% coverings $\tilde{f}$ and $\tilde{f'}$ are equivalent.
% \end{definition}
% %
% %
% %

 
% When the covering space is orientable, generalized branched coverings are in natural bijection with branched coverings. Indeed, when $\mathcal{S}$ is orientable the associated
% orientation double cover is simply
% $\widetilde{\mathcal{S}}= \mathcal{S}\uplus\mathcal{S}$,
% %
% %
% %
% %and the mapping $\tilde{f}:\widetilde{\mathcal{S}}\to \Sphere$ subject
% %to $f \circ \pi_{\mathcal{S}} = p\circ\tilde{f}$ is determined by its
% %behaviour on a single copy of $\mathcal{S}$. Therefore $f = p \circ
% %\widetilde{f}$, where $\widetilde{f}$ is a branched covering of the
% %sphere by $\mathcal{S}$ (and we conclude by using the fact that for a given
% %$f$ the corresponding branched covering $\widetilde{f}$ is unique (up
% %to equivalence). This uniqueness will be proved in \cref{prop:Correspondence2}.
% %
% %
% and the branched covering $\tilde{f}:\widetilde{\mathcal{S}}\to \Sphere$ subject
% to $f \circ \pi_{\mathcal{S}} = p\circ\tilde{f}$ is determined by its
% behaviour on a single copy of $\mathcal{S}$,
% %.
% %Therefore $f = p \circ
% %\widetilde{f}$, where $\widetilde{f}$ is a branched covering of the
% %sphere by $\mathcal{S}$,
% which gives the desired bijection (the uniqueness of $\widetilde{f}$ up to equivalence will be proved in the proof of \cref{prop:Correspondence2}). 



% We will be interested in enumerative properties of the
% generalized branched coverings of the sphere $\Sphere$.
% First note that for a
% generalized branched covering $f:\mathcal{S}\to \Sphere_+$
% and for each $s \in \Sphere$, if $s$ is a ramification point of $\tilde{f}$,
%  then necessarily $s$ lies on the equator $\partial\Sphere_+$ (note the assumption that $f$ restricts to a \emph{covering} of $\Sphere_+\setminus\partial\Sphere_+$, and not to a \emph{branched covering}).
%  The profile of the associated branched
% covering $\tilde{f}$ over $s$ is a partition of
% the form $[2]\la =
%  (\lambda_1,\lambda_1,\lambda_2,\lambda_2,\dots,\lambda_\ell, \lambda_\ell) \vdash 2n$ for some partition $\la
%  \vdash n$.  We will call $p(s)\in \Sphere_+$ a \emph{ramification point} of $f$. We denote the partition $\lambda$ by $\la(s)$ and we will call it the \emph{profile} over
%  $s$. The \emph{full profile} of the generalized branched covering is given
%  by the multiset $\{\la(s):s \text{ is a ramification point}\}$. As
%  before, if ramification points are numbered, the full profile will be
%  an ordered tuple, and we may allow trivial ramification points. Also, when ramification points are numbered, the equivalence between $\tilde{f}$ and $\tilde{f}'$ in Definition~\ref{def:generalizedBC} is understood between branched coverings with \emph{numbered} ramification points.
% The integer $n$, which is half the degree of the branched covering $\tilde{f}$, is called the \emph{degree} of the generalized branched covering $f$. All these definitions are compatible with the standard definitions in the case where $\mathcal{S}$ is orientable, via the natural bijection of the previous paragraph.

% \bigskip

% For each covering $f: \mathcal{S} \rightarrow \Sphere_+$  contributing to~\eqref{eq:logIntro}, the homeomorphism type of the covering surface $\mathcal{S}$ is fully determined by the quantity $\kappa(f) t^{|f|} b^{\nu_\rho(f)}$. Indeed, orientability is controlled by the parameter $\nu_\rho(f)$, and the Euler characteristic is deduced from the Riemann-Hurwitz formula, see~\eqref{eq:EulerCharacteristic}. In particular, \eqref{eq:logIntro} implicitly contains a full topological expansion -- see also Remark~\ref{rem:genus}.

% When $\mathcal{S}$ is orientable, generalized
% branched coverings are in bijection with (usual) branched
% coverings.
%  Therefore for $b=0$, our result recovers the classical
% interpretation of the tau-function~\eqref{eq:SchurIntro} in terms of
% branched coverings (see
% e.g.~\cite{GouldenJackson2008,AlexandrovChapuyEynardHarnad2020}). For
% $b=1$, our theorem says that $\tau^{(k)}_b$ counts generalized
% branched coverings of the sphere by arbitrary surfaces, without any
% $b$-weighting. This fact could probably be proved by (now standard)
% ideas close to the one used by Goulden,
% Jackson~\cite{GouldenJackson1996} and Hanlon, Stanley,
% Stembridge~\cite{HanlonStanleyStembridge1992} which cover the case
% $k=1$  using the connection with representation theory of the Gelfand pair $(\mathfrak{S}_{2n},\mathbb{H}_n)$. However, for $b\not \in\{0,1\}$ our result is inaccessible by these methods, due to the lack of a well-adapted representation theoretic connection to Jack polynomials.


% \bigskip
% Let $\mathcal{S}$
%   be a compact connected surface, orientable or not, and let $\Sphere$
%   denote the two-dimensional sphere.
%   %
% %
% %
% %
% %
% Let $\widetilde{\mathcal{S}}$ be the orientation-double-cover of $\mathcal{S}$.
% A \emph{generalized branched
%   covering of $\Sphere$ by $\mathcal{S}$} is a continuous function 
%   $f: \mathcal{S} \rightarrow\Sphere_+$ from $\mathcal{S}$ to the closed upper hemisphere $\Sphere_+$, 
% which can be lifted to a branched covering 
% $\tilde{f}:\widetilde{\mathcal{S}} \rightarrow \Sphere$ in a certain sense.
% A precise definition, together with the definition of degree, ramification points and
% ramification profiles, is given in~\cref{sec:constellations}.

%  Generalized branched coverings with $k+2$ ramification points are in bijection (\cref{sec:constellations}) with some combinatorial embedded graphs on the surface $\mathcal{S}$ that we call $k$-constellations. These objects come with a natural notion of \emph{rooting} which consists in marking and orienting an angular sector (\cref{sec:constellations}). Our main result can be summarized as follows (see in particular \cref{thm:mainInSection5} page~\pageref{thm:mainInSection5} and \cref{rem:translation} page~\pageref{rem:translation}). 
% In this paper, if $\lambda=(\lambda_1\geq \lambda_2\geq \dots \geq \lambda_\ell)$ is an integer partition and $(p_i)_{i\geq 1}$ is a sequence of variables, we write $p_\lambda= p_{\lambda_1}p_{\lambda_2}\dots p_{\lambda_\ell}$.

% \bigskip


% {\bf Possible developments.}
% Our paper sets the foundations of the study of $b$-Hurwitz
% numbers. Many natural questions arise, which are not the subject of
% this paper. 
% First, the cases $b=0,1$ are related respectively to the integrable hierarchies KP and BKP  -- at least in certain cases, see e.g.\ \cite{AdlervanMoerbeke2001} in the context of $\beta$-ensembles.
% The key role played by Lax structures in this paper indicates that
% some of the properties related to integrability may still be present
% for general $b$. Also, potential links of our work with
% \cite{NatanzonOrlov2017}, where the Hurwitz numbers for (non-generalized)
% branched coverings of the
% projective plane were studied in the context of the BKP hierarchy, could be
% investigated. Further, the
% tau-function for $b=0$ famously satisfies, at least in some special
% cases, the so-called \emph{Virasoro constraints} (e.g.~\cite{KazarianZograf2015}, see also
% \cref{rem:VirasoroPatched}). We plan to address the $b$-deformation of
% these in detail in a forthcoming work (for the case of $\beta$-ensembles, see again \cite{AdlervanMoerbeke2001}). In another direction, although our
% results are not strictly comparable to the Matching-Jack conjecture
% and the $b$-conjecture, they are by far the best partial progress
% towards them. It is conceivable that in the future, results of this
% paper are used in new attacks to these problems. Finally, Hurwitz numbers are classically linked to the moduli space of complex curves in several ways (most famously via the ELSV formula~\cite{EkedahlLandoShapiroVainshtein2001}) and the $b$-deformed \textit{dessins d'enfants} appear for $b=1$ in work of Goulden, Harer, and Jackson on the moduli space of real curves~\cite{GouldenHarerJackson2001}. These authors were the first to ask for the possible significance of the ``$b$ parameter'' in this geometric picture. Our paper is an advance in that direction. Understanding the integer parameter $\nu_\rho$ that we introduce in this paper, in a purely geometric way, seems to be a natural question to consider to go further. It should be related in some sense to a ``stratification'' of the moduli space, yet to be understood.


% \bigskip
% {\bf Overview of the paper and intermediate results.}
% The paper is almost entirely dedicated to the proof of our main result -- only \cref{sec:infinite} is independent and examines the projective limit when $k\rightarrow \infty$.
% %
% However several intermediate concepts and results appearing along the way are interesting in themselves, even in the case of $b\in \{0,1\}$.
% Here is a short overview of our main contributions and a roadmap to our paper.


% In \cref{sec:constellations} we introduce generalized branched
% coverings and their combinatorial counterparts, $k$-constellations. In
% \cref{sec:MonTutte} we introduce the notion of Measure of
% Non-Orien\-tability (MON) and the combinatorial decomposition, that
% together give rise to the $b$-weigths and the parameter $\nu_\rho$.
% We also state that the generating function of generalized branched coverings (or constellations) satisfies an explicit equation reflecting the combinatorial decomposition. This is \cref{thm:TutteConnected} page~\pageref{thm:TutteConnected}.

% In \cref{subsec:proofTutte} we prove the decomposition
% equation by analysing carefully the combinatorial decomposition. As far as we
% know, this equation is interesting even for $b=0$ as it did not appear
% earlier in full generality. \cref{subsec:Commutation} contains
% the key idea of the paper: the combinatorial operators appearing in
% the decomposition equation can alternatively be defined by recursive commutation relations (\cref{thm:commut1,thm:commut2} pages~\pageref{thm:commut1}--\pageref{thm:commut2}) or equivalently by two Lax equations (\cref{prop:Lax}).
% Proving this claim is the hardest part of the paper. \cref{subsec:heuristic} sketches a combinatorial proof for $b\in\{0,1\}$, which serves as an inspiration for the general proof, given in \cref{subsec:proofCommut1,subsec:proofCommut2}.
 
% \cref{sec:Jack} deals with Jack polynomials and shows that
% the function $\tau^{(k)}_b$ is annihilated by the operators
% constructed in the previous section. This makes the connection with
% generalized branched coverings and constellations, and proves our main theorem. Interestingly, and as far as we know, this proof is also new in the classical case $b=0$: it is the first proof of the Schur function expansion of the generating function of coverings that does not rely on representation theory. The same is true of course for $b=1$. 
	
% In \cref{sec:infinite} we show how to take a projective limit
% of our results in order to build a non-orientable, $b$-weighted,
% analogue of the weighted Hurwitz numbers. All results of the paper are
% extended to this setting, including $b$-weights, decomposition
% equations, $b$-polynomiality. In
% \cref{subsec:bHurwitz} we study the case of (simple or double)
% $b$-weighted Hurwitz numbers, which correspond to the case where all
% ramification points except the first two are simple. We prove a deformed version of the Cut-And-Join equation, and piecewise polynomiality. In \cref{subsec:betaEnsembles} we discuss dessins d'enfants and $\beta$-ensembles, and we make a detailed account of the $b$- and Matching-Jack conjectures of Goulden and Jackson, and how they relate to our results. In \cref{subsec:HCIZ} we discuss monotone Hurwitz numbers and the HCIZ integral.

% Finally, the appendix contains the proof of two lemmas relying on computations that present no difficulty, but are included for completeness.

% \bigskip

% Lando and Zvonkine introduced in their book~\cite{LandoZvonkin2004} a
% particular set of vertex-coloured maps, subject to local coloring
% constraints, called \emph{constellations}, that are in bijection with
% branched coverings of the sphere $\Sphere$. The constellation
% associated to a covering $f:\mathcal{S}\rightarrow\Sphere$  with $k+2$
% numbered ramification points, is the map on $\mathcal{S}$ formed by
% the preimage of a ``base graph'' drawn on the sphere going through
% some of these points. The standard choice of base graph is a star
% centered at a generic point, and connected in cyclic order to the
% points numbered $0, 1, \dots, k$. These maps satisfy some simple local
% colouring constraints that fully characterize them. Different choices
% of base graph lead to different definitions which are easily seen to be equivalent, see e.g.~\cite[Figure 1.34]{LandoZvonkin2004} or~\cite{BousquetSchaeffer2000, AlexandrovChapuyEynardHarnad2020}.

% To construct generalized constellations we will use a similar
% principle but it will be important to choose a base graph that does
% not depend on an orientation of the sphere. For this reason we will
% use a path going through the ramification points rather than a star, see~\cref{fig:Coverings}. We leave to combinatorialist readers the pleasure of designing a direct bijection, in the orientable case, between the model we introduce and the one of~\cite{LandoZvonkin2004}.


% \bigskip

% \begin{figure}
% \center
% \includegraphics[width=0.9\linewidth]{Coverings.pdf}
% \caption{The correspondence between generalized branched coverings and
% 	constellations in the case of $k+2=4$ ramification
%         points. The base graph $P$, which is a path, and the corresponding
%         constellations are drawn in fat black. Blue lines give the
%         triangulations described in the proof of \cref{prop:Correspondence2}. On this example, $\widetilde{\mathcal{S}}$ is the
%         sphere and $\mathcal{S}$ is the projective plane.
%         %
% 	}
% \label{fig:Coverings}
% \end{figure}



% We can now state the correspondence between coverings and
%   constellations. The proof uses a classical
%   result in the orientable case.

% \begin{proposition}[see
%   \cref{fig:Coverings}-Left]\label{prop:Correspondence2}
%   	Let $k\geq 1$ be an integer and let $f:
%         \mathcal{S}\rightarrow \Sphere_+$ be a generalized
%         branched covering of the sphere with $k+2$
%         ramification points as in \cref{def:generalizedBC} monotonically numbered from $-1$ to $k$  along the equator $\partial\Sphere_+$. Let $P$ be a path
%         on the equator  going through the points $0,1,\dots,k$ in this
%         order. Then the  preimage $f^{-1}(P)\subset \mathcal{S}$ is a $k$-constellation on $\mathcal{S}$.

	
% 	This construction gives a one-to-one correspondence between
%         equivalence classes of generalized branched
% coverings of the sphere with  $k+2$ monotonically numbered ramification points and full profile
% 	$(\la^{-1},\la^0,\la^1,\dots,\la^k)$, and $k$-constellations with
% the same full profile.
% \end{proposition}

% \begin{proof}
%   The fact that the embedded graph $f^{-1}(P)$ on $\mathcal{S}$
%   satisfies the local constraints of constellations (with vertex
%   colours given by the pull-back of the numbering of ramification
%   points) is clear. The fact that it is a well-defined map comes from
%   the fact that each region delimited by this graph on $\mathcal{S}$
%   retracts to the neighbourhood of a preimage of the ramification point~$-1$. 
% 	Each such neighbourhood is homeomorphic to a disk by
%   definition of a generalized branched covering. We need to prove that
%   $f^{-1}(P)$ gives the same constellation for equivalent
%   generalized coverings.
%   First note that $P$ is naturally identified with
%   $p^{-1}(P)$, since $p$ is the identity on $\partial\Sphere_+ =
%   \Sphere_+\cap \Sphere_-$, so we will denote both paths simply by
% 	$P$. In particular, using~\eqref{eq:compatibility} we have 
% 	$\tilde{f}^{-1}(P) = (p \tilde{f})^{-1}(P) = (f \pi_S)^{-1}(P)$, 
% 	 which does not depend on the
%     choice of $\widetilde{f}$. This means that the constellation
% 	$\tilde{f}^{-1}(P)$ on $\tilde{\mathcal{S}}$ is determined by $f$. It is a standard fact in the orientable case (see~\cite[Chapter 1]{LandoZvonkin2004}, adapted to our slightly different choice of base graph)
%     that the constellation $\tilde{f}^{-1}(P)$ determines
%     $\tilde{f}$ uniquely up to equivalence. Now, let $f$ and $g$
% 	be equivalent generalized coverings, and consider the (unique up to equivalence) associated branched coverings 
% 	$\tilde{f}$, $\tilde{g}$.
% 	%
% %
% %
% %
% 	By definition  $\tilde{f}$ and $\tilde{g}$ are equivalent, therefore $\tilde{f}^{-1}(P)$ and $\tilde{g}^{-1}(P)$ are
% 	the same constellations. Using~\eqref{eq:compatibility} again shows that $f^{-1}(P)=\pi_S (\tilde{f}^{-1}(P))$ (and the analogue statement for $g$), therefore the two constellations $f^{-1}(P)$ and $g^{-1}(P)$ are also the same, as we wanted to prove.
	

% 	Now let $\bM$ be a constellation of $\mathcal{S}$. Then
%         $\widetilde{\bM}=\pi_{\mathcal{S}}^{-1}(\bM)$ is a map on the
% 	orientable surface $\widetilde{\mathcal{S}}$. Using the standard arguments of
% 	the orientable case~\cite[Chapter 1]{LandoZvonkin2004} (adapted again to our
% 	choice of base graph),  we can  construct from $\tilde{\bM}$
%         a branched covering $\tilde{f}: \widetilde{\mathcal{S}}\rightarrow
%         \Sphere$ as follows. Triangulate $\Sphere$ by triangles with
%         vertices given by the ramification points labeled by $-1,i,i-1$ for $i \in [1..k]$. In this way, we obtain $k$ triangles on the upper hemisphere $\Sphere_+$
%         and $k$ corresponding (through $p$) triangles on the lower
% 	hemisphere, and such that the equator $\Sphere_+\cap\Sphere_-$
%         is a cycle $(-1,0,\dots,k)$, see \cref{fig:Coverings}. Triangulate each face of $\bM$ by putting a new
%         vertex $-1$ inside each face and connecting it to all the
%         corners of the corresponding face. Pick an orientation on
%         $\widetilde{\mathcal{S}}$ to send triangles with the set of
%         vertices $-1,i,i-1$ visited in this order to the corresponding
%         triangle in $\Sphere_+$ and visited in the reverse order to the corresponding
%         triangle in $\Sphere_-$. Note that applying
%         $\pi_{\mathcal{S}}$ to the triangulation of
%         $\widetilde{\mathcal{S}}$ we obtain a triangulation of
%           $\mathcal{S}$, which allows us to construct $f$ by sending
%           triangles of the form $-1,i,i-1$ into corresponding
%           triangles in $\Sphere_+$. The compatibility relation $f
%           \circ \pi_{\mathcal{S}} = p\circ\tilde{f}$ is satisfied
%           since the triangulations of $\Sphere_+$ and $\Sphere_-$ are
%           identified by $p$.

% 	The fact that the two constructions are inverse of each other, and that the full profile is preserved, is direct by construction.
%       \end{proof}


% \bigskip

% Indeed, this formula is true for branched coverings and by construction this immediately implies that it holds for general branched coverings as well.
% We remark that~\eqref{eq:EulerCharacteristic} only involves the length of each partition $\lambda^i$. In this paper we will enumerate generalized branched coverings of the sphere without controlling the full profile, but with enough control to keep track of these lengths, hence of the Euler characteristic of the underlying surface. 

% \begin{remark}\label{rem:translation}
% 	Now that the correspondence between generalized branched coverings and
%         constellations is established, in the rest of the paper, we
%         will work with $k$-constellations, which are more convenient to
%         enumerative purposes.
% 	The theorem stated in the introduction (\cref{thm:abbreviated}) will be proved in the language of constellations (Theorem~\ref{thm:mainInSection5}). The sum over rooted coverings $f$ in this theorem is understood as the sum over rooted constellations $(\bM,c)$ in~\eqref{eq:mainInSection5}. The integer parameter $\nu_\rho(f)$ in that theorem is understood as the parameter $\nu_\rho(\bM,c)$ that we introduce in the next section, while the quantities $|f|, \kappa(f), v_i(f)$ in the theorem are understood respectively as the quantities  $|\bM|, \kappa(\bM,c), v_i(\bM)$ defined in \cref{sec:constellations}.
% 	%
% \end{remark}

% \begin{remark}
% 	Some authors may prefer to call $(k+1)$-constellations what we call $k$-constellations here. This is related to the fact that in our main function~\eqref{eq:JackIntro} we have \emph{two} sets of ``time'' parameters $\pp$ and $\qq$. In many applications one studies the specialization $q_i=\bI{i=1}$, which on coverings corresponds to the case where the second marked ramification point is trivial (this is the same as viewing single Hurwitz numbers as special cases of double ones). Among these two natural choices of terminology, we kept the one that was shorter and more convenient for our purposes.
% \end{remark}

% 	%
% 	We conclude this section by introducing the notion of duality. 
% 	\begin{definition}
% 	\emph{Duality} is the involution on $k$-constellations defined
%         as follows. Given a $k$-constellation $\bM$, add a new vertex of colour $-1$ inside each face and
% link it by a new edge to all corners of label $k$. Then remove all vertices of $\bM$ of colour $0$ and edges
% incident to them. Finally, exchange colours $-1\leftrightarrow 0$ and $k+1-i\leftrightarrow i$ for each $1 \leq i \leq k$. The map $\tilde \bM$ thus obtained is called the \emph{dual of $\bM$.} 
% 	\end{definition}
% The fact that duality is an involution is clear from the interpretation on coverings, since it can be interpreted as a reflection of $\Sphere_+$ together with a renumbering of ramification points. 
% There is a natural correspondence between corners of colour $0$ in $\bM$ and $\tilde \bM$, which enables to lift duality at the level of labelled and/or rooted objects.

% \bigskip
% Our main results also require us to define integral and coherent MONs.
% \begin{definition}[Integral MON, Coherent MON; see \cref{fig:niceMON}]\label{def:niceMON}

% 		A MON $\rho$ is \emph{integral} if $\rho(\bM,e)$ belongs to
% 	$\{1,b\}$ for any $\bM$ and $e$, and if the following is true: for every every pair $(\bM,e)$ which is in case (1)(b) of Definition~\ref{def:MON} and such that $\bM$ is orientable, we have $\rho(\bM,e)=1$ and $\rho(\tilde{\bM},\tilde{e})=b$, in the notation of Definition~\ref{def:MON}. 
 

% 	A MON $\rho$ is \emph{coherent} if for any colored map $\bM$, for any corner $c$ of $\bM$ of color $j$, and for any face $f$ of $\bM$ having an even number of corners of color $j+1$, the following is true. 
% 	Choose an arbitrary orientation of $f$, and number  $c_1,\dots,c_{2d}$ the corners of color $j+1$ in $f$. Also choose an arbitrary orientation for $c$. 
% 	For $i\in [1..2d]$ let $e_i, \tilde{e}_i$ be the two possible edges connecting $c$ to $c_i$, where $e_i$ is the one that respect the corner orientations, and $\tilde{e}_i$ is its twist.
% 	Then for any $i$: 
% 	$$\rho(\bM\cup\{e_i\},e_i)+\rho(\bM\cup\{\tilde e_{i+1}\},\tilde{e}_{i+1})=(1+b).$$
% \end{definition}


% It is easy to see that MONs exist and to construct them. In fact, we have
% \begin{lemma}\label{lemma:MONexist}
% There exist MONs that are both coherent and integral.
% \end{lemma}
% \begin{proof}
% The only choices that we have to make to construct a MON are
%         the values of $\rho(\bM,e)$ and $\rho(\tilde{\bM},\tilde{e})$
%         for pairs $(\bM,e)$ that are in case 1(b) of
%         \cref{def:MON}. Indeed, all other values are imposed
%         by the axioms. If we only wanted to respect Axiom (1), we
%         could choose these values arbitrarily in $\{1,b\}$. Here we
%         also have to be careful to perform  these choices
%         simultaneously to respect Axiom (2) and to ensure
%         coherence and integrality. For this we adapt the arguments of \cite[Section
%         5.1]{Dolega2017a} to our settings.

% 	We first equip every connected vertex-coloured map with a fixed orientation of all its faces, given by a global orientation if the surface is orientable, and chosen arbitrarily for each face otherwise.
% 	Given an edge $e$ in a vertex-coloured map $\bM$ such that the pair $(\bM,e)$ is in case 1(b) of \cref{def:MON}, we let $\bN$ be the connected component of $\bM$ containing $e$.
% Removing $e$ from $\bN$ creates a smaller map with two marked corners. We let $\rho(M,e)=1$ if the edge $e$ respects the fixed orientation of these corners, and $\rho(M,e)=b$ otherwise. By construction this choice respects Axiom (2). Coherence is clear, because the edges $e_i$ and $\tilde e_{i+1}$ in \cref{def:niceMON} have opposite conventional orientations along their faces and therefore are associated with the two values $1$ and $b$ (in some order). The fact that this MON is integral comes from the fact that we have chosen the orientation of faces from a global surface orientation whenever the map is orientable.
% \end{proof}

% %

% \begin{remark}\label{rem:rhoSYM}
% 	We can construct a MON by choosing the two values in case 1(b) of \cref{def:MON} to be both equal to $(1+b)/2$. We obtain a MON denoted by $\rho_{SYM}$, which is not integral, but is coherent. Introducing $\rho_{SYM}$ is not necessary, strictly speaking, to prove the results of this paper, but it played a role in our discovery of the ``heuristic proofs'' given in \cref{subsec:proofCommut1}. Also, since we prove in this paper that the $b$-weighted enumeration of constellations is independent of the choice of a coherent MON (\cref{cor:independentOfMONPatched} page~\pageref{cor:independentOfMONPatched}), it is natural to expect that further works on the subject use the possibility to work with the coherent MON $\rho_{SYM}$, which is simple and canonical. 
% \end{remark}



% \begin{definition}[Right-path](see \cref{fig:rightPathExample})
% 	Let $\bM$ be a $k$-constellation and $c$ be an oriented corner of color $0$ in $\bM$, lying in some face $f$. The sequence of $k$ edges $e_1,e_2,\dots,e_k$ that follow $c$ around $f$, in the orientation of $c$, is called the \emph{right-path} of $c$. Note that the edge $e_i$ has color $\{i-1,i\}$. 
% \end{definition}
% \begin{figure}%
% \center\includegraphics[height=45mm]{rightPathExample.pdf}
% 	\caption{The $3$-constellation of \cref{fig:constellation}. The right-path $(e_1,e_2,e_3)$ of the (oriented) root is shown in (bold) red. In (dotted) green, we show the right-path $(f_1,f_2,f_3)$ of the same corner but with opposite orientation.}
% \label{fig:rightPathExample}
% \end{figure}


% From the local coloring constraints that define $k$-constellations, we clearly have:
% \begin{lemma}
% 	If $\bM$ is a $k$-constellation and $P$ is a right-path in $\bM$, then $\bM\setminus P$ is again a $k$-constellation (with possibly more connected components than $\bM$).
% \end{lemma}

%  We now introduce the \emph{combinatorial decomposition}%
%  %
%  \footnote{Along the years more and more complicated algorithms to
%    decompose rooted maps by edge deletion were found, and the present example may be the most complicated to date. Sometimes the name ``Tutte decomposition'' is used generically for them, here we prefer ``combinatorial decomposition''. 
%    Tutte's original work~\cite{Tutte1962a} dealt with planar maps. Lehman and Walsh~\cite{WalshLehman1972} were the first to write a decomposition for the case of higher genera, and many works followed in the context of enumeration of orientable or non-orientable maps, see e.g.~\cite{BenderCanfield1991, Gao1993}. A combinatorial decomposition  for $k$-constellations in the orientable case appears in Fang's PhD thesis~\cite{Fang:PhD} in the case were one only tracks the face degrees (and not vertices of colour $0$). The decomposition presented in this paper contains these examples as special cases.
%  The equations obtained by analyzing these decompositions are often called Tutte equations, but we prefer to use ``decomposition equations'' below, see \cref{subsec:TutteEq}.
%  In the context of mathematical physics, similar equations are often called Dyson-Schwinger equations or loop equations, see e.g.~\cite{LandoZvonkin2004, Eynard:book}, although not all loop equations directly reflect a combinatorial decomposition, see e.g.~\cite{AlexandrovChapuyEynardHarnad2020}.}%
%  %
% %
% which consists in removing right-paths recursively from a connected $k$-constellation until the whole map has been exhausted.

%  In this definition we assume that some underlying MON $\rho$ has been fixed.
% \begin{definition}[Combinatorial decomposition]\label{def:randomDecomposition}
% 	Let $(\bM,c)$ be a rooted connected $k$-constellation.
% 	The combinatorial decomposition of $(\bM,c)$ is the recursive algorithm defined as follows:
% \begin{itemize}
% 		\item We let  $\mathbf{M}_0=\mathbf{M}$ and we let $c_0:=c$;
% 		\item
% 			For each $i$ from $1$ to $m$, we let $P_i=(e^{(i)}_1,e^{(i)}_2,\dots,e^{(i)}_k)$ be the right-path of the corner $c_{i-1}$ in $\bM_{i-1}$. We let $\mathbf{M}_i:=\mathbf{M}_{i-1}\setminus P_i$. For $i<m$ we let $c_{i}$ be the oriented corner induced by $c_{i-1}$ in the map~$\bM_i$.
% 		 \item We let $L$ be the ordered list of edges 	$$
% 	L=(e^{(1)}_1,e^{(1)}_2, \dots, e^{(1)}_k, e^{(2)}_1, e^{(2)}_2, \dots, e^{(2)}_k,\dots,\dots,e^{(m)}_1, e^{(m)}_2,\dots,e^{(m)}_k).
% 	$$
% We say that the weight $\rho(\bM;L)$ has been \emph{collected} by the algorithm.
% %
% \item If the map $\bM_m$ is empty, then stop. If it is not, let $\bM^{(1)}, \dots, \bM^{(i)}$ be its connected components, each rooted at the first corner of colour $0$ following the last corner from which an edge was deleted in the execution of the algorithm. Let $\tilde{\bM}^{(1)}, \dots, \tilde{\bM}^{(i)}$ be their dual maps. 
% 			\item Run recursively the algorithm on each of the maps $\tilde{\bM}^{(1)}, \dots, \tilde{\bM}^{(i)}$.
% 		\end{itemize}
% \end{definition}
% The need to alternate between primal and dual maps in the decomposition may seem unnatural to the reader. As we will see, it comes from the fact that the differential equations we use to make the connection with Jack polynomials mix two sets of differential variables ($\pp$ and $\qq$, see Proposition~\ref{prop:mixDiffEqs}).
% %
% %
% %
% %
% %
% %
% %
% %
% %
% %
% %
% %
% %
% \begin{definition}
% 		Fix a MON $\rho$, and let  $(\bM,c)$ be a rooted connected $k$-constellation. We define the \emph{weight} $\vec\rho(\bM,c)$ of $(\bM,c)$ as the product of all the weights collected during the combinatorial decomposition of $(\bM,c)$.
% %
% 	\end{definition}

% 	\subsection{$b$-weights and $p_{\cdot}q_{\cdot}u_{\cdot}y_{\cdot}$-markings}

% 	In this paper we will consider generating  functions of constellations, and we will  be able to keep track of many parameters of these combinatorial objects in our formulas. In order to make our discussions as clear and readable as possible, we fix some terminology and notation now.

% 	To a constellation $\bM$ (possibly rooted, or labelled), we will associate several sorts of ``weights'':
% 	\begin{itemize}
% 		\item a \emph{$b$-weight}, which is a quantity in
%                   $\mathbb{Q}[b]$, a priori dependent on the choice of
%                   an underlying MON $\rho$.
% 			Example of $b$-weights
%                   are the quantities $\vec\rho(\bM,c)$ or
% 			$b^{\nu_\rho(\bM,c)}$ defined above. We will restrict the word \emph{weight} to these quantities.
% 	  \item a monomial weight in the variables $p_i,q_i,u_i,y_i$, which serves as a marking keeping track of parameters of the map, such as the face or vertex degrees. To avoid confusion with the $b$-weights we will use the word \emph{marking} instead of \emph{weight} for these quantities.
% 	\end{itemize}

% 	For the rest of this paper we fix indeterminates $b$,
%         $\pp=(p_i)_{i\geq 1}$, $\qq=(q_i)_{i\geq 1}$,
%         $\yy=(y_i)_{i\geq 0}$, $\uu = (u_i)_{i\geq 1}$.
% 	If $\bM$ is a constellation we denote by $cc(\bM)$ its number
%         of connected components, and by $F(\bM)$ the set of its
%         faces. For $i\geq 0$ we denote by $V_i(\bM)$ the set of
%         vertices of color $i$ and by $v_i(\bM)$ its cardinality.
% Recall also that $|\bM|$ is the size, \textit{i.e.} the number of corners of colour $0$, of $\bM$.
% 	\begin{definition}[Markings]
% 		Let $\bM$ be a $k$-constellation. The \emph{marking} of $\bM$ is the monomial
% 		\begin{align}\label{eq:marking}
% 		 \kappa(\bM):=\prod_{f\in F(\bM)} p_{\deg(f)}\prod_{v\in V_0(\bM)} q_{\deg(v)} \prod_{i=1}^k u_i^{v_i(\bM)}.
% 		\end{align}
% 		Let $(\bM,c)$ be a rooted $k$-constellation, and $f_c$ be its root face. The \emph{marking} of $(\bM,c)$ is the monomial
% 		$$
% 		 \vec\kappa(\bM,c):= y_{\deg (f_c)}\prod_{f\in F(\bM)\setminus\{f_c\}} p_{\deg(f)}\prod_{v\in V_0(\bM)} q_{\deg(v)} \prod_{i=1}^k u_i^{v_i(\bM)}
% 		 = \frac{y_{\deg (f_c)}}{p_{\deg (f_c)}}\kappa(\bM).
% 		$$
% 	\end{definition}
% 		In other words, our marking uses variables $p_i$ to record a non-root face of degree $i$, $q_i$ to record a vertex of colour $0$ and degree $i$, variables $u_i$ to record a vertex of color $i$, and $y_i$ to record the fact that the root face has degree $i$. %
% \Gremoved{
% For $m\geq 1$, we also denote by $\vec{H}_\rho^{[m]}$ the contribution to $\vec{H}_\rho$ of maps whose root vertex has degree $m$
% \begin{equation}
%   \label{eq:DefHRhoM}
% 	\vec{H}_\rho^{[m]}(t;\pp,\qq, \yy,u_1,\dots,u_k) := q_m^{-1}\sum_{n\geq 1}\sum_{(\bM,c)\atop \deg v_c=m} t^{n} \vec{\rho}(\bM,c) \vec\kappa(\bM,c),
% \end{equation}
% where the second sum is now taken over rooted connected (unlabelled) $k$-constellations of size~$n$ whose root vertex $v_c$ has degree $m$. Note that we do not count the root vertex in the marking (hence the factor $q_m^{-1}$). By definition one has:
% \begin{align*}
% 	\vec{H}_\rho = \sum_{m\geq 1} q_m \cdot \vec{H}_\rho^{[m]}.
% \end{align*}
% We also consider the variant where the root face is marked with $\pp$-variables, namely
% \begin{align*}
% 	H_\rho^{[m]} =\Theta_Y \vec H_\rho^{[m]}
% 	=q_m^{-1}\sum_{n\geq 1}\sum_{(\bM,c)\atop \deg v_c=m} t^{n} \vec{\rho}(\bM,c) \kappa(\bM),
% \end{align*}
% with
% \begin{align}
%      \Theta_Y &:= \sum_{i\geq 1}p_i\frac{\partial}{\partial y_i}.
% \end{align}
% Finally, we let $\pi$ be the operator that exchanges the sets
%   of variables $\pp \leftrightarrow \qq$ and $u_{i}\leftrightarrow
%   u_{k+1-i}$ for $1 \leq i \leq k$, and we let  
% \begin{align}\label{eq:dualFunction}
% \widetilde{H}_\rho^{[m]}:= \pi H_\rho^{[m]}.
% \end{align}
% Note that, by applying duality, we have
% \begin{align}\label{eq:dualFunction2}
% \widetilde{H}_\rho^{[m]}:= 
% 	p_m^{-1}\sum_{n\geq 1}\sum_{(\bM,c)\atop \deg f_c =m} t^{n} \vec\rho {(\tilde{\bM}, \tilde{c})} \kappa(\bM),
% \end{align}
% where the second sum is now taken over rooted connected (unlabelled) $k$-constellations of size~$n$ whose root \emph{face} has degree $m$. Note that in this sum the $b$-weight is computed  on the dual rooted map $(\tilde{\bM}, \tilde{c})$ of $(\bM,c)$, and that we used that $\kappa(\tilde{\bM}) = \pi \kappa (\bM)$. }


% \Gremoved{
% \begin{theorem}[Decomposition equations]\label{thm:TutteConnected}
% 	Let $\rho$ be any coherent MON.
% Then	
% 	the family of generating series $\vec H_\rho^{[m]}\equiv \vec H_\rho^{[m]}(t;\pp,\qq,\yy,u_1,\dots,u_k)$
% 	satisfies the following set of equations, for $m\geq 1$:
% 	\begin{equation}\label{eq:TutteConnected}
% \vec H_\rho^{[m]} =t^m  \cdot
% 		\Big(Y_+\prod_{l=1}^{k}\big(\Lambda_{Y}+u_l + \sum_{i,j\geq 1} y_{j+i-1}
% 		\widetilde{H}_\rho^{[i]} \frac{\partial}{\partial y_{j-1}}
% 		\big)\Big)^{m}  (y_0).
% \end{equation}
% \end{theorem}


% \begin{corollary}\label{cor:TutteThetaConnected}
% 		Let $\rho$ be any coherent MON.
% Then	
% 	the family of generating series $(H_\rho^{[m]})_{m\geq 1}$ 
% 	is fully characterized by the following set of equations, for $m\geq 1$:
% 	\begin{equation}\label{eq:TutteThetaConnected}
% H_\rho^{[m]}(t;\pp,\qq,u_1,\dots,u_k) =t^m  \cdot
% 		\Theta_Y \Big(Y_+\prod_{l=1}^{k}\big(\Lambda_{Y}+u_l + \sum_{i,j\geq 1} y_{j+i-1}
% 		\widetilde {H}_\rho^{[i]} \frac{\partial}{\partial y_{j-1}}
% 		\big)\Big)^{m}  (y_0),
% \end{equation}
% 	together with~\eqref{eq:dualFunction}.
% \end{corollary}
% \begin{proof}[Proof of the corollary]
% 	Equation~\eqref{eq:TutteConnected} implies \eqref{eq:TutteThetaConnected} by applying the operator $\Theta_Y$ to both sides. It is clear that this set of equations characterizes the functions since coefficients can be computed inductively from the equations, order by order in $t$.
% \end{proof}

% In particular, we observe that
% \begin{corollary}\label{cor:independentOfMONPatched}
% 	The functions $\vec H_\rho$, $H_\rho^{[m]}$, $\widetilde{H}_\rho^{[m]}$ do not depend on the coherent MON $\rho$.
% \end{corollary}

% %
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% %

% 	\subsection{Unconnected functions}

% Let us consider the function $\Theta_Y \vec H_\rho$ of rooted connected constellations.
% We let $F_\rho$ be the function defined by
% \begin{align}\label{eq:newDefFrho}
% (1+b) t \frac{\partial}{\partial t}\ln F_\rho =\Theta_Y \vec H_\rho, 
% \end{align}
% \Gnew{which does not depend on the choice of MON assuming it has some
%   special properties, see~\cref{ChapuyDolega2020}}\Gremoved{and we note that by Corollary~\ref{cor:independentOfMONPatched} it does not depend on the coherent MON $\rho$}.
% Equivalently, let us consider the following antiderivative of $\Theta_Y \vec H_\rho$:
% $$H_\rho :=
% \sum_{n\geq 1} \frac{1}{n} \sum_{(\bM,c)} t^{n} \vec{\rho}(\bM,c) \kappa(\bM),
% $$
% where the second sum is taken over rooted connected (unlabelled) $k$-constellations of size~$n$.
% Then the series $F_\rho$ is defined by 
% $$F_\rho = \exp \frac{1}{1+b} H_\rho.$$

% We now want to give a combinatorial interpretation to coefficients of $F_\rho$. Because each connected rooted constellation has $n! 2^{n-1}$ different labellings, we can also write 
% $$
% H_\rho =
% \sum_{n\geq 1} \frac{1}{n} \sum_{(\bM,c)} \frac{t^{n}}{2^{n-1} n!} \vec{\rho}(\bM,c) \kappa(\bM)
% $$
% where the second sum is now taken over \emph{labelled} and rooted connected  $k$-constellations of size~$n$. This can also be rewritten
% $$
% H_\rho =
% \sum_{n\geq 1} \sum_{\bM} \frac{t^{n}}{2^{n-1} n!} \kappa(\bM) \cdot \mathbf{E}_{c\in \mathbf{M}} [\vec{\rho}(\bM,c)] ,
% $$
% where the second sum is now taken over \emph{labelled} (but no more rooted) connected  $k$-constellations of size~$n$, and where $\mathbf{E}_{c\in \mathbf{M}}$ now denotes expectation with respect to a corner $c$ of colour $0$ chosen uniformly at random among those of $\bM$.

% \begin{definition}
% 	Define the weight  of a labelled connected constellation $\bM$ as $\tilde \rho(\bM):=\mathbf{E}_{c\in \mathbf{M}} [\vec{\rho}(\bM,c)]$. Extend this definition multiplicatively to unconnected constellations.
% \end{definition}

% Because $\tilde{\rho}(\bM)$ is multiplicative on connected components
% by definition, and $\kappa(\bM)$ and  $2^{n|\bM|-cc(\bM)}$ also are,
% the generating function $F_\rho = \exp \frac{1}{1+b} H_\rho$ can be
% directly interpreted as the generating function of unconnected objects with these markings. More precisely:}
% \Gremoved{      
% \begin{remark}
% 	We will prove (Theorem~\ref{thm:mainInSection5}) that $F_\rho$ is in fact equal to the function $\tau^{(k)}_b$ defined in~\eqref{eq:JackIntro}. Thus the last theorem gives an explicit interpretation of the coefficients of $\tau^{(k)}_b$. We will also show (Lemma~\ref{lem:MultiJack}) that $F_\rho=\tau^{(k)}_b$ satisfies the following equation
% 	\begin{align}\label{eq:TutteUnconnected}
% 		m \frac{q_m\partial}{\partial q_m} F_\rho (t;\pp,\qq,u_1,\dots,u_k) = \Theta_Y t^m \cdot q_m \cdot
% 	\big(Y_+\prod_{l=1}^{k} (\Lambda_{Y}+u_l)\big)^{m}  \frac{y_0}{1+b} F_\rho(t;\pp,\qq,u_1,\dots,u_k),
% 	\end{align}
% 	and similarly, in Corollary~\ref{cor:equalInSection5}, we will show that 
% 	$$
% 	 m \frac{\partial }{\partial q_m} H_\rho = H_\rho^{[m]}.
% 	$$
% 	This equation has the following interpretation. Fix a monomial $\mathbf{m}=p_\lambda q_\mu u_1^{v_1}\dots u_k ^{v_k}$ such that the number of parts in $\mu$ equal to $m$ is nonzero, and that there exists at least one connected $k$-constellation with marking $\mathbf{m}$. Then, if $\bM$ denotes a random connected $k$-constellation such that $\kappa(\bM)=\mathbf{m}$, chosen uniformly at random, one has
% 	$$
% 	\mathbf{E}_{\mathbf{M}} 
% 	\mathbf{E}_{c\in \mathbf{M}}  [\vec{\rho}(\bM,c)]
% =
% \mathbf{E}_{\mathbf{M}} 
% 	\mathbf{E}_{c\in \mathbf{M}}  [\vec{\rho}(\bM,c) | \deg v_c = m].
% 	$$
% %
% 	This ``symmetry'' is not directly apparent on the combinatorial model.
% \end{remark}}


% 	\begin{remark}[Other relations and deformed Virasoro constraints]\label{rem:VirasoroPatched}
% 	In the special cases of $b\in \{0,1\}$ we can obtain other
%         equations, which do not involve differential operators
% 	with respect to the variables $(q_i)_{i \geq 1}$. Indeed, it is easy to see that for $b\in\{0,1\}$, the coefficient of $y_\ell$ in $\vec F_\rho (\mathbf{p},\mathbf{q},\mathbf{y}; u_1,\dots,u_k)$ is equal to $\frac{\ell\partial}{\partial p_\ell} F_\rho (\mathbf{p},\mathbf{q}; u_1,\dots,u_k)$. Therefore for $b\in \{0,1\}$ we get
% 	\begin{align}\label{eq:Virasoro3}
% 	\frac{\ell\partial}{\partial p_\ell}
%           F_\rho =
%           [y_\ell] \sum_{m \geq 1}t^m \cdot q_m \cdot
% 	\big(Y_+\prod_{l=1}^{k} (\Lambda_{Y}+u_l)\big)^{m}  \frac{y_0}{1+b} F_\rho, 
% 	\end{align}
% 	where $[y_\ell]$ is coefficient extraction. In the special case $q_i=\delta_{1,i}$ and $k=2$, this equation is precisely the $\ell$-th Virasoro constraint for dessins d'enfants for $b=0$ (or its non-orientable generalisation for $b=1$), see~\cite{KazarianZograf2015}. In the general case, possible $b$-analogues of the differential equations~\eqref{eq:Virasoro3} and their links with the Virasoro algebra and its central extensions deserve further interest and will be studied in further works.
% \end{remark}




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