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\title[Cyclic Sieving for one-face maps]{Counting unicellular maps under cyclic symmetries}

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\author[Theo Douvropoulos]{Theo Douvropoulos\thanks{\href{mailto:tdouvropoulos@brandeis.edu}{tdouvropoulos@brandeis.edu}}\addressmark{1}}

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\address{\addressmark{1}Department of Mathematics, Brandeis University, Waltham USA}

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\abstract{
We count unicellular maps (matchings of the edges of a $2n$-gon) of arbitrary genus with respect to the $2n$-rotation symmetries of the polygon. An associated generating function that keeps track of the number of symmetric vertices of the resulting map generalizes the celebrated Harer-Zagier formula.

The answer to this enumerative question is not in the form of the usual cyclic sieving phenomenon (CSP), but does recover in the leading terms (genus-$0$ maps) a well known CSP for the Catalan numbers. The approach is representation theoretic, in that we relate symmetric unicellular maps with factorizations of the Coxeter element in a reflection group of type $G(m,1,n)$.}


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\keywords{Harer-Zagier formula, unicellular maps, reflection groups, cyclic sieving}

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\begin{document}

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\section{Introduction}

Unicellular maps are the $3$-constellations of the form $\sigma\alpha c=\bm{1}$ where $\sigma,\alpha,c\in S_{2n}$, $\sigma$ is a fixed point free involution, $\alpha$ an arbitrary permutation, and $c:=(1,2,\ldots, 2n)$ the long cycle. This corresponds to gluing the edges of a $2n$-gon (the gluing pattern is encoded in the involution $\sigma$).

The \defn{genus} $g$ of a unicellular map is given as $2g=n+1-\op{cyc}(\alpha)$ (see also \cite[p.~23]{LZ_book}). The Harer-Zagier numbers $\varepsilon_g(n)$ count the unicellular maps with $n$ edges and genus $g$ and they have a very nice generating function formula:
\begin{equation}
\dfrac{1}{(2n-1)!!}\sum_g\varepsilon_g(n)\Phi_{n+1-2g}(X)=\dfrac{(1+X)^n}{(1-X)^{n+2}},\label{eq:e_g-thm}
\end{equation}
where the polynomials $\Phi_n(X)$ are essentially the Eulerian polynomials; they are defined as follows:
\begin{equation}
\Phi_n(X)=\dfrac{\sum_{k=0}^{n-1} A(n,k)X^k}{(1-X)^{n+1}}\quad\quad \text{or equivalently}\quad\quad\Phi_n(X)=\sum_{k=0}^{\infty}(k+1)^nX^k,\label{eq:defn:eul_pol_type_A}
\end{equation}
where $A(n,k)$ is an Eulerian number (i.e., the number of permutations in $S_n$ with $k$ descents).

\begin{definition}[Rotation of constellations]\label{defn:rotation_of_constellations}
There is a natural cyclic action $\Psi$ of order $2n$ on unicellular maps that corresponds to rotating the polygon. In terms of the constellation, the action is given as 
\[
\Psi\big[(\sigma,\alpha,c)\big]=(c^{-1}\sigma c,c^{-1}\alpha c,c).
\]
\end{definition}

To count symmetric $3$-constellations, we essentially need to count the factorizations $\sigma\alpha c=\bm{1}$ that are fixed by \emph{simultaneous} conjugation by some power $c^N$ of $c$. Equivalently this means counting factorizations $\sigma\alpha c=\bm{1}$ in $S_{2n}$ all of whose factors $\sigma,\alpha,c$ also belong to the centralizer $Z_{S_{2n}}(c^N)$. Now, the centralizer  $Z_{S_{2n}}(c^N)$ is just the reflection group\footnote{Note that the reflections of $G(m,1,N)$ do not come from transpositions of $S_{2n}$; they come from some elements of type $\big(2^m,1^{2n-2m}\big)$ (the transposition-like ones) and some other ones --multiple cycle types-- for the diagonal-like reflections; see Example~\ref{ex:d_m(sigma)} and Remark~\ref{rem:num_diag_posns}.} $G(m,1,2n/m)$ where $m$ is the order of $c^N$ (i.e. $m=2n/\gcd(2n,N)$). From now on, we will always assume that $N$ divides $2n$ and we will always have $mN=2n$. 

That is, the problem of counting $3$-constellations fixed under $\Psi^r$ is equivalent to counting factorizations $\sigma\alpha c=\bm{1}$ in $G(m,1,N)=Z_{S_{2n}}(c^r)$ where $\sigma$ belongs to the conjugacy classes of $G(m,1,N)\leq S_{2n}$ into which the class $S_{2n}$ of fixed point free involutions has been decomposed.
This problem turns out to be particularly easy because $c=(1,2,\ldots,2n)$ is a Coxeter element \emph{also} in $G(m,1,N)$.

There is however a caveat: In the Harer-Zagier formula \eqref{eq:e_g-thm}, the genus is directly related to the reflection length of $\alpha$ so we can keep track of it with representation theory. Here, the genus of a symmetric constellation is related to the length of $\alpha$ as an element in $S_{2n}$ but this is not the same as (or a multiple of) its length as an element in $G(m,1,N)$. There are two natural approaches here; track the length as an element in $G(m,1,N)$ and interpret it as a \emph{combinatorial} statistic on the map (this succeeds with Theorem~\ref{thm:main}) or define a new length function to track the genus and attempt to express it representation-theoretically (a first attempt here fails; we discuss it in Section~\ref{ssec:maps:genus_version}).

We present the first approach in Section~\ref{ssec:maps:length_version}, where we interpret the usual length function for $G(m,1,N)$ as a combinatorial (but sadly not topological) statistic on the maps. Then, Zagier's proof \cite{zagier} of the Harer-Zagier formula \eqref{eq:e_g-thm} generalizes essentially out of the box; we have existing theorems that replace all the ingredients of the proof and we prove Theorem~\ref{thm:main} which is a direct generalization of \eqref{eq:e_g-thm}. 

In Section~\ref{ssec:maps:genus_version} we define a new length function for $G(m,1,N)$ that corresponds to the topological genus; it is a class invariant and is even somewhat compatible with a factorization in the group algebra of $G(m,1,N)$ which gives us some control over the formulas coming from the Frobenius lemma. It is not clear though what the analog of the Eulerian polynomials $\Phi_n(X)$ of \eqref{eq:defn:eul_pol_type_A} should be in this case (nor whether such an analog should a priori exist!). 


We first start with a mini review of Zagier's proof of the Harer-Zagier formula \eqref{eq:e_g-thm} to set up a pattern of how the proofs would go in these two approaches.

\section{Main ingredients of Zagier's proof of the Harer-Zagier formula}
\label{ssec:maps_zagier_review}


We give in this section the main ingredients in Zagier's proof (or a re-imagining of Zagier's proof relying more on Jucys-Murphy elements). We will generalize each of them in the next section.

The first is a direct application of the Frobenius lemma from representation theory (recall: $n+1-2g=\op{cyc}(\alpha)=2n-\ell_R(\alpha)$).

\begin{equation}
\sum_g\varepsilon_g(n)X^{n+1-2g}=\dfrac{(2n-1)!!}{(2n)!}\cdot\sum_{\chi\in\widehat{S_{2n}}}\chi(\sigma)\chi(c)\cdot \widetilde{\chi}\left( \sum_{w\in S_{2n}}w X^{2n-\ell_R(w)} \right),\label{eq:ingred_A}\tag{A} 
\end{equation}
where $\sigma$ is any fixed point free involution in $S_{2n}$, $c$ any fixed long cycle, and $\widetilde{\chi}$ denotes the normalized character $\chi$ (i.e. $\widetilde{\chi}(a):=\chi(a)/\chi(1)$ for an element $a\in \CC[S_{2n}]$).

The second ingredient is a well known factorization in the symmetric group algebra:

\begin{equation}
\sum_{w\in S_{2n}}wX^{2n-\ell_R(w)}=X(X+J_2)(X+J_3)\cdots (X+J_{2n}),\label{eq:ingred_B1}\tag{B1}    
\end{equation}
where $J_i:=(1i)+\cdots+(i-1 i)$ is the $i$-th Jucys-Murphy element. As an application of this factorization we know for instance that the normalized traces appearing in \eqref{eq:ingred_A} are just binomials:

\begin{equation}
\dfrac{1}{(2n)!}\cdot \widetilde{\chi_k}\left( \sum_{w\in S_{2n}}w X^{2n-\ell_R(w)} \right)= \binom{X+2n-1-k}{2n},\label{eq:ingred_B2}\tag{B2}
\end{equation}
where $\chi_k$ is the $k$-th exterior power of the reflection representation of $S_{2n}$ (it is a direct application of the Murnaghan-Nakayama rule that only these irreducible characters are non-zero on the long cycle $c$). 

The third ingredient is that the eulerian polynomials of \eqref{eq:defn:eul_pol_type_A} give exactly the change-of-basis between the binomials in $X$ that appear above and the monomials $X^n$:
\begin{equation}
\sum_{k=1}^n \varepsilon_k X^k=\sum_{k=1}^nb_k\binom{X+n-k}{k}\quad\text{if and only if}\quad  (1-X)^{n+1}\sum_{k=1}^n \varepsilon_k\Phi_k(X)=\sum_{k=1}^nb_kX^{k-1}.\label{eq:ingred_C}\tag{C}   
\end{equation}
This has many proofs but it is very conveniently stated in  Theorems~2.5 and~2.10 in \cite{miller}.


The final ingredient is the usual relation (as in \cite{CD_MT_thms} or \cite{galashin2022rational}) between the characters $\chi$ such that $\chi(c)\neq 0$, the Coxeter numbers $c_{\chi}=k(2n)$, the exterior powers $\chi_k$, and hence the matrix of an element in the reflection representation of $S_{2n}$:

\begin{equation}
\sum_{\chi\in\widehat{S_{2n}}}\chi(\sigma)\chi(c)X^{\frac{c_{\chi}}{2n}}=\sum_{k=0}^{2n-1}\chi_k(\sigma)(-1)^kX^k=\dfrac{\bm{\mathfrak{p}}(\sigma;X)}{1-X}\label{eq:ingred_D}\tag{D},    
\end{equation}
where $\bm{\mathfrak{p}}(\sigma;X)$ is the characteristic polynomial of $\sigma$ in the \emph{standard} $(2n)$-dimensional representation of $S_{2n}$. Together \eqref{eq:ingred_A},\eqref{eq:ingred_B2},\eqref{eq:ingred_C},\eqref{eq:ingred_D} give us the Harer-Zagier formula \eqref{eq:e_g-thm} because $\bm{\mathfrak{p}}(\sigma;X)=(1-X^2)^n$.


\section{Counting symmetric maps keeping track of $G(m,1,N)$-length}
\label{ssec:maps:length_version}

In this section we generalize the Harer-Zagier formula \eqref{eq:e_g-thm} in a way that has all of the ingredients of Zagier's proof from the previous section working out of the box. To have a \emph{meaningful interpretation} of the theorem however we will give first a combinatorial interpretation of the $G(m,1,N)$-length.

Recall that the for the $3$-constellation $\bm\pi=(\sigma,\alpha,c)$ the number $\op{cyc}(\alpha)$ of cycles of $\alpha$ equals the number of vertices $\bm v(\bm\pi)$ of the combinatorial map $\bm\pi$ and also that 
\[
n+1-2g=2n-\ell_{S_{2n}}(\alpha)=\op{cyc}(\alpha)=\bm v(\bm\pi).
\]
So, then the Harer-Zagier formula \eqref{eq:e_g-thm} can be rephrased as 

\begin{equation}
\dfrac{1}{(2n-1)!!}\sum_{v}\mathcal{E}_v(n)\Phi_{v}(X)=\dfrac{(1+X)^n}{(1-X)^{n+2}},\label{eq:harer_zagier:v(pi)}
\end{equation}
where $\mathcal{E}_v(n)=\varepsilon_{(n+1-2v)/2}(n)$ counts the number of unicellular maps $\bm\pi$ witn $n$ edges and $v$ vertices.

Now, we will give an explicit definition of unicellular maps with rotational symmetry at least $m$:
\begin{definition}
Let $n,m,N$ be positive integers such that $mN=2n$. We denote by $C^m(N)$ the number of $3$-constellations $\bm\pi=(\sigma,\alpha,c)$ with factors from $S_{2n}$ that are fixed by the operation $\Psi^N$ (i.e. have symmetry at least $m$):
\[
C^m(N)=\Big\{(\sigma,\alpha)\in S_{2n}^2\ |\  \sigma\alpha c=\sigma^2=\bm{1},\ \ell_{S_{2n}}(\sigma)=n,\ c^{-N}\sigma c^N=\sigma,\ c^{-N}\alpha c=\alpha\Big\}.
\]
\end{definition}

As we mentioned earlier, we can enumerate $C^m(N)$ by counting certain factorizations in $G(m,1,N)$. The factors $\sigma, \alpha, c$ are still elements of $G(m,1,N)$ and $c$ is its Coxeter element, but the class in $S_{2n}$ of fixed point free involutions $\sigma$ breaks into multiple conjugacy classes (see Remark~\ref{rem:num_diag_posns}) and the new length $\ell_{G(m,1,N)}(\alpha)$ is not a function of $g$ (or equivalently $\bm v(\bm\pi)$). For this reason we define these two statistics:

\begin{definition}\label{defn:dm(sigma)}
Let $n,m,N$ be positive integers such that $mN=2n$ and let $\sigma$ be a fixed point free involution of $S_{2n}$ such that $c^{-N}\sigma c^N=\sigma$. We write ${\color{blue}\bm{d_m(\sigma)}}$ for the number of $\Psi^N$-orbits of \emph{centrally symmetric} $2$-cycles of $\sigma$.  (A \defn{centrally symmetric} transposition is one of the form $(i,n+i)$.)
\end{definition}



\begin{example}\label{ex:d_m(sigma)}
Consider the involution $\sigma:=(1,12)(2,8)(3,4)(5,11)(6,7)(9,10)$ of $S_{12}$. There are two \emph{centrally symmetric} $2$-cycles: $(2,8)$ and $(5,11)$. The involution is symmetric both under $\Psi^3$ (conjugation by $c^3$ or rotation of order $m=4$) and under $\Psi^6$ (conjugation by $c^6$ or rotation of order $m=2$). But the cycles $(2,8)$ and $(5,11)$ form two orbits under $\Psi^6$ but only one orbit under $\Psi^3$. See Figure~\ref{fig:invol_d_values}.
\begin{figure}
    \centering
    \includegraphics[scale=0.5]{invol.png}
    \caption{For the involution $\sigma$ of the figure, we have $d_4(\sigma)=1$ but $d_2(\sigma)=2$.}
    \label{fig:invol_d_values}
\end{figure}
\end{example}
\begin{remark}[$d_m$ detects conjugacy class in $G(m,1,N)$]\label{rem:num_diag_posns}
The point of this definition is that it detects the conjugacy class of the involution $\sigma$ as an element of $G(m,1,N)$. \textbf{The number $d_m(\sigma)$ counts on how many indices from $1$ to $N$ the involution $\sigma$ acts diagonally-like (maps $i$ to $-i$).} For Example~\ref{ex:d_m(sigma)} above, the centralizer $Z_{S_{2n}}(c^3)$ is isomorphic to the group $G(4,1,3)$ where the coordinates of the ($3$-dimensional ambient space) correspond to the three sets $\{1,4,7,10\}$, $\{2,5,8,11\}$, $\{3,6,9,12\}$. In this case $\sigma$ becomes $(1,3^{-i})(2,\bar{2})$: the first 2-cycle $(1,3^{-i})$ corresponds to the part $(1,12)(4,3)(7,6)(10,9)$ and the $2$-cycle $(2,\bar{2})$ corresponds to the part $(2,8)(5,11)$. Then, the $d_4$ value here is $d_4(\sigma)=1$ because the involution $\sigma$ has a single \emph{diagonal} position in $G(4,1,3)$. 

Similarly the centralizer $Z_{S_{2n}}(c^6)$ is isomorphic to the group $G(2,1,6)$ with coordinates corresponding to the three sets $\{1,7\}$, $\{2,8\}$, $\{3,9\}$, $\{4,10\}$, $\{5,11\}$, $\{6,12\}$. In this case $\sigma$ becomes $(1,\bar{6})(2,\bar{2})(3,4)(5,\bar{5})$ and thus $d_2(\sigma)=2$ since $\sigma$ has two \emph{diagonal} positions in $G(2,1,6)$.
\end{remark}

We need to also replace the quantity $\bm v(\bm\pi)$ (the number of vertices of the map $\bm\pi$) with a new object that keeps track of the rotational symmetry of the vertices of the polygon that were identified into vertices of the map. 

\begin{definition}\label{defn:vfree}
For any $3$-constellation $\bm\pi=(\sigma,\alpha,c)$ in $S_{2n}$, and any numbers $m,N$ such that $mN=2n$, we define  $\vfree^m(\bm\pi)$ to be the number of vertices of $\bm\pi$ (equivalently cycles of $\alpha$) that are not fixed by \emph{any} power of $\Psi^N$ (apart from of course $\Psi^{Nm}=\op{Id}$).
\end{definition}

\begin{proposition}\label{prop:length_vs_vfree(pi)}
If a $3$-constellation $\bm\pi=(\sigma,\alpha,c)$ in $S_{2n}$ is fixed under some power $\Psi^N$, then if $m$ is such that $mN=2n$, 
\[
\ell_{G(m,1,N)}(\alpha)=\dfrac{2n-\vfree^m(\bm\pi)}{m}.
\]
\end{proposition}


Before finally stating the main theorem of this section, we need to define the generalizations of the polynomials $\Phi_n(X)$ of \eqref{eq:defn:eul_pol_type_A}. We will be using a well known generalization of Eulerian polynomials for $G(m,1,N)$ that encodes the notion of descent due to Steingr{\'i}msson \cite{einar}.

\begin{definition}\label{defn:Phi_pol:(m,1,n)}
For any two positive integers $m,N$ we define the polynomials
\[
\Phi_{m,N}(X)=\dfrac{\sum_{k=0}^NA(m,N,k)X^k}{(1-X)^{N+1}}\quad \text{or equivalently}\quad \Phi_{m,N}(X)=\sum_{k=0}^{\infty}(mk+1)^NX^k,
\]
where $A(m,N,k)$ is the number of elements in $G(m,1,N)$ with $k$ descents, see \cite[Thm.~17]{einar}.
\end{definition}


With these interpretations, we are ready to state and give a (sketch of the) proof of the following generalization of the Harer-Zagier theorem \eqref{eq:e_g-thm} that counts maps that remain invariant under a given rotation of the initial polygon.

\begin{theorem}\label{thm:main}
For any $n,m,N,k\in \ZZ_{>0}$ such that $2n=mN$, the numbers $\mathcal{E}_{k,v}(m,N)$ of $3$-constellations $\bm\pi=(\sigma,\alpha,c)$ in $S_{2n}$ with $d_m(\sigma)=k$ and $\vfree^m(\bm\pi)=mv$ (see Defn.~\ref{defn:dm(sigma)} and Defn.~\ref{defn:vfree}) such that $\Psi^N(\bm\pi)=\bm\pi$ (see Defn.~\ref{defn:rotation_of_constellations}) can be calculated via:
\[
\dfrac{1}{\binom{N}{k}\cdot (N-k-1)!!\cdot m^{\frac{N-k}{2}}}\sum_{v}\mathcal{E}_{k,v}(m,N)\cdot \Phi_{m,v}(X)=\dfrac{1}{1-X}\cdot \left(\dfrac{1+X}{1-X}\right)^{\frac{N-k}{2}},
\]
where the polynomials $\Phi_{m,v}(X)$ are as in Defn.~\ref{defn:Phi_pol:(m,1,n)}.
\end{theorem}


\begin{proof}[Sketch]
All the ingredients \eqref{eq:ingred_A},\eqref{eq:ingred_B2},\eqref{eq:ingred_C},\eqref{eq:ingred_D} are readily available. \eqref{eq:ingred_A} is just the Frobenius lemma. For \eqref{eq:ingred_B2} see \cite[Prop.~3.2]{miller} but it can also be shown using the following version of \eqref{eq:ingred_B1}:
\[
\sum_{w\in G(m,1,N)}wX^{N-\ell_{{G(m,1,N)}}(w)}=(X+J_1)(X+J_2)\cdots (X+J_n),
\]
where $J_i=(1,i)+\cdots +(i-1,i^{\bar{\xi}})+(i,i^{\xi})+\ldots+(i,i^{\bar{\xi}})$ are a version of the JM elements. The approach of \cite[Prop.~4.8]{Ram_seminormal} expresses the character values on these generalized Jucys-Murphy elements as certain content calculations, see also \cite[Section~4.2]{fusion} or \cite{zinn-justin}.

The change-of-basis \eqref{eq:ingred_C} is in Theorems~3.17 and~3.18 of \cite{miller}. The final ingredient \eqref{eq:ingred_D} comes from our previous work, joint with Chapuy, in \cite[Section~9.5.2]{CD_MT_thms} where we prove an \emph{equality} in $G(m,1,N)$ between $\sum\chi(c)\chi$ and a virtual character that involves the exterior powers of certain $N$-dimensional representations that are analogues of the standard representation of $S_N$.
\end{proof}


\begin{remark}
The genus $0$ case, or equivalently $\op{cyc}(\alpha)=n+1$, appears only if $\vfree(\bm\pi)=n+1$ (no symmetry) or $\vfree(\bm\pi)=n$ ($\bm \pi$ has some symmetry). In this way, Theorem~\ref{thm:main} recovers the known symmetry count in the form of a CSP \cite[\S7]{Reiner_CSP} in the genus-$0$ case (there the matchings must be non-crossing and determine a (different) noncrossing partition of the odd vertices $1,3,\ldots 2n-1$; it is this object that is studied in \cite{Reiner_CSP}).
\end{remark}


\begin{remark}
The approach described above can give a complete version of Zagier's main theorem from \cite{zagier} (i.e. for any conjugacy class of $G(m,1,N)$ not just the fixed point free involutions).
\end{remark}



\begin{remark}
The approach of this section can be generalized to other factorization counting questions, where conjugation by the long cycle is a natural symmetry. For instance, in works of Goupil-Schaeffer \cite{goupil-schaeffer} and Bernardi-Morales \cite{bernardi-morales}, one could try to count symmetric factorizations by transfering the question to some $G(m,1,n)$ group. Factorizations of the Coxeter element $c\in G(m,1,n)$ have been extensively studied by Lewis-Morales, where the authors also observe \cite[\S8.2]{lewis-morales} in their setting that the $G(m,1,n)$-factorizations cannot keep track of the topological genus of a corresponding map.
\end{remark}


\section{Counting symmetric maps keeping track of genus} 
\label{ssec:maps:genus_version}

The main disadvantage of Theorem~\ref{thm:main} is that the enumeration cannot keep track of the topological genus of the map $\bm\pi$. We discuss in this section a partial attempt to resolve this. We define a new length function in $G(m,1,N)$ given as 
\[
\defn{\ell_{sp}(w)}:=\ell_{S_{mN}}(w),
\]
that is the \defn{symmetric length} of $w\in G(m,1,N)$ is its length as an element of $S_{mN}$. Notice that this is a class function since if two elements are conjugate in $G(m,1,N)$ then they are also conjugate in $S_{mN}$ hence have the same length.



Then, a generalization of \eqref{eq:e_g-thm} in the spirit of Theorem~\ref{thm:main} but using $\ell_{sp}(w)$ instead of $\ell_{G(m,1,N)}(w)$ would rely on understanding
\[
\sum_{\chi\in\widehat{G(m,1,N)}}\chi(\sigma_k)\chi(c)\cdot\sum_{w\in G(m,1,N)}\dfrac{\chi(w)}{\chi(1)}X^{\ell_{sp}(w)},
\]
where $\sigma_k$ is any involution with $d_m(\sigma_k)=k$.

It is not difficult to see that there is a factorization 
\begin{align*}
\sum_{w\in G(m,1,N)}wX^{\ell_{sp}(w)}=&\big[\bm{1}+X^{m-1}(11^{\xi})+\ldots+X^{m-1}(11^{\bar{\xi}})\big]\times \\
&\times \big[\bm{1}+X^m(12)+\ldots+X^m(12^{\xi})+\ldots+X^{m-1}(22^{\bar{\xi}})\big]\cdots
\end{align*}
where each reflection $\tau$ contributes the term $X^{\ell_{sp}}(\tau)$.

This factorization might be seen as an analogue of $(B1)$ and we can certainly calculate the corresponding traces for irreducible characters (either manually or by the techniques of \cite[Lemma~3.7]{wang}, or even by following Jucys original argument  \cite[Section~4]{jucys} and relying on existing determinations of the eigenvalues of these generalized Jucys-Murphy elements on eigenvectors indexed by tuples of Young tableau as for instance \cite{Ram_seminormal}). 

However, we have no analogue of $(B2)$: Even though Sage experiments suggest that we always have nice formulas for $\displaystyle \sum_{w\in G(m,1,N)}\widetilde{\chi}(w)X^{\ell_{sp}}(w)$, it is not clear that there exists a change of basis analogous to \eqref{eq:ingred_C} (or even that one \emph{might exist}: we need to transform more than $n$ polynomials; the corresponding polynomials with $X^{\ell_{G(m,1,N)}}$ depend only on the Coxeter number of $\chi$ when $\chi(c)\neq 0$ but with $\ell_{sp}$ this is no longer true.

\acknowledgements{I would like to thank Vic Reiner who, back in 2014 when teaching a topics course on \cite{LZ_book}, had asked for a cyclic sieving phenomenon on unicellular maps of a fixed genus, generalizing the $q$-Catalan numbers. I would also like to thank Guillaume Chapuy for some very helpful earlier advice on this problem.}

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