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

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%% define your title in the usual way
\title[A Catalanimal formula for Macdonald polynomials]{A Catalanimal formula for Macdonald polynomials}

%% define your authors in the usual way
%% use \addressmark{1}, \addressmark{2} etc for the institutions, and use \thanks{} for contact details
% \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.}\addressmark{1}, \and you\addressmark{2}}

% %% then use \addressmark to match authors to institutions here
% \address{\addressmark{1}Department of This, University of That, There \\ \addressmark{2}School of Something, University of Something Else, Somewhere}

   \author{
   J. Blasiak\thanks{\href{mailto:jblasiak@gmail.com}{jblasiak@gmail.com}. Supported by NSF Grant DMS-1855784}\addressmark{1}, 
   M. Haiman\thanks{\href{mailto:mhaiman@math.berkeley.edu}{mhaiman@math.berkeley.edu}.}\addressmark{2}, 
   J. Morse\thanks{\href{mailto:morsej@virginia.edu}{morsej@virginia.edu}. Supported by NSF Grant DMS-1855804 and Simons Foundation-821999.}\addressmark{3}, 
   A. Pun\thanks{\href{mailto:ayp6e@virginia.edu}{ayp6e@virginia.edu}. Supported by NSF Grant DMS-1855804.}\addressmark{3}, \and 
   G. H. Seelinger\thanks{\href{mailto:ghseeli@umich.edu}{ghseeli@umich.edu}. Supported by NSF Grant DMS-1855804.}\addressmark{4}
   }

\address{
\addressmark{1}Department of Mathematics, Drexel University, Philadelphia, PA\\
\addressmark{2}Department of Mathematics, University of California,Berkeley, CA\\
\addressmark{3}Department of Mathematics, University of Virginia, Charlottesville, VA\\
\addressmark{4}Department of Mathematics, University of Michigan, Ann Arbor, MI}

   % \thanks{Authors were supported by NSF Grants DMS-1855784 (J.~B.)
   %  and DMS-1855804 (J.~M., A.~P. and G.~S.) and Simons Foundation-821999 (J.~M.).}

%% put the date of submission here
\received{\today}

%% leave this blank until submitting a revised version
%\revised{}

%% put your English abstract here, or comment this out if you don't have one yet
%% please don't use custom commands in your abstract / resume, as these will be displayed online
%% likewise for citations -- please don't use \cite, and instead write out your citation as something like (author year)
\abstract{Catalanimals are rational functions encoding virtual \(GL_l\)
  character series. When truncated to the polynomial characters, they
  have been shown to recover many important symmetric function
  quantities with coefficients in \(\QQ(q,t)\) that arise from the
  study of diagonal harmonics, including \(\nabla^m
  e_k\) and, more generally, \(\nabla^m s_\lambda\). Providing
  Catalanimal representatives of these quantities gave the necessary
  tools to show \(\nabla^m s_\lambda\) is essentially a positive \(q,t\)-weighted sum of distinguished LLT
  polynomials, thereby resolving the Loehr-Warrington
  conjecture. Missing from this story was a Catalanimal description of
  the modified Macdonald polynomials \(\Htild_\mu\), which are
  intimately linked to the \(\nabla\) operator. In
  this abstract, we give a Catalanimal style expression for the modified
Macdonald polynomials and provide a positivity conjecture on the
entire \(GL_l\) character series.}

%% put your French abstract here, or comment this out if you don't have one
%\resume{}

%% put your keywords here, or comment this out if you don't have them yet
\keywords{Symmetric functions, Macdonald polynomials, Catalanimals}

%% you can include your bibliography however you want, but using an external .bib file is STRONGLY RECOMMENDED and will make the editor's life much easier
%% regardless of how you do it, please use numerical citations; i.e., [xx, yy] in the text

%% this sample uses biblatex, which (among other things) takes care of URLs in a more flexible way than bibtex
%% but you can use bibtex if you want
\usepackage[backend=bibtex]{biblatex}
\addbibresource{references}
%\addbibresource{sample.bib}
%% note the \printbibliography command at the end of the file which goes with these biblatex commands

\begin{document}

\maketitle

\section{Introduction}
Macdonald polynomials are a basis of symmetric functions with
coefficients in \(\QQ(q,t)\) exhibiting deep connections to representation
theory and algebraic geometry. In particular, specific specializations
of the \(q,t\) parameters recover various widely-studied bases of
symmetric functions, such as Hall-Littlewood polynomials, Jack
polynomials, \(q\)-Whittaker functions, and Schur functions, among
others.



Since their introduction, the Macdonald Positivity Conjecture was
central to the study of Macdonald polynomials. Precisely, the Macdonald
Positivity Conjecture was that the Schur expansion coefficients of the
modified Macdonald polynomials \(\Htild_\mu(X;q,t)\) lie in \(\ZZ_{\geq
0}[q,t]\). In~\cite{GarsiaHaiman93}, it was shown that the Macdonald
polynomials are doubly graded
characters of certain symmetric group modules, known as Garsia-Haiman
modules. In~\cite{Haiman01}, the Macdonald Positivity Conjecture was
proved by resolving the \(n!\)-Conjecture for the Garsia-Haiman modules.

Out of this study arose the space of diagonal harmonics, a bigraded
\(S_k\)-module containing all the Garsia-Haiman modules associated to
partitions of \(k\) as submodules. It was conjectured in~\cite{GaHa96}
and proven in~\cite{Haiman02} that the doubly graded character of the
diagonal harmonics can be succinctly described as \(\nabla e_k\) for
\(e_k\) the \(k\)th elementary symmetric function and \(\nabla\) an
operator on symmetric functions with eigenfunctions
given by the Macdonald polynomials. Precisely,
% In the study of Macdonald
% polynomials,~\cite{BeGaHaTe99} described an operator \(\nabla\)
% on symmetric functions, with eigenfunctions given by the modified
% Macdonald polynomials; precisely,
\begin{equation}
  \label{eq:9}
  \nabla \Htild_\mu = t^{n(\mu)}q^{n(\mu^*)} \Htild_\mu \text{ where
  }n(\mu)= \sum_i (i-1)\mu_i \,.
\end{equation}

Haglund et.\ al~\cite{HaHaLoReUl05} conjectured a combinatorial identity expressing
\(\nabla e_k\) as a
\(q,t\)-weighted sum of LLT polynomials, which are \(q\)-analogues of
skew Schur functions related to Kazhdan-Lusztig polynomials and Fock
space representations of \(\Ucal_q(\hat{\sl}_n)\).
This identity, now known as the Shuffle Theorem, was proven in~\cite{CarlMell18}.
After being
conjectured in 2005, this formula was generalized in many
directions, including the Rational Shuffle Theorem
(conjectured in~\cite{BeGaSeXi16}, proven in~\cite{Mellit21}), the Compositional Shuffle Theorem
(conjectured in~\cite{HagMoZa12}, proven in~\cite{CarlMell18}), the
Delta Theorem (conjectured in~\cite{HagRemWil18},
proven in~\cite{DaddMell22}), and the Loehr-Warrington Conjecture (\cite{LoehWarr08}) for \(\nabla
s_\mu\), where \(s_\mu\) is the Schur function indexed by any partition \(\mu\).

In several previous works, we provide a general class of functions
called \emph{Catalanimals}, special cases of which give quantities
like \(\nabla^m s_\mu\) and expressions in the Rational
Shuffle Theorem. Catalanimals are defined with \(3\) ``root
ideals'' and a weight, denoted \(H(R_q, R_t, R_{qt}, \gamma)\).
Catalanimals are properly thought of as infinite series of
virtual \(GL_l\) characters, i.e.\ linear combinations of terms
\(\chi_\lambda\) for \(\lambda \in \ZZ^l\) satisfying \(\lambda_1
\geq \cdots \geq \lambda_l\) for \(\chi_\lambda\) an irreducible
\(GL_l\)-character. To recover symmetric functions, we define
\(\pol_X\) to be the truncation of
this series to terms \(\chi_\lambda\) satisfying \(\lambda_l \geq
0\), called polynomial terms. This produces an element of \(\Lambda\) via
the identification \(\chi_\lambda \leftrightarrow s_\lambda\).

Using Catalanimals, we gave an alternative proof of the Rational Shuffle
Theorem and provided a generalization whose combinatorics are
controlled by Dyck paths below a line of irrational slope
in~\cite{BHMPS-paths}. In~\cite{BHMPS-delta}, we provide a proof of the Extended
Delta Conjecture by expressing it as a simple sum of Catalanimals. In~\cite{BHMPS-lw}, we used the Catalanimals
introduced in~\cite{BHMPS-llt} to resolve and generalize the
Loehr-Warrington Conjecture for \(\nabla^m s_\lambda\). Missing from
these works was any treatment of Macdonald polynomials, the foundation
upon which these shuffle theorems lie. In this extended abstract, we
now fill in this missing piece by providing a Catalanimal style formula for a
modified Macdonald polynomial. Our Catalanimal-style formula
reads
\begin{equation}
  \label{eq:macanimal series}
 \hspace{-0.02175in}\HS_{\mu}(\zz;q,t) = \sigmabold \Bigg(
\frac{
\prod_{\alpha_{ij} \in R_\mu \setminus \widehat{R}_\mu }
\big(1- q^{\arm(\bx{\mu}{i})+1}\spa t^{-\leg(\bx{\mu}{i})} z_i/z_j \big)
\prod_{\alpha_{ij} \in \widehat{R}_{\mu}}
\big(1-q\spa  t\, z_i/z_j \big)} {\prod_{\alpha_{ij} \in R_+} \big(1-q \, z_i/z_j\big)
\prod_{\alpha_{ij} \in R_\mu} \big(1-t \, z_i/z_j\big)}\Bigg) \,,
\end{equation}
where \(\sigmabold\) is the Weyl symmetrization operator defined in
\cref{ss:Weyl-etc} and \(R_\mu,\widehat{R}_\mu\) are subsets of the set of
positive roots for \(GL_l\) explained in \cref{s catalanimals}.
Under polynomial truncation, our formula
recovers \(\omega \Htild_\mu(X;q,t)\).

The proof begins with the Haglund-Haiman-Loehr formula \cite{HaHaLo05}
for $\Htild_{\mu }(X;q,t)$ as a weighted sum of LLT polynomials.  
We then apply the operator $\nabla$ and combine this with a formula for $\nabla$ on an LLT
polynomial established in our recent work~\cite{BHMPS-llt}.
%We then apply the operator $\nabla$, which has $\Htild_{\mu }(X;q,t)$ as
%an eigenfunction, and use the formula for $\nabla$ on an LLT
%polynomial established in our recent work~\cite{BHMPS-llt}.

This work serves as an extended abstract for a future paper in which
we will elaborate on and further generalize this result. 

\section{Background}
\label{s:background}


\subsection{Partitions and symmetric functions}
\label{ss:background}

The (French style) diagram of a partition $\mu = (\mu_1 \ge \cdots \ge
\mu_k > 0)$ is the set
$\{(i,j) \in \ZZ_{+}^2  :  1\leq j\leq
k,\;1\leq i \leq \mu _{j}\}$.
We identify $(i,j)$ with the lattice
square or \emph{box} whose northeast corner has coordinates $(x,y) =
(i,j)$, and refer to this box as being in {\em column} $i$ and {\em
row} $j$.
We set $|\mu| = \mu _{1}+\cdots +\mu _{k}$ and let $\ell(\mu )=k$ be the
number of non-zero parts.
We write $\mu^*$ for the transpose of a partition $\mu$.
The {\em arm} and {\em leg} of a box $b\in \mu$ are the number of
boxes in $\mu$ strictly east of $b$ and strictly north of $b$,
respectively.

\begin{example}
  Below we give the diagram of the partition \(\mu=(2,2,1)\) and we
  fill box \(b\) with \((\arm(b),\leg(b))\):\[
    \ytableausetup{aligntableaux=bottom}
    \begin{ytableau}
      {\scriptstyle 0,0}\\
      \scriptstyle 1,1& \scriptstyle 0,0\\
      \scriptstyle 1,2&\scriptstyle 0,1
    \end{ytableau} \,.
  \]
  For this \(\mu\), we have \(|\mu|=5\) and \(\ell(\mu) = 3\).
\end{example}


Let $\Lambda = \Lambda(X)$ be the algebra of symmetric functions in
infinitely many variables $X = x_{1},x_{2},\ldots$, with coefficients
in the field $\kk = \QQ (q,t)$.  We follow Macdonald's notation
\cite{Macdonald95} for the graded bases of $\Lambda $, and the
automorphism $\omega \colon \Lambda \rightarrow \Lambda $
given on Schur functions by
$\omega s_{\lambda } = s_{\lambda^*}$.  We also work with series and
symmetric functions in finitely many variables $\zz = z_1,\dots,z_l$.
If $f(X)\in \Lambda$ is a formal symmetric function, then $f(\zz )$ or
$f(z_1, \ldots, z_l)$ denotes its specialization with $X = z_1,
\ldots, z_l,0,0,\ldots$.

Given a symmetric function $f\in \Lambda$ and any expression $A$
involving indeterminates, the plethystic evaluation $f[A]$ is defined
by writing $f$ as a polynomial in the power-sums $p_{k}$ and
evaluating with $p_{k}\mapsto p_{k}[A]$, where $p_{k}[A]$ is the
result of substituting $a^{k}$ for every indeterminate $a$ occurring
in $A$.  The variables $q, t$ from our ground field $\kk $ count as
indeterminates.

By convention, the name of an alphabet $X = x_{1},x_{2},\ldots$ stands
for $x_{1}+x_{2}+\cdots $ inside a plethystic evaluation.  Then $f[X]
= f[x_{1}+x_{2}+\cdots ] = f(x_{1},x_{2},\ldots) = f(X)$.  For
example, the evaluation $f[X/(1-t^{-1})]$ is the image of $f(X)$ under
the $\kk $-algebra automorphism of $\Lambda $ that sends $p_{k}$ to
$p_{k}/(1-t^{-k})$.

The {\em modified Macdonald polynomials} $\Htild_\mu = \Htild _{\mu
}(X;q,t)$ of \cite{GarsHaim96} are defined in terms of the Macdonald
polynomials $Q_{\mu }(X;q,t)$ \cite[VI (4.12)]{Macdonald95} or their
integral forms $J_{\mu }(X;q,t)$ \cite[VI (8.3)]{Macdonald95} by
\begin{equation}\label{e:H-tilde}
\Htild _{\mu }(X;q,t) = t^{\mathsf{n}(\mu )}
J_{\mu}[\frac{X}{1-t^{-1}};q,t^{-1}] = t^{\mathsf{n}(\mu)} \bigl( \prod_{b \in
\mu}(1-q^{\arm(b)+1}t^{-\leg(b)}) \bigr)
Q_{\mu}[\frac{X}{1-t^{-1}};q,t^{-1}],
\end{equation}
where $\mathsf{n}(\mu ) = \sum _{i} (i-1)\mu_{i}$.  The $\tilde
H_\mu(X;q,t)$ also have a direct combinatorial
description~\cite{HaHaLo05}, which we will recall in \cref{t
HHL formula}.

When $q=0$, the modified Macdonald polynomials reduce to the
\emph{modified Hall-Littlewood polynomials}
\begin{equation}
 \Htild_\mu(X;0,t) = t^{\mathsf{n}(\mu )} Q_{\mu}[\frac{X}{1-t^{-1}};t^{-1}],
\end{equation}
where the Hall-Littlewood polynomials $Q_{\mu}(X;t)$ are as defined in~\cite[III (2.11)]{Macdonald95}.  At $t=1$ and $t = \infty $, the
$\Htild_\mu(X;0,t)$ specialize to the complete homogeneous symmetric
functions $\Htild_\mu(X;0,1) = h_\mu(X)$ and Schur functions
$\Htild_\mu(X;0,\infty)= s_\mu(X)$.

\begin{example}
  The Macdonald polynomial \(\Htild_{22}(X;q,t)\) has the following
  expansion into the basis of Schur functions.
  \begin{equation}
    \label{eq:8}
    \Htild_{22}(X;q,t) = s_4 + (q+t+qt)s_{31} + (q^2+t^2)s_{22}+(qt +
    q^2 t + qt^2)s_{211} + q^2 t^2 s_{1111}
  \end{equation}
\end{example}

\subsection{Weyl symmetrization and related operators}
\label{ss:Weyl-etc}

The \emph{Weyl symmetrization operator} $\sigmabold$ for $\GL
_{l}$ is defined by
\begin{equation}\label{e:Weyl-symmetrizer}
\sigmabold (f(z_1,\dots,z_l))
= \sum _{w\in \SS_{l}} w \bigg(
\frac{f(z_1,\dots,z_l)}{\prod _{i < j} (1 - z_j/z_i)}
\bigg)
= \sum _{w\in \SS_{l}} w \bigg(
\frac{f(z_1,\dots,z_l)}{\prod _{\alpha_{ij} \in R_{+}} (1 - z_j/z_i)}
\bigg)
\,,
\end{equation}
where $f \in \kk[z_1^{\pm1},\dots, z_l^{\pm1}]$ is a Laurent
polynomial, $\SS_l$ acts by permuting the variables $z_1, \dots, z_l$,
and $R_+= R_+(GL_l) = \{\alpha _{ij}  :  1\leq i < j\leq l \}$
denotes the set of positive roots for $\GL _{l}$, with $\alpha_{ij} =
\epsilon_i -\epsilon_j \in \ZZ^l$.

When $\zz^\nu = z_1^{\nu_1} \cdots z_l^{\nu_l}$ for a dominant weight
$\nu$ (a weight $\nu \in \ZZ^l$ is dominant if $\nu_1 \ge \cdots \ge
\nu_l$), $\sigmabold (\zz ^{\nu }) = \chi _{\nu }$ is an irreducible
$\GL _{l}$ character.  For an arbitrary weight $\gamma \in \ZZ ^{l}$,
either $\sigmabold (\zz ^{\gamma }) = \pm \chi _{\nu }$ for a suitable
dominant weight $\nu $, or $\sigmabold (\zz ^{\gamma }) = 0$.  We
extend $\sigmabold$ to an operator on formal $\kk$-linear combinations
$\sum_{\gamma \in \ZZ^l} c_\gamma \zz^\gamma$ by applying it term by
term, giving an infinite formal linear combination of irreducible
$GL_l$ characters $\sum_{\nu} a_\nu \chi_\nu = \sum_{\gamma \in \ZZ^l}
c_\gamma \sigmabold(\zz^\gamma) $.
%
%\Mark{added a sentence} looks good
%
This makes sense because for each dominant weight $\nu $, the
set of monomials $\zz ^{\gamma }$ such that $\sigmabold (\zz ^{\gamma
}) = \pm \chi _{\nu }$ is finite.


Recall that the {\em polynomial characters} of $\GL _l$ are the
irreducible characters $\chi _{\nu }$ for which $\nu $ is a partition,
that is, $\nu _l\geq 0$.  Given any formal $\kk$-linear combination
$\sum _{\nu} a_{\nu } \chi _{\nu }$ of irreducible $\GL _{l}$
characters, we define its \emph{polynomial truncation} by
\begin{align}
\label{ed pol}
\pol_X \! \Big(\sum _{\nu}a_{\nu }\spa \chi _{\nu }\Big)
= \sum _{ \nu_l \ge 0 }a_{\nu }\spa s_{\nu }(X).
\end{align}
In principle, the right hand side is an infinite formal sum of
symmetric functions, but, for instance, if $\sum a_{\nu } \chi _{\nu
}$ is homogeneous of degree $d$, then the right hand side is an
ordinary symmetric function, homogeneous of degree $d$.

%\Mark{added explanation on the series expansion and rephrased a little} looks good
% We define a related operator $\mathbf{h}_X$
% on Laurent polynomials $f(\zz )$
% by
% \begin{equation}\label{e:Weyl-symmetrizer 2}
% \mathbf{h}_X(f(\zz)) =
% \pol_X  \sigmabold
% \bigg(\frac{f(\zz)}{\prod_{\alpha_{ij} \in R_+} (1-z_i/z_j)} \bigg),
% \end{equation}
% where the factors $(1- z_{i}/z_{j})^{-1}$ are expanded as
% geometric series before applying  $\sigmabold$.
% %
% When $f$ is a monomial, it is well known \cite{Thomas81} that
% \begin{align}
% \label{e hX equals h}
% \mathbf{h}_X(\zz^\gamma) &= h_\gamma(X),
% \end{align}
% where for any integer vector $\gamma \in \ZZ^l$, we define $h_\gamma =
% h_{\gamma_1}\cdots h_{\gamma_l}$ to be the product of complete
% homogeneous symmetric functions, with $h_d$ for $d\leq 0$ interpreted
% as $h_0 =1$, or $h_d = 0$ for $d< 0$.

% We again extend the definition to formal linear combinations of
% monomials, so that $\mathbf{h}_X(\sum_{\gamma \in \ZZ^l} c_\gamma
% \zz^\gamma) = \sum_{\gamma \in \ZZ^l} c_\gamma h_\gamma (X)$.
% %
% %\Mark{swapped last two sentences in this \P} looks good
% %
% With this interpretation, \eqref{e:Weyl-symmetrizer 2} still remains
% valid when $f$ is a power series in $z_i/z_j$ for $i<j$.
% %
% As with $\pol_X$, in principle $\mathbf{h}_X(\sum_{\gamma} c_\gamma
% \zz^\gamma)$ is an infinite formal sum of symmetric functions, but,
% for instance, if $\sum_{\gamma} c_\gamma \zz^\gamma $ is homogeneous
% of degree $d$, then $\mathbf{h}_X(\sum_{\gamma } c_\gamma \zz^\gamma)$
% is an ordinary symmetric function, homogeneous of degree $d$.


\subsection{LLT polynomials}
\label{ss:LLT}

We recall the definition and basic properties of LLT polynomials
\cite{LaLeTh97}, using the `attacking inversions' formulation from
\cite{HaHaLoReUl05}.

A {\it skew diagram} is a difference $\nu = \lambda / \mu$
of partition diagrams $\mu\subseteq\lambda$.
The {\em content} of a box $b = (i,j)$ in row $j$, column $i$ of a
(skew) diagram is $c(b) = i-j$.

Let $\nubold = (\nu_{(1)},\ldots,\nu_{(k)})$ be a tuple of skew
diagrams.  We consider the set of boxes in $\nubold $ to be the
disjoint union of the sets of boxes in the $\nu _{(i)}$, and define
the {\em adjusted content} of a box $a\in \nu_{(i)}$ to be $\ctild (a)
= c(a)+i\epsilon $, where $\epsilon $ is a fixed positive number such
that $k \epsilon <1$.

A {\em diagonal} in $\nubold $ is the set of boxes of a fixed adjusted
content, that is, a diagonal of fixed content in one of the $\nu_{(i)}$.

The {\em reading order} on $\nubold $ is the total ordering $<$ on the
boxes of $\nubold $ such that $a<b \Rightarrow \ctild (a)\leq
\ctild(b)$ and boxes on each diagonal increase from southwest to
northeast.
See \cref{ex:skew shape tuple}.  An {\em attacking
pair} is an ordered pair of boxes $(a,b)$ in $ \nubold$ such that
$a<b$ in reading order and $0<\ctild(b)-\ctild(a)<1$.

A {\em semistandard tableau} on the tuple $\nubold $ is a map $T\colon
\nubold \rightarrow \ZZ _{+}$ which restricts to a semistandard Young
tableau on each component $\nu_{(i)}$.  The set of these is denoted
$\SSYT (\nubold )$.  An {\em attacking inversion} in $T$ is an
attacking pair $(a,b)$ such that $T(a)>T(b)$.  The number of attacking
inversions in $T$ is denoted $\inv (T)$.

\begin{defn}\label{def:G-nu}
The {\em LLT polynomial} indexed by a tuple of skew diagrams $\nubold
$ is the generating function
\begin{equation}\label{e:G-nu}
\Gcal_{\nubold  }(X;q) = \sum _{T\in \SSYT (\nubold)}q^{\inv (T)}\xx ^{T},
\end{equation}
where $\xx ^{T} = \prod _{a\in \nubold  } x_{T(a)}$.
By~\cite{HaHaLoReUl05,LaLeTh97},  $\Gcal_{\nubold  }(X;q) $ is known to be symmetric.
\end{defn}
\begin{example}\label{ex:skew shape tuple}
Let $\nubold = ((32)/(10),(33)/(11))$ and let
\(\nubold[1],\ldots,\nubold[8]\) denote the boxes of \(\nubold\) in
increasing reading order as illustrated in~\eqref{eq:reading order tab
ex}. 
We can visualize this by aligning the shapes of \(\nubold\) along content
diagonals and reading diagonals top to bottom and bottom left to top
right along each diagonal. 
  \begin{equation}
    \label{eq:reading order tab ex}
    \begin{tikzpicture}[scale=0.55, baseline=(current  bounding  box.center)]
\draw [dashed,gray] (-0.5,-1.5)--(3.5,2.5); \draw [dashed,gray]
      (-0.5,0.5)--(3.5,4.5); \draw [dashed,gray]
      (-0.5,-0.5)--(3.5,3.5);
      \draw [dashed,gray]
      (0.5,-1.5)--(4.5,2.5);
      \drawskewdg{3/1,2/0}{0};
      \drawskewdg{3/1,3/1}{2};
      \setcounter{boxnum}{1};
      \foreach \x\y in {1/2,2/2,4/4,2/1,4/3,5/4,3/1,5/3} {
        \node at (\x-0.5,\y-0.5) {\scriptsize \(\nubold[\theboxnum]\)};
        \addtocounter{boxnum}{1};
      }
    \end{tikzpicture}
  \end{equation}
 The tuple \(\nubold\) has attacking pairs
 \begin{multline}
   \{(\nubold[2],\nubold[3]),(\nubold[3],\nubold[4]),(\nubold[4],\nubold[5]),(\nubold[4],\nubold[6]),\\(\nubold[5],\nubold[7]),(\nubold[6],\nubold[7]),(\nubold[7],\nubold[8])\} \,.
 \end{multline}
 Let \(T \in
  \SSYT(\nubold)\) be as follows:
  \begin{equation}
    \label{eq:6}
    T =
    \begin{tikzpicture}[scale=0.55,baseline=(current  bounding  box.center)]
\draw [dashed,gray] (-0.5,-1.5)--(3.5,2.5); \draw [dashed,gray]
      (-0.5,0.5)--(3.5,4.5); \draw [dashed,gray]
      (-0.5,-0.5)--(3.5,3.5);
      \draw [dashed,gray]
      (0.5,-1.5)--(4.5,2.5);
      \drawskewdg{3/1,2/0}{0};
      \drawskewdg{3/1,3/1}{2};
      \foreach \x\y\l in {1/2/1,2/2/2,4/4/2,2/1/1,4/3/1,5/4/4,3/1/4,5/3/3} {
        \node at (\x-0.5,\y-0.5) {\footnotesize \l};
      }
    \end{tikzpicture} \,.
  \end{equation}
  \(T\) has attacking inversions \(\{(\nubold[3],\nubold[4]),(\nubold[7],\nubold[8])\}\), so
  \(\inv(T) = 2\). Also, \(\xx^T = x_1^3 x_2^2 x_3 x_4^2\).
\end{example}
\section{Catalanimals}\label{s catalanimals}
\setcounter{subsection}{1}
In~\cite{BHMPS-llt}, we introduced
{\it Catalanimals}---infinite series of irreducible
$\GL _{l}$ characters in variables $\zz = z_{1},\ldots,z_{l}$, defined by
\begin{equation}\label{e d bigraded catalan}
H(R_q,R_t,R_{qt},\lambda) = \sigmabold \bigg(\frac{z_1^{\lambda_1}\cdots
z_l^{\lambda_l} \prod_{\alpha_{ij} \in R_{qt}} (1-q\, t\, z_i/z_j )}
{ \prod_{\alpha_{ij} \in R_q} (1-q
\, z_i/z_j) \prod_{\alpha_{ij} \in R_t} (1-t \, z_i/z_j)}\bigg),
\end{equation}
depending on a weight $\lambda \in \ZZ^l$ and subsets $R_q,R_t,R_{qt}$
of the set of positive roots $R_+ = \{\epsilon_i-\epsilon_j \mid 1 \le
i < j \le l\}$ for $\GL _{l}$.
Our convention is always to expand
the denominator factors \((1-c \, z_i/z_j)\) for \(c \in \kk\) and \(i
< j\) as geometric series $(1 - c z_i/z_j)^{-1} = 1 + c z_i/z_j
+ \cdots$ before applying $\sigmabold $.

Distinguished among Catalanimals are the \emph{LLT Catalanimals},
denoted \(H_{\nubold}^{1,1}\) in~\cite{BHMPS-llt},
whose polynomial part is equal to \(\omega \nabla\) applied to the LLT
polynomial \(\Gcal_\nubold\), up to a \(q,t\)-monomial scalar factor. The Catalanimal
$H_{\nubold }^{1,1}$ has root sets $R_+ \supseteq R_q \supseteq R_t
\supseteq R_{qt}$, determined as follows using the same attacking
inversion combinatorics as in the definition of the LLT polynomial
$\Gcal_{\nubold }$:
\begin{itemize}
\item $R_+ \setminus R_q  \xleftrightarrow{\ } $ pairs of boxes in the
same diagonal in \(\nubold\),
\item $R_q \setminus R_t  \, \,  \xleftrightarrow{\ } $ the attacking
  pairs in \(\nubold\),
\item $R_t \setminus R_{qt}  \,  \xleftrightarrow{\ } $ pairs going
between adjacent diagonals \(\nubold\),
\end{itemize}
where the boxes of $\nubold $ are numbered $1,\dots, l$ in
reading order (see \cref{ex:skew shape tuple}).
The weight
$\lambda$ is obtained by filling each diagonal $D$ of $\nubold $ with
the value \\ $1+ \chi({\small \text{$D$ contains a row start}}) -
\chi({\small \text{$D$ contains a row end}})$, where $\chi(P) = 1$ if $P$ is true or $0$ if $P$ is false, and then reading this
filling in the reading order---see \cref{fig:Catalanimals intro}.

\begin{figure}[t!]
\[
\begin{array}{@{\hspace{1.2mm}}c@{\qquad\qquad\quad\quad\quad}cr}
{\fontsize{7pt}{6pt}\selectfont \tableau{2&2&1\\2&1&0\\1&0&1& 0\\} }
&
\raisebox{-5.4mm}{
\begin{tikzpicture}
\begin{scope}[scale = .84*.5]
\draw[thick] (0,1) grid (2,2);
\draw[thick] (1,0) grid (3,1);
%
\draw[thin, black!60]  (0,0) grid (1,1);
\node at (0.5, 1.5) { \tiny 2};
\node at (1.5, 1.5) { \tiny 0};
\node at (1.5, 0.5) { \tiny 2};
\node at (2.5, 0.5) { \tiny 0};
\end{scope}
\begin{scope}[xshift = 50, scale = .84*.5]
\draw[thick] (1,0) grid (3,2);
%
\draw[thin, black!60] (0,0) grid (1,2);
\node at (1.5, 1.5) {\tiny 2};
\node at (2.5, 1.5) { \tiny 1};
\node at (1.5, 0.5) { \tiny 1};
\node at (2.5, 0.5) { \tiny 0};
\end{scope}
\end{tikzpicture} }
& \\[7mm]
{\text{  $\lambda$, as a filling of $\mu$} }
&
{\text{ $\lambda$, as a filling of $\nubold $}}
& \\[5mm]
\begin{tikzpicture}[scale = .47]
\begin{scope}
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\draw[thin] (10,-10) -- (11,-10);
\draw[thin] (11,-10) -- (11,-11);
\node at (3/2,-3/2) {\tiny $2 $};
\node at (5/2,-5/2) {\tiny $2 $};
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\node at (9/2,-9/2) {\tiny $1 $};
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\node at (13/2,-13/2) {\tiny $1 $};
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\node at (17/2,-17/2) {\tiny $0 $};
\node (vv) at (19/2,-19/2) {\tiny $1 $};
\node at (21/2,-21/2) {\tiny $0 $};
\node [left=13.4mm of vv] {\small $H_{(\mu)}^{1,1}$};
\end{scope}
\end{tikzpicture}
&
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\node at (3/2,-3/2) {\tiny $2 $};
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\node at (7/2,-7/2) {\tiny $2 $};
\node at (9/2,-9/2) {\tiny $2 $};
\node at (11/2,-11/2) {\tiny $1 $};
\node at (13/2,-13/2) {\tiny $1 $};
\node at (15/2,-15/2) {\tiny $0 $};
\node (vv) at (17/2,-17/2) {\tiny $0 $};
\node[left=11.4mm of vv] {\small \raisebox{2mm}{$H_{\nubold }^{1,1}$}};
\end{scope}
\end{tikzpicture} }
&
\ \ \
\raisebox{16mm}{
\begin{tikzpicture}[scale = .39]
\begin{scope}
\draw[draw = none, fill = black!100] (2,-1) rectangle (3,-2);
\node[anchor = west] at (3,-1.5) {\scriptsize  \  $R_+ \setminus R_q$};
\end{scope}
\begin{scope}[yshift = -34*1]
\draw[draw = none, fill = \mymidgray] (2,-1) rectangle (3,-2);
\node[anchor = west] at (3,-1.5) {\scriptsize  \  $R_q \setminus R_t$};
\end{scope}
\begin{scope}[yshift = -34*2]
\draw[thin, black!0] (2,-1) -- (3,-1);
\draw[thin, black!0] (2,-1) -- (2,-2);
\draw[thin, black!0] (2,-2) -- (3,-2);
\draw[thin, black!0] (3,-2) -- (3,-1);
\draw[draw = none, fill = gray!100] (2+0.5, -1-0.5) circle (.2);
\node[anchor = west] at (3,-1.5) {\scriptsize  \  $R_t \setminus R_{qt}$};
\end{scope}
\begin{scope}[yshift = -34*3]
\draw[draw = none, fill = black!14] (2,-1) rectangle (3,-2);
\node[anchor = west] at (3,-1.5) {\scriptsize \  $R_{qt}$};
\end{scope}
\end{tikzpicture}}
\\[2mm]
\omega\nabla s_{433}= (q\, t)^9 (H_{(\mu)}^{1,1})_{\pol} & \
\omega\nabla \Gcal_{\nubold }= -(q\, t)^4 q^7 (H_{\nubold
}^{1,1})_{\pol } &
\end{array}
\]
\caption{{(i) The Catalanimal
$H_{(\mu)}^{1,1}$ for $\mu = (433)$.  (ii) The Catalanimal
$H_{\nubold}^{1,1}$ for $\nubold = ((32)/(10),(33)/(11))$.  These are
illustrated by drawing the root sets in an $\ell\times \ell$ grid
labeled by matrix-style coordinates, with the sets $R_+ \setminus
R_q$, $R_q \setminus R_t$, $R_t \setminus R_{qt}$, $R_{qt}$ specified
according to the legend on the right; the weight $\lambda$ is written
on the diagonal with $\lambda_1$ in the upper left.}}
\label{fig:Catalanimals intro}
\end{figure}

\begin{example}
  Consider \(\nubold=(\nu^{(1)})\) for \(\nu^{(1)} = (111)\). To
  construct \(H^{(1,1)}_\nubold\), we note that
  every box is on a different diagonal in \(\nubold\) and no boxes
  attack each other, so \(R_q = R_t = R_+\). Furthermore, in reading
  order, boxes 1 and 2 are adjacent and boxes 2 and 3 are
  adjacent, so \(R_{qt} = \{\alpha_{13}\}\). Finally, every box is
  both a row start and a row end, so \(\lambda = (111)\).
  Therefore, \(H_\nubold^{(1,1)}\) is given by \begin{equation}
    H(R_+, R_+, \{\alpha_{13}\},(111)) = \sigmabold \left(
      \frac{\zz^{111}(1-q\,t\,z_1/z_3)}{\prod_{1 \leq i < j \leq 3}(1-q\, z_i/z_j)(1-t\, z_i/z_j)} \right)
  \end{equation}
  Taking the polynomial part, this becomes
  %Furthermore, the polynomial part becomes
  \begin{equation}
  \begin{split}
    \label{eq:10}
    &\pol_X H(R_+,R_+,\{\alpha_{13}\},(111))   \\ &=  s_{111} +
                                                      (q+t+q^2+qt+t^2)s_{21}+(qt+q^3+q^2t+qt^2+t^3)s_3\\
    &= \omega \nabla e_3
  \end{split}
\end{equation}
Note that the LLT polynomial with
  \(\nubold=(\nu^{(1)})\) satisfying \(\nu^{(1)} = (111)\) is equal to
  \(e_3\).
\end{example}

% \begin{figure}
% \[
% \begin{array}{@{\hspace{1.2mm}}c@{\qquad\qquad\quad\quad\quad}cr}
% {\fontsize{7pt}{6pt}\selectfont \tableau{2&2&1\\2&1&0\\1&0&1& 0\\} }
% &
% \raisebox{-5.4mm}{
% \begin{tikzpicture}
% \begin{scope}[scale = .84*.5]
% \draw[thick] (0,1) grid (2,2);
% \draw[thick] (1,0) grid (3,1);
% %
% \draw[thin, black!60]  (0,0) grid (1,1);
% \node at (0.5, 1.5) { \tiny 2};
% \node at (1.5, 1.5) { \tiny 0};
% \node at (1.5, 0.5) { \tiny 2};
% \node at (2.5, 0.5) { \tiny 0};
% \end{scope}
% \begin{scope}[xshift = 50, scale = .84*.5]
% \draw[thick] (1,0) grid (3,2);
% %
% \draw[thin, black!60] (0,0) grid (1,2);
% \node at (1.5, 1.5) {\tiny 2};
% \node at (2.5, 1.5) { \tiny 1};
% \node at (1.5, 0.5) { \tiny 1};
% \node at (2.5, 0.5) { \tiny 0};
% \end{scope}
% \end{tikzpicture} }
% & \\[7mm]
% {\small \text{  $\lambda$, as a filling of $\mu$} }
% &
% {\small \text{ $\lambda$, as a filling of $\nubold $}}
% & \\[5mm]
% \begin{tikzpicture}[scale = .47]
% \begin{scope}
% \draw[draw = none, fill = black!14] (4,-1) rectangle (5,-2);
%  \draw[draw = none, fill = black!14] (5,-1) rectangle (6,-2);
%  \draw[draw = none, fill = black!14] (6,-1) rectangle (7,-2);
%  \draw[draw = none, fill = black!14] (7,-1) rectangle (8,-2);
%  \draw[draw = none, fill = black!14] (7,-2) rectangle (8,-3);
%  \draw[draw = none, fill = black!14] (7,-3) rectangle (8,-4);
%  \draw[draw = none, fill = black!14] (8,-1) rectangle (9,-2);
%  \draw[draw = none, fill = black!14] (8,-2) rectangle (9,-3);
%  \draw[draw = none, fill = black!14] (8,-3) rectangle (9,-4);
%  \draw[draw = none, fill = black!14] (9,-1) rectangle (10,-2);
%  \draw[draw = none, fill = black!14] (9,-2) rectangle (10,-3);
%  \draw[draw = none, fill = black!14] (9,-3) rectangle (10,-4);
%  \draw[draw = none, fill = black!14] (9,-4) rectangle (10,-5);
%  \draw[draw = none, fill = black!14] (9,-5) rectangle (10,-6);
%  \draw[draw = none, fill = black!14] (9,-6) rectangle (10,-7);
%  \draw[draw = none, fill = black!14] (10,-1) rectangle (11,-2);
%  \draw[draw = none, fill = black!14] (10,-2) rectangle (11,-3);
%  \draw[draw = none, fill = black!14] (10,-3) rectangle (11,-4);
%  \draw[draw = none, fill = black!14] (10,-4) rectangle (11,-5);
%  \draw[draw = none, fill = black!14] (10,-5) rectangle (11,-6);
%  \draw[draw = none, fill = black!14] (10,-6) rectangle (11,-7);
%  \draw[draw = none, fill = black!14] (10,-7) rectangle (11,-8);
%  \draw[draw = none, fill = black!14] (10,-8) rectangle (11,-9);
%  \draw[draw = none, fill = gray!100] (2+0.5, -1-0.5) circle (.2);
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% \draw[thin, black!31] (1,-1) -- (11,-1);
% \draw[thin, black!31] (2,-1) -- (2,-1);
% \draw[thin, black!31] (2,-2) -- (11,-2);
% \draw[thin, black!31] (3,-2) -- (3,-1);
% \draw[thin, black!31] (3,-3) -- (11,-3);
% \draw[thin, black!31] (4,-3) -- (4,-1);
% \draw[thin, black!31] (4,-4) -- (11,-4);
% \draw[thin, black!31] (5,-4) -- (5,-1);
% \draw[thin, black!31] (5,-5) -- (11,-5);
% \draw[thin, black!31] (6,-5) -- (6,-1);
% \draw[thin, black!31] (6,-6) -- (11,-6);
% \draw[thin, black!31] (7,-6) -- (7,-1);
% \draw[thin, black!31] (7,-7) -- (11,-7);
% \draw[thin, black!31] (8,-7) -- (8,-1);
% \draw[thin, black!31] (8,-8) -- (11,-8);
% \draw[thin, black!31] (9,-8) -- (9,-1);
% \draw[thin, black!31] (9,-9) -- (11,-9);
% \draw[thin, black!31] (10,-9) -- (10,-1);
% \draw[thin, black!31] (10,-10) -- (11,-10);
% \draw[thin, black!31] (11,-10) -- (11,-1);
% \draw[draw = none, fill = black!100] (3,-2) rectangle (4,-3);
%  \draw[draw = none, fill = black!100] (5,-4) rectangle (6,-5);
%  \draw[draw = none, fill = black!100] (6,-4) rectangle (7,-5);
%  \draw[draw = none, fill = black!100] (6,-5) rectangle (7,-6);
%  \draw[draw = none, fill = black!100] (8,-7) rectangle (9,-8);
%  \draw[thin] (1,-1) -- (2,-1);
% \draw[thin] (2,-1) -- (2,-2);
% \draw[thin] (2,-2) -- (3,-2);
% \draw[thin] (3,-2) -- (3,-3);
% \draw[thin] (3,-3) -- (4,-3);
% \draw[thin] (4,-3) -- (4,-4);
% \draw[thin] (4,-4) -- (5,-4);
% \draw[thin] (5,-4) -- (5,-5);
% \draw[thin] (5,-5) -- (6,-5);
% \draw[thin] (6,-5) -- (6,-6);
% \draw[thin] (6,-6) -- (7,-6);
% \draw[thin] (7,-6) -- (7,-7);
% \draw[thin] (7,-7) -- (8,-7);
% \draw[thin] (8,-7) -- (8,-8);
% \draw[thin] (8,-8) -- (9,-8);
% \draw[thin] (9,-8) -- (9,-9);
% \draw[thin] (9,-9) -- (10,-9);
% \draw[thin] (10,-9) -- (10,-10);
% \draw[thin] (10,-10) -- (11,-10);
% \draw[thin] (11,-10) -- (11,-11);
% \node at (3/2,-3/2) {\tiny $2 $};
% \node at (5/2,-5/2) {\tiny $2 $};
% \node at (7/2,-7/2) {\tiny $2 $};
% \node at (9/2,-9/2) {\tiny $1 $};
% \node at (11/2,-11/2) {\tiny $1 $};
% \node at (13/2,-13/2) {\tiny $1 $};
% \node at (15/2,-15/2) {\tiny $0 $};
% \node at (17/2,-17/2) {\tiny $0 $};
% \node (vv) at (19/2,-19/2) {\tiny $1 $};
% \node at (21/2,-21/2) {\tiny $0 $};
% \node [left=13.4mm of vv] {\small $H_{(\mu)}^{1,1}$};
% \end{scope}
% \end{tikzpicture}
% &
% \raisebox{9mm}{
% \begin{tikzpicture}[scale=.47]
% \begin{scope}
% \draw[draw = none, fill = black!14] (3,-1) rectangle (4,-2);
%  \draw[draw = none, fill = black!14] (4,-1) rectangle (5,-2);
%  \draw[draw = none, fill = black!14] (5,-1) rectangle (6,-2);
%  \draw[draw = none, fill = black!14] (5,-2) rectangle (6,-3);
%  \draw[draw = none, fill = black!14] (6,-1) rectangle (7,-2);
%  \draw[draw = none, fill = black!14] (6,-2) rectangle (7,-3);
%  \draw[draw = none, fill = black!14] (7,-1) rectangle (8,-2);
%  \draw[draw = none, fill = black!14] (7,-2) rectangle (8,-3);
%  \draw[draw = none, fill = black!14] (7,-3) rectangle (8,-4);
%  \draw[draw = none, fill = black!14] (8,-1) rectangle (9,-2);
%  \draw[draw = none, fill = black!14] (8,-2) rectangle (9,-3);
%  \draw[draw = none, fill = black!14] (8,-3) rectangle (9,-4);
%  \draw[draw = none, fill = black!14] (8,-4) rectangle (9,-5);
%  \draw[draw = none, fill = gray!100] (2+0.5, -1-0.5) circle (.2);
% \draw[draw = none, fill = gray!100] (4+0.5, -2-0.5) circle (.2);
% \draw[draw = none, fill = gray!100] (5+0.5, -3-0.5) circle (.2);
% \draw[draw = none, fill = gray!100] (6+0.5, -3-0.5) circle (.2);
% \draw[draw = none, fill = gray!100] (7+0.5, -4-0.5) circle (.2);
% \draw[draw = none, fill = gray!100] (8+0.5, -5-0.5) circle (.2);
% \draw[draw = none, fill = gray!100] (8+0.5, -6-0.5) circle (.2);
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%  \draw[draw = none, fill = \mymidgray] (6,-4) rectangle (7,-5);
%  \draw[draw = none, fill = \mymidgray] (7,-5) rectangle (8,-6);
%  \draw[draw = none, fill = \mymidgray] (7,-6) rectangle (8,-7);
%  \draw[draw = none, fill = \mymidgray] (8,-7) rectangle (9,-8);
%  \draw[thin, black!31] (1,-1) -- (9,-1);
% \draw[thin, black!31] (2,-1) -- (2,-1);
% \draw[thin, black!31] (2,-2) -- (9,-2);
% \draw[thin, black!31] (3,-2) -- (3,-1);
% \draw[thin, black!31] (3,-3) -- (9,-3);
% \draw[thin, black!31] (4,-3) -- (4,-1);
% \draw[thin, black!31] (4,-4) -- (9,-4);
% \draw[thin, black!31] (5,-4) -- (5,-1);
% \draw[thin, black!31] (5,-5) -- (9,-5);
% \draw[thin, black!31] (6,-5) -- (6,-1);
% \draw[thin, black!31] (6,-6) -- (9,-6);
% \draw[thin, black!31] (7,-6) -- (7,-1);
% \draw[thin, black!31] (7,-7) -- (9,-7);
% \draw[thin, black!31] (8,-7) -- (8,-1);
% \draw[thin, black!31] (8,-8) -- (9,-8);
% \draw[thin, black!31] (9,-8) -- (9,-1);
% \draw[draw = none, fill = black!100] (6,-5) rectangle (7,-6);
%  \draw[thin] (1,-1) -- (2,-1);
% \draw[thin] (2,-1) -- (2,-2);
% \draw[thin] (2,-2) -- (3,-2);
% \draw[thin] (3,-2) -- (3,-3);
% \draw[thin] (3,-3) -- (4,-3);
% \draw[thin] (4,-3) -- (4,-4);
% \draw[thin] (4,-4) -- (5,-4);
% \draw[thin] (5,-4) -- (5,-5);
% \draw[thin] (5,-5) -- (6,-5);
% \draw[thin] (6,-5) -- (6,-6);
% \draw[thin] (6,-6) -- (7,-6);
% \draw[thin] (7,-6) -- (7,-7);
% \draw[thin] (7,-7) -- (8,-7);
% \draw[thin] (8,-7) -- (8,-8);
% \draw[thin] (8,-8) -- (9,-8);
% \draw[thin] (9,-8) -- (9,-9);
% \node at (3/2,-3/2) {\tiny $2 $};
% \node at (5/2,-5/2) {\tiny $0 $};
% \node at (7/2,-7/2) {\tiny $2 $};
% \node at (9/2,-9/2) {\tiny $2 $};
% \node at (11/2,-11/2) {\tiny $1 $};
% \node at (13/2,-13/2) {\tiny $1 $};
% \node at (15/2,-15/2) {\tiny $0 $};
% \node (vv) at (17/2,-17/2) {\tiny $0 $};
% \node[left=11.4mm of vv] {\small \raisebox{2mm}{$H_{\nubold }^{1,1}$}};
% \end{scope}
% \end{tikzpicture} }
% &
% \ \ \
% \raisebox{16mm}{
% \begin{tikzpicture}[scale = .39]
% \begin{scope}
% \draw[draw = none, fill = black!100] (2,-1) rectangle (3,-2);
% \node[anchor = west] at (3,-1.5) {\scriptsize  \  $R_+ \setminus R_q$};
% \end{scope}
% \begin{scope}[yshift = -34*1]
% \draw[draw = none, fill = \mymidgray] (2,-1) rectangle (3,-2);
% \node[anchor = west] at (3,-1.5) {\scriptsize  \  $R_q \setminus R_t$};
% \end{scope}
% \begin{scope}[yshift = -34*2]
% \draw[thin, black!0] (2,-1) -- (3,-1);
% \draw[thin, black!0] (2,-1) -- (2,-2);
% \draw[thin, black!0] (2,-2) -- (3,-2);
% \draw[thin, black!0] (3,-2) -- (3,-1);
% \draw[draw = none, fill = gray!100] (2+0.5, -1-0.5) circle (.2);
% \node[anchor = west] at (3,-1.5) {\scriptsize  \  $R_t \setminus R_{qt}$};
% \end{scope}
% \begin{scope}[yshift = -34*3]
% \draw[draw = none, fill = black!14] (2,-1) rectangle (3,-2);
% \node[anchor = west] at (3,-1.5) {\scriptsize \  $R_{qt}$};
% \end{scope}
% \end{tikzpicture}}
% \\[2mm]
% \omega\nabla s_{433}= (q\, t)^9 (H_{(\mu)}^{1,1})_{\pol} & \
% \omega\nabla \Gcal_{\nubold }= -(q\, t)^4 q^7 (H_{\nubold
% }^{1,1})_{\pol } &
% \end{array}
% \]
% \caption{\label{fig:Catalanimals intro} {\small (i) The Catalanimal
% $H_{(\mu)}^{1,1}$ for $\mu = (433)$.  (ii) The Catalanimal
% $H_{\nubold}^{1,1}$ for $\nubold = ((32)/(10),(33)/(11))$.  These are
% illustrated by drawing the root sets in an $\ell\times \ell$ grid
% labeled by matrix-style coordinates, with the sets $R_+ \setminus
% R_q$, $R_q \setminus R_t$, $R_t \setminus R_{qt}$, $R_{qt}$ specified
% according to the legend on the right; the weight $\lambda$ is written
% on the diagonal with $\lambda_1$ in the upper left.  }}
% \end{figure}

For the purposes of this work, we will need only LLT Catalanimals for
\(\nubold\) a tuple of ribbon shapes.
A \emph{ribbon} is a connected skew shape containing no $2 \times 2$
block of boxes. If we only consider ribbons with \(m\) boxes such
that the bottom-right box has content \(-1\), then the contents of all the
boxes are consecutive integers \(-1,-2,\ldots,-m\). Define the
\emph{descent set} of a ribbon \(\nu\) to be the set of contents
\(c(u)\) of the boxes \(u = (i,j) \in \nu\) such that the box \(v = (i,j-1)\)
directly below \(u\) also belongs to \(\nu\).

Then, to a partition \(\mu\) and a subset \(D \subset
\{(i,j) \in \mu \mid j > 1\}\), we associate a tuple of ribbons
\(\nubold(\mu,D) = (\nu^{(1)}, \ldots, \nu^{(k)})\) where \(k=\mu_1\)
is the number of columns of \(\mu\),
and \(\nu^{(i)}\) has size
\(\mu_i^*\), box contents \(\{-1,-2,\ldots,-\mu_i^*\}\), and descent set
\(Des(\nu^{(i)}) = \{-j \mid (i,j) \in D\}\). See \cref{ex:
  full HHL ex} for examples of \(\nubold(\mu,D)\) for fixed \(\mu\) as \(D\) varies. Let
\(\mubold[1],\ldots,\mubold[l]\) denote the boxes of
\(\nubold(\mu,D)\) in increasing reading order. Also, let \(A_\mu\)
denote the number of attacking pairs in \(\nubold(\mu,D)\). Note that
both the reading order of the boxes of \(\nubold(\mu,D)\)  and the
number of attacking pairs
depend only on \(\mu\) and not on \(D\). Then, for any box \(b = (i,j) \in
\mu\), let \(\south(b) = (i,j-1)\) and let \(V_\mu =
\{(\mubold[i],\mubold[j]) \mid \south(\mubold[i]) = \mubold[j]\}\) be
the set of vertical dominoes in \(\mu\).

In this case, the root sets for the LLT Catalanimal indexed by \(\nubold(\mu,D)\) become \(R_q = R_+\), and
\begin{eqnarray}
  \label{eq:4}
  R_t = R_\mu & \defeq \{\alpha_{ij} \in R_+ \mid
  \ctild(\mubold[i])+1 \leq \ctild(\mubold[j])\} \\
  R_{qt} = \widehat{R}_\mu & \defeq \{\alpha_{ij} \in R_+ \mid
  \ctild(\mubold[i])+1 < \ctild(\mubold[j])\}
\end{eqnarray}
Then, we have the following special case
of~\cite[Corollary 8.4.1]{BHMPS-llt}.
\begin{thm}
  For a partition \(\mu\) and a subset \(D \subset \{(i,j) \in \mu \mid i
  > 1\}\), let \(\nubold =
  \nubold(\mu,D)\) and \(S = \{(\mubold[i],\mubold[j]) \in V_\mu \mid \mubold[i] \in D\}\). Then, we have the following
  formula for the operator \(\nabla\) applied to the LLT polynomial \(\Gcal_{\nubold}(X;q)\):
  \begin{multline}
    \label{eq:nabla llt catalanimal}
      \nabla \Gcal_{\nubold}(X;q)
      = \omega \pol_X \sigmabold
    \left( (-qt)^{|\mu|-\mu_1-|D|} q^{A_\mu}
      \phantom{\frac{\prod}{\prod}}\right.\\\left. \times \frac{z_1 \cdots z_l \prod_{(\mubold[i],\mubold[j])\in
          V_\mu\setminus D} z_i/z_j \prod_{\alpha_{ij} \in
          \widehat{R}_\mu} (1-qt z_i/z_j)}{\prod_{\alpha_{ij} \in R_+}
        (1-q z_i / z_j) \prod_{\alpha_{ij} \in R_\mu} (1-t z_i/z_j)}
    \right)
  \end{multline}
\end{thm}
\begin{example}
  For $\mu = (2,2,1,1)$, $l = 6$, the root sets \(R_\mu\) and
  \(\widehat{R}_\mu\) are visualized below drawn in an \(l \times l\)
  grid, labeled by matrix-style coordinates and specified by the legend.
\begin{align*}
\raisebox{-5mm}{
\begin{tikzpicture}[xscale=.77, yscale = .59]
\draw[thick] (0,0) grid (1,4);
\draw[thick] (0,0) grid (2,2);
\node at (0.5, 3.5) { \footnotesize $\bx{\mu}{1}$};
\node at (0.5, 2.5) { \footnotesize $\bx{\mu}{2}$};
\node at (0.5, 1.5) { \footnotesize $\bx{\mu}{3}$};
\node at (1.5, 1.5) { \footnotesize $\bx{\mu}{4}$};
\node at (0.5, 0.5) { \footnotesize $\bx{\mu}{5}$};
\node at (1.5, 0.5) { \footnotesize $\bx{\mu}{6}$};
\node at (1,-.57) {\small reading order };
\end{tikzpicture}}
\ \ \
\quad \quad  \quad \quad \quad
\begin{tikzpicture}[scale = .42]
\draw[draw = none, fill = black!14] (2,-1) rectangle (3,-2);
 \draw[draw = none, fill = black!14] (3,-1) rectangle (4,-2);
 \draw[draw = none, fill = black!14] (3,-2) rectangle (4,-3);
 \draw[draw = none, fill = black!14] (4,-1) rectangle (5,-2);
 \draw[draw = none, fill = black!14] (4,-2) rectangle (5,-3);
 \draw[draw = none, fill = black!14] (5,-1) rectangle (6,-2);
 \draw[draw = none, fill = black!14] (5,-2) rectangle (6,-3);
 \draw[draw = none, fill = black!14] (5,-3) rectangle (6,-4);
 \draw[draw = none, fill = black!14] (6,-1) rectangle (7,-2);
 \draw[draw = none, fill = black!14] (6,-2) rectangle (7,-3);
 \draw[draw = none, fill = black!14] (6,-3) rectangle (7,-4);
 \draw[draw = none, fill = black!14] (6,-4) rectangle (7,-5);
 \draw[draw = none, fill = gray!100] (3+0.5, -1-0.5) circle (.2);
\draw[draw = none, fill = gray!100] (4+0.5, -1-0.5) circle (.2);
\draw[draw = none, fill = gray!100] (4+0.5, -2-0.5) circle (.2);
\draw[draw = none, fill = gray!100] (5+0.5, -1-0.5) circle (.2);
\draw[draw = none, fill = gray!100] (5+0.5, -2-0.5) circle (.2);
\draw[draw = none, fill = gray!100] (6+0.5, -1-0.5) circle (.2);
\draw[draw = none, fill = gray!100] (6+0.5, -2-0.5) circle (.2);
\draw[draw = none, fill = gray!100] (6+0.5, -3-0.5) circle (.2);
\draw[thin, black!31] (1,-1) -- (7,-1);
\draw[thin, black!31] (2,-1) -- (2,-1);
\draw[thin, black!31] (2,-2) -- (7,-2);
\draw[thin, black!31] (3,-2) -- (3,-1);
\draw[thin, black!31] (3,-3) -- (7,-3);
\draw[thin, black!31] (4,-3) -- (4,-1);
\draw[thin, black!31] (4,-4) -- (7,-4);
\draw[thin, black!31] (5,-4) -- (5,-1);
\draw[thin, black!31] (5,-5) -- (7,-5);
\draw[thin, black!31] (6,-5) -- (6,-1);
\draw[thin, black!31] (6,-6) -- (7,-6);
\draw[thin, black!31] (7,-6) -- (7,-1);
\draw[thin] (1,-1) -- (2,-1);
\draw[thin] (2,-1) -- (2,-2);
\draw[thin] (2,-2) -- (3,-2);
\draw[thin] (3,-2) -- (3,-3);
\draw[thin] (3,-3) -- (4,-3);
\draw[thin] (4,-3) -- (4,-4);
\draw[thin] (4,-4) -- (5,-4);
\draw[thin] (5,-4) -- (5,-5);
\draw[thin] (5,-5) -- (6,-5);
\draw[thin] (6,-5) -- (6,-6);
\draw[thin] (6,-6) -- (7,-6);
\draw[thin] (7,-6) -- (7,-7);
\end{tikzpicture}
\ \ \
\raisebox{6mm}{
\begin{tikzpicture}[scale = .42]
\begin{scope}
\draw[draw = none, fill = black!14] (2,-1) rectangle (3,-2);
\node[anchor = west] at (3,-1.5) {\scriptsize   $R_\mu$};
\end{scope}
\begin{scope}[yshift = -34*1]
\draw[thin, black!0] (2,-1) -- (3,-1);
\draw[thin, black!0] (2,-1) -- (2,-2);
\draw[thin, black!0] (2,-2) -- (3,-2);
\draw[thin, black!0] (3,-2) -- (3,-1);
\draw[draw = none, fill = gray!100] (2+0.5, -1-0.5) circle (.2);
\node[anchor = west] at (3,-1.5) {\scriptsize    $\widehat{R}_\mu$};
\end{scope}
\end{tikzpicture}}
\end{align*}

\end{example}
\section{Macdonald Catalanimals}
\subsection{The Haglund-Haiman-Loehr formula}
Haglund-Haiman-Loehr \cite{HaHaLo05} gave a formula for the modified
Macdonald polynomials $H_\mu(X;q,t)$ as a positive sum of LLT
polynomials indexed by a tuple of ribbons. We now
recall this formula.


\begin{thm}\label{t HHL formula}
  For partition \(\mu\),
  \begin{equation}
    \label{eq:HHL formula}
    \Htild_\mu(X;q,t) = \sum_D \left( \prod_{u \in D} q^{-\arm(u)} t^{\leg(u)+1}\right)
    \Gcal_{\nubold(\mu,D)}(X;q)\,,
  \end{equation}
  where the sum runs over all subsets \(D \subset \{(i,j) \in \mu \mid
  j > 1\}\).
\end{thm}

\begin{example}\label{ex: full HHL ex}
 We illustrate the theorem for \(\mu = (2,2,1)\). Below we draw \(\nubold(\mu,D)\) for the 8
 possible values of \(D\) with boxes labeled in reading order and with
 the corresponding coefficient \(\prod_{u \in D} q^{-\arm(u)}
 t^{\leg(u)+1}\).
\begin{equation*}
\begin{minipage}{0.16\linewidth}
  \begin{center}
    \begin{tikzpicture}[scale=0.7]
      \drawDg{3,2} \setcounter{boxnum}{1}; \foreach \x\y in
      {1/3,1/2,2/2,1/1,2/1} { \node at (\x-0.5,\y-0.5) {\footnotesize
          $\bx{\mu}{\theboxnum}$}; \addtocounter{boxnum}{1}; };
      \node at (1,-.58) {$\mu$};
    \end{tikzpicture}
  \end{center}
  % \vspace{1mm}
  \begin{center}
        \begin{tikzpicture}[scale=0.5]
      \drawDg{3,2} \foreach \x\y in {1/3,2/2} { \node at
        (\x-0.5,\y-0.5) {$0$}; } \node at (0.5,1.5) {$1$};
      \node at (1,-0.62) {\footnotesize \phantom{l}arms\phantom{g}};
    \end{tikzpicture}
    \
    \begin{tikzpicture}[scale=0.5]
      \drawDg{3,2} \foreach \x\y in {1/3,2/2} { \node at
        (\x-0.5,\y-0.5) {$0$}; } \node at (0.5,1.5) {$1$};
      \node at (1,-0.62) {\footnotesize legs};
    \end{tikzpicture}
  \end{center}
\end{minipage}
\begin{minipage}{0.8\linewidth}
  \begin{center}
    \begin{tabular}{rccl}
    \begin{tikzpicture}[scale=0.4]
      \draw [dashed,gray] (-0.5,1.5)--(3.5,5.5); \draw [dashed,gray]
      (-0.5,0.5)--(3.5,4.5); \draw [dashed,gray]
      (-0.5,-0.5)--(3.5,3.5); \drawskewdg{0/1,0/1,0/1}{0}
      \drawskewdg{0/1,0/1}{2}
      \setcounter{boxnum}{1};
      \foreach \x\y in {1/3,1/2,3/4,1/1,3/3} {
        \node at (\x-0.5,\y-0.5) {\tiny \theboxnum};
        \addtocounter{boxnum}{1};
      }
      \node at (1.5,-1.5) {\footnotesize\(D = \{\mubold[1],\mubold[2],\mubold[3]\}\)};
      \node at (3.5,0.5) {\(q^{\shortminus 1}t^4\)};
    \end{tikzpicture}&
    \begin{tikzpicture}[scale=0.4]
      \draw [dashed,gray] (-0.5,1.5)--(2.5,4.5); \draw [dashed,gray]
      (-0.5,0.5)--(3.5,4.5); \draw [dashed,gray]
      (-0.5,-0.5)--(3.5,3.5); \drawskewdg{0/1,-1/1}{0}
      \drawskewdg{0/1,0/1}{2}
      \setcounter{boxnum}{1};
      \foreach \x\y in {0/2,1/2,3/4,1/1,3/3} {
        \node at (\x-0.5,\y-0.5) {\tiny \theboxnum};
        \addtocounter{boxnum}{1};
      }
      \node at (1.5,-1.5) {\footnotesize \(D = \{\mubold[2],\mubold[3]\}\)};
      \node at (3.5,0.5) {\(q^{\shortminus 1}t^3\)};
    \end{tikzpicture}&
    \begin{tikzpicture}[scale=0.4]
      \draw [dashed,gray] (-0.5,1.5)--(3.5,5.5); \draw [dashed,gray]
      (-0.5,0.5)--(3.5,4.5); \draw [dashed,gray]
      (-0.5,-0.5)--(4.5,4.5); \drawskewdg{0/1,0/1,0/1}{0}
      \drawskewdg{-1/1}{3}
      \setcounter{boxnum}{1};
      \foreach \x\y in {1/3,1/2,3/4,1/1,4/4} {
        \node at (\x-0.5,\y-0.5) {\tiny \theboxnum};
        \addtocounter{boxnum}{1};
      }
      \node at (1.5,-1.5) {\footnotesize \(D = \{\mubold[1],\mubold[2]\}\)};
      \node at (3.5,0.5) {\(q^{\shortminus 1}t^3\)};
    \end{tikzpicture}&
    \begin{tikzpicture}[scale=0.4]
      \draw [dashed,gray] (-1.5,0.5)--(3.5,5.5); \draw [dashed,gray]
      (-1.5,-0.5)--(3.5,4.5); \draw [dashed,gray]
      (-0.5,-0.5)--(3.5,3.5); \drawskewdg{-1/1,-1/0}{0}
      \drawskewdg{0/1,0/1}{2}
      \setcounter{boxnum}{1};
      \foreach \x\y in {0/2,0/1,3/4,1/1,3/3} {
        \node at (\x-0.5,\y-0.5) {\tiny \theboxnum};
        \addtocounter{boxnum}{1};
      }
      \node at (1.5,-1.5) {\footnotesize \(D = \{\mubold[1],\mubold[3]\}\)};
      \node at (3.5,0.5) {\(t^2\)};
    \end{tikzpicture}\\
    \begin{tikzpicture}[scale=0.4]
      \draw [dashed,gray] (-0.5,1.5)--(2.5,4.5); \draw [dashed,gray]
      (-0.5,0.5)--(3.5,4.5); \draw [dashed,gray]
      (-0.5,-0.5)--(4.5,4.5); \drawskewdg{0/1,-1/1}{0}
      \drawskewdg{-1/1}{3}
      \setcounter{boxnum}{1};
      \foreach \x\y in {0/2,1/2,3/4,1/1,4/4} {
        \node at (\x-0.5,\y-0.5) {\tiny \theboxnum};
        \addtocounter{boxnum}{1};
      }
      \node at (1.5,-1.5) {\footnotesize \(D = \{\mubold[2]\}\)};
      \node at (3.5,0.5) {\(q^{\shortminus 1}t^2\)};
    \end{tikzpicture}&
    \begin{tikzpicture}[scale=0.4]
      \draw [dashed,gray] (-1.5,0.5)--(2.5,4.5); \draw [dashed,gray]
      (-1.5,-0.5)--(3.5,4.5); \draw [dashed,gray]
      (-0.5,-0.5)--(3.5,3.5); \drawskewdg{-2/1}{0}
      \drawskewdg{0/1,0/1}{2}
      \setcounter{boxnum}{1};
      \foreach \x\y in {-1/1,0/1,3/4,1/1,3/3} {
        \node at (\x-0.5,\y-0.5) {\tiny \theboxnum};
        \addtocounter{boxnum}{1};
      }
      \node at (1.5,-1.5) {\footnotesize \(D = \{\mubold[3]\}\)};
      \node at (3.5,0.5) {\(t\)};
    \end{tikzpicture}&
    \begin{tikzpicture}[scale=0.4]
      \draw [dashed,gray] (-1.5,0.5)--(2.5,4.5); \draw [dashed,gray]
      (-1.5,-0.5)--(2.5,3.5); \draw [dashed,gray] (-0.5,-0.5)--(3.5,3.5);
      \drawskewdg{-1/1,-1/0}{0} \drawskewdg{-1/1}{2}
      \setcounter{boxnum}{1};
      \foreach \x\y in {0/2,0/1,2/3,1/1,3/3} {
        \node at (\x-0.5,\y-0.5) {\tiny \theboxnum};
        \addtocounter{boxnum}{1};
      }
      \node at (1.5,-1.5) {\footnotesize \(D = \{\mubold[1]\}\)};
      \node at (3.5,0.5) {\(t\)};
    \end{tikzpicture}&
    \begin{tikzpicture}[scale=0.4]
      \draw [dashed,gray] (-1.5,0.5)--(1.5,3.5); \draw [dashed,gray]
      (-1.5,-0.5)--(2.5,3.5); \draw [dashed,gray] (-0.5,-0.5)--(3.5,3.5);
      \drawskewdg{-2/1}{0} \drawskewdg{-1/1}{2}
      \setcounter{boxnum}{1};
      \foreach \x\y in {-1/1,0/1,2/3,1/1,3/3} {
        \node at (\x-0.5,\y-0.5) {\tiny \theboxnum};
        \addtocounter{boxnum}{1};
      }
      \node at (1.5,-1.5) {\footnotesize \(D = \varnothing\)};
      \node at (3.5,0.5) {\(1\)};
    \end{tikzpicture}
    \end{tabular}
  \end{center}
\end{minipage}
\end{equation*}
\end{example}

\subsection{Proving the Macdonald Catalanimal formula}\label{ss: the proof}
Applying \(\nabla\) to~\eqref{eq:HHL formula} and substituting our
Catalanimal formula~\eqref{eq:nabla llt catalanimal}
for \(\nabla \Gcal_\nubold\) for each summand, the
resulting sum simplifies
and yields the following formula for the 
modified
Macdonald polynomials $\Htild_\mu(X;q,t)$
 (previewed in \eqref{eq:macanimal series}): 
%to the following formula.
%\begin{equation}
%  \label{eq:2}
%  \omega \pol_X
%\sigmabold \Bigg(
%\frac{
%\prod_{\alpha_{ij} \in R_\mu \setminus \widehat{R}_\mu }
%\big(1- q^{\arm(\bx{\mu}{i})+1}\spa t^{-\leg(\bx{\mu}{i})} z_i/z_j \big)
%\prod_{\alpha_{ij} \in \widehat{R}_{\mu}}
%\big(1-q\spa  t\, z_i/z_j \big)} {\prod_{\alpha_{ij} \in R_+} \big(1-q \, z_i/z_j\big)
%\prod_{\alpha_{ij} \in R_\mu} \big(1-t \, z_i/z_j\big)}\Bigg) \,.
%\end{equation}
%
%Thus, we recover our Catalanimal-style formula for the modified
%Macdonald polynomials.
\begin{thm}
  For every partition \(\mu\),
  \begin{equation}
    \label{eq:11}
    \Htild_\mu(X;q,t) = \omega \pol_X \Hbold_\mu(\zz;q,t)
  \end{equation}
  for \(\Hbold_\mu\) given by
  \begin{equation}\label{eq:macdonald series final}
     \hspace{-0.022in} \HS_{\mu}(\zz;q,t) = \sigmabold \Bigg(
\frac{
\prod_{\alpha_{ij} \in R_\mu \setminus \widehat{R}_\mu }
\big(1- q^{\arm(\bx{\mu}{i})+1}\spa t^{-\leg(\bx{\mu}{i})} z_i/z_j \big)
\prod_{\alpha_{ij} \in \widehat{R}_{\mu}}
\big(1-q\spa  t\, z_i/z_j \big)} {\prod_{\alpha_{ij} \in R_+} \big(1-q \, z_i/z_j\big)
\prod_{\alpha_{ij} \in R_\mu} \big(1-t \, z_i/z_j\big)}\Bigg) \,.
  \end{equation}
\end{thm}
The work of~\cite{HaHaLo05, HaHaLo08} actually gives many formulae for \(\Htild_\mu\),
  one for each rearrangement of the columns of \(\mu\). In turn,
  following the same proof technique, 
  we obtain a Catalanimal style formula for \(\Htild_\mu\) for each rearrangement of
  the columns of \(\mu\).  
  More details will be given in the full version of this paper.
    %we can recover a
%  Catalanimal representing \(\Htild_\mu\) for each rearrangement of
%  the columns of \(\mu\). In the full version of this paper, we plan
%  to explain these formulae in detail with proof.
\subsection{Positivity of the Macdonald Catalanimal}
Finally, we provide a conjecture on the series \(\Hbold_\mu\)
in~\eqref{eq:macdonald series final}.
It is a theorem of~\cite{Haiman01} that the expansion \(\Htild_\mu(X;q,t) =
\sum_\lambda \tilde{K}_{\lambda,\mu}(q,t) s_\lambda\) has positive
coefficients \(\tilde{K}_{\lambda,\mu}(q,t) \in \ZZ_{\geq
  0}[q,t]\). However, computer experimentation suggests that
\(\Hbold_\mu(\zz;q,t)\) exhibits a stronger positivity as a series of \(GL_l\)-characters.
\begin{conj}
\label{conj:Hmu-positivity}
For every partition $\mu $ of $l$, the series $\Hbold_\mu(\zz;q,t)$ is
a positive sum of irreducible $\GL_l$ characters; that is, the
coefficients in
\begin{equation}\label{e:Hmu-positivity}
\Hbold_\mu(\zz;q,t) = \sum_{\nu } \Kbold _{\nu , \mu}(q,t)\, \chi_\nu
\end{equation}
are polynomials $\Kbold _{\nu , \mu}(q,t)\in \ZZ_{\geq 0}[q,t]$ with
non-negative integer coefficients.
\end{conj}
% \subsection{Remarks about Macdonald Catalanimals}\label{ss final remarks}

% In a future work, the authors will expand upon the following facts.
% \begin{enumerate}
% \item The work of~\cite{BHMPS-llt} gives, for each pair of coprime
%   integers \((m,n)\), an LLT Catalanimal for every tuple of skew
%   shapes \(\nubold\). For fixed \((m,n)\), collecting the LLT
%   Catalanimals associated to the ribbons arising from \(\mu\), we get
%   an \((m,n)\)-Macdonald Catalanimal. However, when both \(m > 1\) and
%   \(n>1\), the polynomial part of these \((m,n)\)-Macdonald
%   Catalanimals fails to be Schur-positive.
% \item The Macdonald Catalanimal expressions can be algebraically
%   manipulated in a straightforward manner to give similar expressions
%   for the integral form of the unmodified Macdonald polynomials \(J_\mu(X;q,t)\).
% \end{enumerate}


\printbibliography



\end{document}

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