%% if you are submitting an initial manuscript then you should have submission as an option here
%% if you are submitting a revised manuscript then you should have revision as an option here
%% otherwise options taken by the article class will be accepted
\documentclass[finalversion]{FPSAC2023}
\articlenumber{68}
%% but DO NOT pass any options (or change anything else anywhere) which alters page size / layout / font size etc

%% note that the class file already loads {amsmath, amsthm, amssymb}



\usepackage{tikz,caption,subcaption}
\usepackage{genyoungtabtikz}
\usepackage{ytableau,tikz,varwidth,cases}
\usetikzlibrary{calc}
%\usepackage[francais]{babel}




\usetikzlibrary{calc}
\usepackage{ytableau}
%\usepackage{dsfont}
\usepackage{multicol}
%\captionsetup[sub]{font=small,labelfont={sf}}
%\usepackage[francais]{babel}

\usepackage{here}
  \usepackage{soul}
\setstcolor{green}



\def\bbbr{\mathbb R}
\def\bbbq{\mathbb Q}
\def\bbbc{\mathbb C}
\def\bbbf{\mathbb F}
\def\bbbz{\mathbb Z}
\def\bbbn{\mathbb N}
\def\bbbg{\mathbb G}
\def\cch{\mathcal H}
\def\ccsc{\mathcal {SC}}
\def\ccdd{\mathcal {DD}}
\def\ccp{\mathcal P}
\def\ccc{\mathcal C}
\DeclareMathOperator{\BGP}{BG}
\DeclareMathOperator{\WLP}{WL}


\def\bbba{\overline{\bbbq}}
\def\bolda{\bm{a}}
\def\boldb{\bm{b}}
%\def\span{{\rm Span}}
\def\det{{\rm det}}
\def\Gal{{\rm Gal}}
\def\inprod#1{\left<#1\right>}
\def\qed{\hfill {$\Box$}}
\def\tref#1{\ref{#1}}
\def\fref#1{(\ref{#1})}
\def\ker{{\rm ker}}
\def\im{{\rm im}}
\def\is{\equiv}
\def\mod#1{({\rm mod}\ #1)}
\def\sgn{{\rm sgn}}
\def\rank{{\rm rank}}
%\def\v#1{{\bf #1}}
\def\rd#1{\left(\left(#1\right)\right)}
\def\Det#1{\left|\matrix{#1}\right|}
\def\R{\bbbf_q[\theta]}
\def\Sol{{\rm Sol}}

\newtheorem{thm}{Theorem}
\newtheorem{lem}{Lemma}
\newtheorem{proposition}[thm]{Proposition}
\newtheorem{definition}[thm]{Definition}
\newtheorem{corollary}[thm]{Corollary}

\theoremstyle{remark}
\newtheorem{remark}[thm]{Remark}
\newtheorem{ex}{Example}

\usepackage{mathtools, stmaryrd}
\usepackage{xparse} 
\DeclarePairedDelimiterX{\Iintv}[1]{\llbracket}{\rrbracket}{\iintvargs{#1}}
\NewDocumentCommand{\iintvargs}{>{\SplitArgument{1}{,}}m}
{\iintvargsaux#1} %
\NewDocumentCommand{\iintvargsaux}{mm} {#1\mkern1.5mu..\mkern1.5mu#2}

\renewcommand{\sp}{\mathrm{sp}}
\DeclareMathOperator{\oo}{so}
\renewcommand{\oe}{\mathrm{o}^{\text{even}}}
\DeclareMathOperator{\I}{I} 


%\captionsetup[sub]{font=small,labelfont={sf}}


%% define your title in the usual way
\title[Combinatorial interpretations of the Macdonald identities for affine root systems]{Combinatorial interpretations of the Macdonald identities for affine root systems}

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


\author[David Wahiche]{David Wahiche\addressmark{1}}
%\address{Institut Camille Jordan, Universit\'e Claude Bernard Lyon 1,
%69622 Villeurbanne Cedex, France}
%\email{wahiche@math.univ-lyon1.fr}

%\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}Univ. Lyon, Universit\'e Claude Bernard Lyon 1, UMR 5208, Institut Camille Jordan, France}

%% 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{We explore some connections between vectors of integers and integer partitions seen as bi-infinite words. This methodology enables us to give a combinatorial interpretation of the Macdonald identities for affine root systems of the seven infinite families in terms of symplectic and special orthogonal Schur functions. From these results, we are able to derive $q$-Nekrasov--Okounkov formulas associated to each family. Nevertheless we only give results for types $\tilde{C}$ and $\tilde{C}^{\vee}$, and give a sketch of the proof for type $\tilde{C}$.}

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

%\resume{\lipsum[2]}

%% put your keywords here, or comment this out if you don't have them yet
\keywords{integer partitions, hook length, Macdonald identities for affine root systems, Littlewood decomposition, $q$-Nekrasov--Okounkov formula.}

%% 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{macdo_ident.bib}
%% note the \printbibliography command at the end of the file which goes with these biblatex commands

\begin{document}

\maketitle
%% note that you DO NOT have to put your abstract here -- it is generated by \maketitle and the \abstract and \resume commands above


\section{Introduction and notations}

Formulas involving hook length abound in combinatorics and representation theory. Between 2006 and 2008, using various methods coming from representation theory \cite{We}, gauge theory \cite{NO} and combinatorics \cite{Ha}, several authors proved the so-called Nekrasov--Okounkov formula which can be stated as follows:
\begin{equation}\label{NOdebut}
\sum_{\lambda\in\ccp}T^{\lvert \lambda\rvert}\prod_{h\in\cch(\lambda)}\left(1-\frac{z}{h^2}\right)=\prod_{k\geq 1}\left(1-T^k\right)^{z-1}.
\end{equation}
Here $T$ is a formal variable, $z \in\mathbb{C}$, $\ccp$ is the set of integer partitions and $\cch(\lambda)$ is the multiset of hook lengths of the partition $\lambda$.

This formula does not only cover the generating series for $\ccp$ obtained by setting $z=0$ in \eqref{NOdebut}: it actually gives a connection between powers of the Dedekind $\eta$ function and integer partitions.
%This formula has been generalized in many ways (see for instance \cite{HJ,Pe}).
Among generalizations of \eqref{NOdebut} that can be found in the literature, a $(q,t)$-extension was proved by Rains--Warnaar \cite{RW}, by using refined skew Cauchy-type identities for Macdonald polynomials. This result was also obtained independently by Carlsson--Rodriguez-Villegas \cite{CRV} by means of vertex operators and the plethystic exponential. As mentioned in \cite{RW}, the special case $q=t$ is a reformulation of a result due to Dehaye--Han \cite{HD} and Iqbal--Nazir--Raza--Salem \cite{INRS} which reads as follows:
\begin{equation}\label{Hande}
\sum_{\lambda\in\ccp}T^{\lvert \lambda\rvert}\prod_{h\in\cch(\lambda)}\frac{(1-uq^h)(1-u^{-1}q^h)}{(1-q^h)^2}=\prod_{k,r\geq 1}\frac{(1-uq^rT^k)^r(1-u^{-1}q^rT^k)^r}{(1-q^{r-1}T^k)^r(1-q^{r+1}T^k)^r}.
\end{equation}
\noindent Here $T$ and $q$ are formal variables and $u \in \mathbb{C}$. Note that taking $u=q^z$ and letting $q\rightarrow 1$ in \eqref{Hande} yields \eqref{NOdebut}, although it is not immediate for the product side. 

Methods used by Han to prove \eqref{NOdebut} and Dehaye--Han for \eqref{Hande} both start from a specialization of the Macdonald formula for affine root systems specialized in type $\tilde{A}$. However, one needs an unspecialized Macdonald identity for type $\tilde{A}$ to get \eqref{Hande}. This can be found for instance in \cite{RS}, where Rosengren and Schlosser give a proof of Macdonald identities for the seven infinite affine root systems with elliptic determinantal evaluation (see also Stanton's reformulation in \cite{Stanton}). The next step in \cite{HD} to prove \eqref{Hande} uses new combinatorial notions such as exploded tableaux and a $V_t$-coding adapted from Garvan--Kim--Stanton \cite{GKS}. These techniques are close to the methodology presented here. However their extensions to other types seem complicated. 


A \textit{partition} $\lambda$ of a positive integer $n$ is a nonincreasing sequence of positive integers $\lambda=(\lambda_1,\lambda_2,\dots,\lambda_\ell)$ such that $\lvert \lambda \rvert := \lambda_1+ \dots+\lambda_\ell = n$. The $\lambda_i$'s are the \textit{parts} of $\lambda$, the number $\ell$ of parts being the \textit{length} of $\lambda$, denoted by $\ell(\lambda)$. Each partition can be represented by its Ferrers diagram, which consists in a finite collection of boxes arranged in left-justified rows, with the row lengths in non-increasing order. The \textit{Durfee square} of $\lambda$ is the maximal square fitting in the Ferrers diagram. Its diagonal $\Delta$ will be called the main diagonal of $\lambda$. It is of size $d=d_\lambda:=\max(s | \lambda_s\geq s)$. Let us introduce a signed statistic $\varepsilon$ already appearing in \cite{MP}. For a box $s$ of $\lambda$ of coordinates $(i,j)$, \textit{$\varepsilon_{s}$} is defined as $-1$ if $s$ is strictly below the main diagonal of the Ferrers diagram and as $1$ otherwise, as depicted in \cref{fig:var}. The partition $\lambda'=(\lambda_1',\lambda_2',\dots,\lambda_{\lambda_1}')$ is the \textit{conjugate} of $\lambda$, where $\lambda_j'$ denotes the number of boxes in the column $j$. 
%As an example, in \cref{fig:ferrers}, the Durfee square of $\lambda=(4,3,3,2)$, which is a partition of $12$ of length $4$, is coloured in red.


Recall that the \textit{hook length} of $s$, denoted by $h_s$, is the number of boxes $v$ such that either $s=v$, or $v$ lies strictly below (respectively to the right) of $s$ in the same column (respectively row). For any $t\in \mathbb{N}^{*}$, the multiset of all hook lengths that are congruent to $0 \pmod t$ is denoted by $\mathcal{H}_t(\lambda)$. Note that $\mathcal{H}(\lambda)=\mathcal{H}_1(\lambda)$. A partition $\omega$ is a \textit{$t$-core} if $\cch_t(\omega)=\varnothing$. For any $\mathcal{A}\subset\ccp$, let $\mathcal{A}_{(t)}$ be the subset of elements of $\mathcal{A}$ that are $t$-cores. For example, the only $2$-cores are the ``staircase'' partitions $(k,k-1,\dots,1)$ for any $k\in \mathbb{N}^*$.

An integer partition $\lambda$ is \textit{self-conjugate} if its Ferrers diagram is symmetric along the main diagonal. Let $\mathcal{SC}$ be the set of self-conjugate partitions. The set of doubled distinct partitions, denoted by $\ccdd$, is that of all partitions $\lambda$ with Durfee square of size $d_\lambda$ such that $\lambda_i=\lambda_i'+1$ for all $i\in\lbrace 1,\dots,d\rbrace$. In \cref{fig:varsc} for instance, $\lambda=(5,3,3,1,1)\in\ccsc$ has its main diagonal $\Delta$ shaded in green while the main diagonal of $\lambda=(6,4,4,1,1)\in\ccdd$ is shaded in green in \cref{fig:vardd}. The strip shaded in yellow corresponds to the boxes added to a self-conjugate partition to obtain a $\ccdd$ partition. These subsets of partitions arise when one expresses the Weyl denominator formula for types $B,C$ and $D$ \cite[p.$79$]{Macbook} and have been of particular interest within the work  of P\'etr\'eolle \cite{MP} where two Nekrasov--Okounkov type formulas for $\ccsc$ and $\ccdd$ are derived. For instance, he proves the following $\ccdd$ Nekrasov--Okounkov type formula, coming from the basic specialization of the Macdonald identity specialized for type $\tilde{C}$, which stands that for a formal variable $T$ and any complex number $z$:
%Another kind of extension of \eqref{NOdebut} was discovered by Pétréolle in \cite{MP}, where he starts from Macdonald's specialization in type $\tilde{C}$ to derive the following Nekrasov--Okounkov formula for doubled distinct partitions denoted by $\ccdd$ for any complex $z$:
%and which stands, for any complex number $z$, that: 
%\begin{equation}\label{formulemathias}
%\sum_{\lambda\in\ccdd} (-1)^{d_\lambda}T^{\lvert\lambda\rvert/2}\prod_{\substack{s\in\lambda\\h_s\in\cch(\lambda)}}\left(1-\frac{2z+2}{\varepsilon_{u}h_u}\right)=\prod_{k\geq 1}\left(1-T^k\right)^{2z^2+z}.
%\end{equation}
\begin{equation}\label{formulemathias}
\sum_{\lambda\in\ccdd} (-1)^{d_\lambda}T^{\lvert\lambda\rvert}\prod_{\substack{s\in\lambda\\h_s\in\cch(\lambda)}}\left(1-\frac{2z+2}{h_s\varepsilon_{s}}\right)=\prod_{k\geq 1}\left(1-T^k\right)^{2z^2+z}.
\end{equation}



\begin{figure}[t]
\centering
\scalebox{0.8}{
%\captionsetup[subfigure]{labelformat=empty}
%\hfill
\begin{subfigure}[t]{.45\textwidth}
\centering

\begin{ytableau}
 *(green!60)+  &+ & + &+ & +\\
 - &*(green!60)+ &+ & \none \\
 - &-  &*(green!60)+&\none \\
  -&\none &\none[\textcolor{green}{\Delta}] &\none  \\
 - &\none &\none &\none \\
% \none% \none
\end{ytableau}
\caption{$\lambda=(5,3,3,1,1)\in\ccsc$}

\label{fig:varsc}
\end{subfigure}
\begin{subfigure}[t]{.5\textwidth}
\centering
%\YFrench
%\yng(4,3,3,2)
%\\\hspace*{6cm}\includegraphics{partitionrapport.png}
%\centering
%\YFrench
%\young(+-+-,-+-,+-+,-+)
\begin{ytableau}
 *(green!60)+  &+ & + &*(yellow!60)+&+ & +\\
 - &*(green!60)+ &+ &*(yellow!60)+ & \none \\
 - &-  &*(green!60)+&*(yellow!60)+ &\none \\
  -&\none &\none[\textcolor{green}{\Delta}] &\none & \none \\
 - &\none &\none &\none & \none \\
% \none% \none
\end{ytableau}
\caption{$\lambda=(6,4,4,1,1)\in\ccdd$}
%\caption{Ferrers diagram filled with signs before summing to a BG-rank of $0$.}
\label{fig:vardd}
\end{subfigure}}
\caption{Self-conjugate and doubled distinct partitions filled with $\varepsilon$}
\label{fig:var}
\end{figure}

% In \cref{fig:hooks}, the hook lengths of all boxes for the partition $\lambda=(4,3,3,2)$ are written in their corresponding boxes, the boxes associated with $\mathcal{H}_3(\lambda)$ are shaded in red. The hook-length multiset $\mathcal{H}(\lambda)$ is here given by $\lbrace 2,1,4,3,1,5,4,2,7,6,4,1\rbrace$ and $\mathcal{H}_3(\lambda)=\lbrace 3,6\rbrace$.
%
%A \emph{rim hook} (or border strip, or ribbon) is a connected skew shape containing no $2\times2$ square. The length of a rim hook is the number of boxes in it, and its height is one less than its number of rows. By convention, the height of an empty rim hook is zero.


%More recently, $q,t$-analogues were discovered independently by Carlsson-- Rodriguez-Villegas (\cite{CRV}) and Rains--Warnaar (\cite{RW}). As remarked in \cite{CRV,RW}, the case $q=t$ can be derived from the Theorem $1$ in \cite{HD}.
%We have for any complex variable $u$:
%\begin{equation}\label{qanalog}
%\sum_{\lambda\in\ccp}^{\lvert\lambda\rvert}\prod_{h\in\cch(\lambda)}\frac{(1-uq^h)(1-u^{-1}q^h)}{(1-q^h)^2}=\prod_{k,r\geq 1}\frac{(1-uq^rT^k)^r(1-u^{-1}q^rT^k)^r}{(1-q^{r-1}T^k)^r(1-q^{r+1}T^k)^r}.
%\end{equation}


Let $a$ and $T$ be formal variables. Recall that the $T$-Pochhammer symbol is defined as
\begin{equation*}
\displaystyle (a;T)_0=1, \quad (a;T)_\infty = \prod_{j\geq 0} (1-aT^j),\quad\text{and}\quad  \displaystyle (a_1,\dots,a_n)_\infty =(a_1;T)_\infty \dots (a_n;T)_\infty.
\end{equation*}


We denote by $\textsf{sp}$ (respectively $\textsf{so}$) the classical symplectic (respectively odd orthogonal) Schur function (see for instance \cite{FH}). Let $t$ be a strictly positive integer, set \linebreak ${\bf x}:=(x_1,\dots,x_t)$ and let us introduce:
$$ K_T(t,{\bf x})=\prod_{1\leq i<j\leq t}\left(Tx_i x_j,Tx_i^{-1} x_j^{-1},Tx_ix_j^{-1},Tx_jx_i^{-1};T\right)_\infty.$$


The goal of this extended abstract is to investigate combinatorially the connections between all the Macdonald identities and the Nekrasov--Okounkov formulas. According to Corollary 6.2 in \cite{RS} (see also \cite{Mac,Stanton}), the Macdonald identity for type $\tilde{C}_t$ is as follows:
\begin{multline}
\Delta_C(\mathbf{x})\left(T;T\right)_\infty^t \prod_{i=1}^t \left(Tx_i^2,Tx_i^{-2};T\right)_\infty K_T(t,\mathbf{x})=\sum_{{\bf m}\in\bbbz^t}\sum_{\sigma\in S_t}\sgn(\sigma)
\prod_{i=1}^t x_i^{(2t+2)m_i}\\\times T^{2(t+1)\binom{m_i}{2}+(t+1)m_i}
\left((x_iT^{m_i})^{\sigma(i)-t-1}-(x_iT^{m_i})^{t+1-\sigma(i)}\right),\label{eq:macdostart}
\end{multline}
where $\Delta_C(\mathbf{x})=\prod_{1\le i \le t}x_i^{-t}(1-x_i^2)\prod_{1\le i<j\le t}(x_j-x_i)(1-x_ix_j)$.
 We will focus on types $\tilde{C}_t$ (denoted $\tilde{C}_t^{(1)}$ in \cite{Kac}), $\tilde{C}_t^{\vee}$ (denoted $\tilde{D}_{t+1}^{(2)}$ in \cite{Kac}) here, but the method is the same for the other types. By computing the Littlewood decomposition to partitions seen as a bi-infinite sequences of ``$0$'' and ``$1$'' (see \cref{lit} for precise definitions and properties), the quadratic form which is to the exponent of $T$ in \eqref{eq:macdostart} can be interpreted as half the weight of a doubled distinct $(2t+2)$-core partition using the ideas of Garvan--Kim--Stanton \cite{GKS}. Introducing the notion of $V_{g,t}$-coding (see \cref{def:vcoding}) which can be thought of as the last indices $\pmod{g}$ of letters ``$0$" in the bi-infinite sequences, and $H_+:=\lbrace h_s<g,\varepsilon_s=1\rbrace$, one can then reinterpret the right-hand side of\eqref{eq:macdostart} as follows:
\begin{thm}\label{prop:MacdotypeC}
Set $t\in\mathbb{N}^*$. The Macdonald identity for type $\tilde{C}_t$ can be rewritten as follows:
\begin{equation}\label{eq:macdoc}
\sum_{\omega\in\ccdd_{(2t+2)}} (-1)^{d_{\omega}+\lvert H_+\rvert}T^{\lvert\omega\rvert/2}\sp_{\mu}({\bf x})=
\left(T\right)_\infty^t K_T(t,\mathbf{x})\prod_{i=1}^t \left(Tx_i^2,Tx_i^{-2};T\right)_\infty ,
\end{equation}

where $\mathbf{v}$ is the $V_{g,t}-$coding corresponding to $\omega$ (see \cref{def:vcoding}) and $\mu\in\ccp$ such that
$\mu_i:=v_i+i-2t-2$ for all $1\leq i \leq t$.
\end{thm}

Similarly we get the following:
\begin{thm}
\label{prop:Macdoothertypes}
Set $t\in\mathbb{N}^*$. The Macdonald identity for type $\tilde{C}_t^{\vee}$ can be rewritten as follows:
\begin{multline*}
\sum_{\omega\in\ccsc_{(2t)}} (-1)^{\lvert H_+\rvert+\lvert H_+\cap \Delta\rvert+d_\omega}T^{\lvert\omega\rvert/2}\oo_{\mu}({\bf x})\\
=\left(T^{1/2};T^{1/2}\right)_\infty\left(T;T\right)_\infty^{t-1}
K_T(t,{\bf x})\prod_{i=1}^t \left(T^{1/2}x_i,T^{1/2}x_i^{-1};T^{1/2}\right)_\infty
\end{multline*}
where $\mathbf{v}$ is the $V_{g,t}-$coding corresponding to $\omega$ (see \cref{def:vcoding}) and $\mu\in\ccp$ is such that
$\mu_i:=v_i+i-2t$ for all $1\leq i \leq t$.
\end{thm}


% sums of certain subsets of $\ccp$ (see \cref{prop:MacdotypeC})  and on the other hand it provides a way enumerating product over the multiset of hook length of these partitions for any function $\tau$ (see \cref{thm:DDpair}). Specializing ${\bf x}$ in \cref{prop:MacdotypeC} and taking $\tau(x)=1-q^x$ in \cref{thm:DDpair} enables us to derive $q$-Nekrasov--Okounkov formulas (see \cref{NOC}).



%To derive $q$-Nekrasov--Okounkov formulas, it remains to specialize \cref{prop:MacdotypeC}. To try to keep notations compact, we define, for any strictly positive integers $r,m$, the following terms which appears in every product on the right hand side of Macdonald identities:
% the following term common to all types, where $T$ is a formal variable, and $u,q$ are complex numbers
%Define for any strictly positive integers $r$ and $m$
%$$
%K_{r,m}(T,u,q):=\frac{ \left(1-u^{2}q^{r+1}T^m\right)^{r-\lfloor r/2\rfloor}\left(1-u^{-2}q^{r}T^m\right)^{r-\lfloor r/2\rfloor}}{\left(1-q^{r}T^m\right)^{r-\lfloor r/2\rfloor}\left(1-q^{r+1}T^m\right)^{r-\lfloor r/2\rfloor}}.


As a consequence of our results, we can prove the following $q$-analogues of Nekra\-sov--Okounkov type identities.
\begin{thm}\label{NOC}
For formal variables $T$, $q$ and any complex number $u$, we have:
\begin{multline}\label{eq:noc}
\sum_{\lambda\in\ccdd} (-u)^{d_\lambda}T^{\lvert \lambda\rvert/2}\prod_{s\in\lambda}\frac{1-u^{-2\varepsilon_s}q^{h_s}}{1-q^{h_s}}\prod_{s\in\Delta}\frac{1+uq^{h_s/2}}{1+u^{-1}q^{h_s/2}}\\=
\prod_{m,r\geq 1}\frac{1+uq^{r-1}T^m}{1+u^{-1}q^{r}T^m}\frac{\left(1-u^{-2}q^{r+2}T^m\right)^{r-\lfloor r/2\rfloor}\left(1-u^2q^{r-1}T^m\right)^{r-\lfloor r/2\rfloor}}{\left(1-q^{r}T^m\right)^{r-\lfloor r/2\rfloor}\left(1-q^{r+1}T^m\right)^{r-\lfloor r/2\rfloor}}
\end{multline}
\begin{multline}\label{eq:nosc}
\sum_{\lambda\in\ccsc} (-1)^{d_\lambda}T^{\lvert \lambda\rvert/2}\prod_{s\in\lambda}\frac{1-u^{-2\varepsilon_s}q^{2h_s}}{1-q^{2h_s}}\prod_{s\in\Delta}\frac{1-uq^{h_s}}{1-u^{-1}q^{h_s}}\\=
\prod_{m,r\geq 1}\frac{\left(1-T^{m/2}\right)}{\left(1-T^{m}\right)}\frac{1-u^{-1}q^{2r-1}T^{m/2}}{1-uq^{2r-1}T^{m/2}}\frac{\left(1-u^{-2}q^{2(r+1)}T^m\right)^{r-\lfloor r/2\rfloor}\left(1-u^2q^{2r}T^m\right)^{r-\lfloor r/2\rfloor}}{\left(1-q^{2r}T^m\right)^{r+1-\lfloor (r+1)/2\rfloor}\left(1-q^{2(r+1)}T^m\right)^{r-\lfloor r/2\rfloor}}.
\end{multline}
%
%\begin{multline}
%\sum_{\lambda\in\ccsc} T^{\lvert \lambda/2\rvert}\prod_{s\in\lambda}\frac{1-u^{-2\varepsilon_s}q^{h_s}}{1-q^{h_s}}\prod_{s\in\Delta}\frac{1-uq^{(1+h_s)/2}}{1-u^{-1}q^{(-1+h_s)/2}}\\=
%\prod_{m\geq 1}\left(1-u^{-1}T^{m+1/2}\right)\prod_{r\geq 1}\frac{\left(1-u^{-1}q^rT^{m+1/2}\right)}{\left(1-u q^{r}T^{m+1/2}\right)}K_{r,m}(T,u,q).\label{eq:nosc}
%\end{multline}

\end{thm}

Note that taking $u=q^z$ and letting $q\rightarrow 1$ in \eqref{eq:noc} gives \eqref{formulemathias}, while in \eqref{eq:nosc} it yields a new Nekrasov--Okounkov type formula. Actually all the  Macdonald identities in the Appendix of \cite{Mac} can be derived from specializations and limits of \cref{prop:MacdotypeC,prop:Macdoothertypes} and their analogues for other types.
In \cref{lit}, we introduce a way of computing the Littlewood decomposition with words, giving an explicit connection between $t$-cores and vectors of integers. In \cref{hooks}, we use this description of the Littlewood decomposition to characterize the product of hook lengths of subsets of $t$-cores such as $\ccdd_{(2t+2)}$. We derive \cref{thm:DDpair} but we only give the lemmas necessary to its proof. In the last section, we sketch the proofs of \cref{prop:MacdotypeC,NOC} for type $\tilde{C}$ only.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Combinatorial properties of the Littlewood decomposition on certain subsets of partitions} \label{lit}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


In this section, we use the formalism of Han--Ji \cite{HJ}. Let $\partial \lambda$ be the border of the Ferrers diagram of $\lambda$. %Each step on $\partial\lambda$ is either horizontal or vertical. 
Encode the walk along the border from the South-West to the North-East as depicted in \cref{fig:wordddcore}: take ``$0$'' for a vertical step and ``$1$'' for a horizontal step. This yields a $0/1$ sequence denoted \textbf{$s(\lambda)$}. This resulting word over the $\lbrace 0,1\rbrace$ alphabet has infinitely many ``$0$'''s at the beginning (respectively ``$1$'''s at the end), is indexed by $\bbbz$, and written $(c_i)_{i\in\mathbb{Z}}$.  
 %It can easily be seen that if for instance we add "$0$" at the beginning of the word or "$1$"'s at the end of the word, we obtain the same partition. 
 
This writing as a sequence is not unique (since for any $k$ sequences, $(c_{k+i})_{i\in\mathbb{Z}}$ define the same partition), hence the necessity to set the index $0$ uniquely (to ensure this encoding is bijective). To tackle that issue, we set the index $0$ when the number of ``$0$'''s to the right of that index is equal to the number of ``$1$'''s to the left. In other words, the number of horizontal steps along $\partial\lambda$ corresponding to a ``$1$'' of negative index in $(c_i)_{i\in\bbbz}$ must be equal to the number of vertical steps corresponding to ``$0$'''s of nonnegative index in $(c_i)_{i\in\bbbz}$ along $\partial\lambda$. The delimitation between the letter of index $-1$ and that of index $0$ is called the \textit{median} of the word, marked by a $\mid$ symbol. The size of the Durfee square is then equal to the number of ``$1$'''s of negative index. Hence the application $s$ bijectively associates a partition to the word:
\begin{align*}
s(\lambda)=(c_i)_{i\in\mathbb{Z}}=\left(\ldots c_{-2}c_{-1}|c_0c_1c_2\ldots\right), \intertext{where $c_i\in\lbrace 0,1\rbrace$ for any $i\in\bbbz$, and such that}
\#\{i\leq-1,c_i=1\}= \#\{i\geq0,c_i=0\}<\infty.
\end{align*}

\begin{figure}[t]
\centering
\scalebox{0.6}{
\begin{tikzpicture}
    [%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
        dot/.style={circle,draw=black, fill,inner sep=1pt},
    ]%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\fill [red!30] (0,0) rectangle (11,-1);
\fill [blue!30] (0,-1) rectangle (1,-10);
\foreach \x in {0,...,1}{
    \node[dot] at (\x,-10){ };
}



\draw[->,very thick,blue!100] (0.2,-10) -- (0.2+.6,-10);
\draw[->,very thick,blue!100] (1.2,-5) -- (1.2+.6,-5);
\draw[->,very thick,blue!100] (4.2,-2) -- (4.2+.6,-2);
\draw[->,very thick,blue!100] (9.2,-1) -- (9.2+.6,-1);

\draw[->,very thick,brown] (1,-9.8) -- (1,-9.8+.6);
\draw[->,very thick,brown] (2,-4.8) -- (2,-4.8+.6);
\draw[->,very thick,brown] (5.2,-2) -- (5.2+.6,-2);
\draw[->,very thick,brown] (10.2,-1) -- (10.2+.6,-1);


\draw[->,very thick,RoyalPurple] (1,-8.8) -- (1,-8.8+.6);
\draw[->,very thick,RoyalPurple] (2,-3.8) -- (2,-3.8+.6);
\draw[->,very thick,RoyalPurple] (6,-1.8) -- (6,-1.8+.6);
\draw[->,very thick,RoyalPurple] (11,-0.8) -- (11,-0.8+.6);

\draw[->,very thick,purple] (1,-7.8) -- (1,-7.8+.6);
\draw[->,very thick,purple] (2.2,-3) -- (2.2+.6,-3);
\draw[->,very thick,purple] (6.2,-1) -- (6.2+.6,-1);

\draw[->,very thick,teal] (1,-6.8) -- (1,-6.8+.6);
\foreach \x in {3.2}
    \draw[->,very thick,teal] (\x,-3) -- (\x+.6,-3);
\draw[->,very thick,teal] (7.2,-1) -- (7.2+.6,-1);


\draw[->,very thick,red] (1,-5.8) -- (1,-5.8+.6);
\draw[->,very thick,red] (4,-2.8) -- (4,-2.8+.6);
\draw[->,very thick,red] (8.2,-1) -- (8.2+.6,-1);

%

\foreach \y in {5,...,9}
	\node[dot] at (1,-\y){};
\node[dot] at (4,-3){};
\foreach \y in {3,...,5}
    \node[dot] at (2,-\y){};

\foreach \x in {4,...,6}
    \node[dot] at (\x,-2){};
\foreach \x in {6,...,11}
    \node[dot] at (\x,-1){};

\foreach \x in {1,...,11}
    \draw (\x,-.1) -- node[above,xshift=-0.4cm,yshift=1mm] {$\lambda_{\x}'$} (\x,+.1);

%\node[above,xshift=0.5cm,yshift=1mm] at (4,0) {$\lambda_5'$};
\node[above,xshift=0.5cm,yshift=1mm] at (12,0) {NE};
\node[above,xshift=-4mm,yshift=1mm] at (0,0) {NW};

\foreach \y in {1,...,10}
    \draw (.1,-\y) -- node[above,xshift=-4mm,yshift=0.2cm] {$\lambda_{\y}$} (-.1,-\y);
%\node[above,xshift=-4mm,yshift=2mm] at (0,-4) {$\lambda_4$};
\node[below,xshift=-4mm] at (0,-11) {SW};
\node at (-0.5,-11+0.5){0};
\node at (11+0.5,0){1};

\foreach \y in {6,...,10}
	\node at (1,-\y+0.5){0};

\foreach \y in {4,...,5}
    \node at (2,-\y+0.5){0};

\node at (4,-3+0.5){0};
\node at (6,-2+0.5){0};
\node at (11,-0.5){0};


\node at (0.5,-10) {1};
\node at (1.5,-5) {1};

\foreach \x in {2,...,3}
    \node at (\x+0.5,-3){1};
\foreach \x in {4,...,5}
    \node at (\x+0.5,-2){1};
\foreach \x in {6,...,10}
    \node at (\x+0.5,-1){1};
\node at (0.5,-10.25){\textcolor{blue}{$\shortparallel$}};
\node at (0.5,-10.5){\textcolor{blue}{\phantom{o}$c_{-v_1+g}$}};

\node at (3.5,-3.25){\textcolor{teal}{$\shortparallel$}};
\node at (3.5,-3.5){\textcolor{teal}{$c_{0}$}};

\node at (11.75,-0.55){\textbf{\textcolor{RoyalPurple}{$=c_{v_1-g}$}}};

\node[dot] at (6,-1){};  
%\node[dot] at (5,0){};
\draw[->,thick,-latex] (0,-11) -- (0,-11);
\draw[thick] (0,-11) -- (0,1);
\draw[->,thick,-latex] (-1,0) -- (12,0);
\node[circle,draw=blue,fill=blue,inner sep=0pt,minimum size=5pt] at (3,-3){};

\foreach \y in {6,...,9}
	\draw[dotted,gray] (0,-\y)--(1,-\y);
\draw[dotted,gray] (0,-5)--(2,-5);
\draw[dotted,gray] (0,-4)--(2,-4);
\draw[dotted,gray] (0,-3)--(3,-3);
\draw[dotted,gray] (0,-2)--(5,-2);
\draw[dotted,gray] (0,-1)--(6,-1);


\draw[dotted,gray] (1,-6)--(1,0);
\draw[dotted,gray] (2,-5)--(2,0);
\draw[dotted,gray] (3,-3)--(3,0);
\draw[dotted,gray] (4,-3)--(4,0);
\draw[dotted,gray] (5,-2)--(5,0);
\draw[dotted,gray] (6,-1)--(6,0);
\foreach \x in {7,...,10}
	\draw[dotted,gray] (\x,-1)--(\x,0);

%\draw[fill=gray!50] plot[smooth, samples=100, domain=1:e] (\x,) -| (0,0) -- cycle;


%\draw[fill=red!60] (0,0)-- (11,0)-- (11,-1)-- (0,-1) --(0,0);

\draw[->,very thick, blue](1.5,-4.5)--(1.5,-2.5);
\draw[->,very thick, red](1.5,-2.5)--(3.5,-2.5);

\end{tikzpicture}}
\caption{$\omega=(11,6,4,2,2,1,1,1,1,1)\in\ccdd_{(6)}$ and its binary correspondence}
\label{fig:wordddcore}
\end{figure}


\begin{lem}\label{lem:indices}
This application maps bijectively a box $s$ of hook length $h_s$ of the Ferrers diagram of $\lambda$ to a pair of indices $(i_s, j_s)\in\bbbz^2$ of $s(\lambda)$ such that $i_s<j_s$, $c_{i_s}=1$ and $c_{j_s}=0$, $j_s-i_s=h_s$.
\end{lem}  
\cref{lem:durf} below allows to characterize the position of a box.
\begin{lem}(\cite[Lemma 2.1]{Wmult})\label{lem:durf}
Set $\lambda\in\ccp$ and $s(\lambda)$ its corresponding word. Let $s$ be a box of the Ferrers diagram of $\lambda$. Let $(i_s,j_s)\in\bbbz^2$ be the indices in $s(\lambda)$ associated with $s$. Then $s$ is a box strictly above the main diagonal in the Ferrers diagram of $\lambda$ if and only if $|i_s|\leq |j_s|$.
\end{lem}
The map below is often called the Littlewood decomposition (see \cite{GKS,HJ} for instance).

\begin{definition}\label{defphi}{\em
Let $t \geq 2$ be an integer and consider:
$$\begin{array}{l|rcl}
\Phi_t: & \mathcal{P} & \to & \mathcal{P}_{(t)} \times \mathcal{P}^t \\
& \lambda & \mapsto & (\omega,\nu^{(0)},\ldots,\nu^{(t-1)}),
\end{array}$$
where if $s(\lambda)=\left(c_i\right)_{i\in\bbbz}$, then for all $k\in\lbrace 0,\dots,t-1\rbrace$ one has $\nu^{(k)}:=s^{-1}\left(\left(c_{ti+k}\right)_{i\in\bbbz}\right)$. The tuple $\underline{\nu}=\left(\nu^{(0)},\ldots,\nu^{(t-1)}\right)$ is the $t$-quotient of $\lambda$, denoted by \textit{$quot_t(\lambda)$}, while $\omega$ is the $t$-core of $\lambda$, denoted by \textit{$core_t(\lambda)$}.}
\end{definition}
Obtaining the $t$-quotient is straightforward from $s(\lambda)=\left(c_i\right)_{i\in\bbbz}$: we just look at subwords with indices congruent to the same values modulo $t$. The sequence $10$ within these subwords are replaced iteratively by $01$ until the subwords are all the infinite sequence of ``$0$'''s before the infinite sequence of ``$1$'''s (in fact it consists in removing all rim hooks in $\lambda$ of length congruent to $0\pmod t$). Then $\omega$ is the partition corresponding to the word which has the subwords $\pmod t$ obtained after the removal of the $10$ sequences.

For example, if we take $\lambda = (4,4,3,2) \text{ and } t=3$, then $s(\lambda)=\ldots \color{red}{0} \color{blue}{0} \color{green}{1} \color{red}{1} \color{blue}{0}\color{green}{1} \color{black}| \color{red}{0} \color{blue}{1} \color{green}{0} \color{red}{0} \color{blue}{1} \color{green}{1}\color{black}\ldots$
\begin{align*}
\begin{array}{rc|rcl}
s\left(\nu^{(0)}\right)=\ldots \color{red} 001 \color{black}| \color{red}001\color{black}\ldots& &s\left(w_{0}\right)=\ldots \color{red} 000 \color{black}| \color{red}011\color{black}\ldots,\\
 s\left(\nu^{(1)}\right)=\ldots \color{blue} 000 \color{black}| \color{blue}111\color{black}\ldots& \longmapsto& s\left(w_{1}\right)=\ldots \color{blue} 000 \color{black}| \color{blue}111\color{black}\ldots , \\
 s\left(\nu^{(2)}\right)=\ldots \color{green} 011 \color{black}| \color{green}011\color{black}\ldots & & s\left(w_{2}\right)=\ldots \color{green} 001 \color{black}| \color{green}111\color{black}\ldots .
\end{array}\\
\end{align*}
Thus $s(\omega)=\ldots \color{red}{0} \color{blue}{0} \color{green}{0} \color{red}{0} \color{blue}{0}\color{green}{1} \color{black}| \color{red}{0} \color{blue}{1} \color{green}{1} \color{red}{1} \color{blue}{1} \color{green}{1}\color{black}\ldots $, and
$$
quot_3(\lambda)=\left(\nu^{(0)},\nu^{(1)},\nu^{(2)}\right)=\left((1,1),\varnothing,(2)\right),\ core_3(\lambda)= \omega=(1)%=\quad \YFrench \yng(1).
$$

One might therefore see $t$-core partitions as partitions whose $t$-quotient in the Littlewood decomposition is empty. Let $\omega$ be a partition, it is a $t$-core if and only if $quot_t(\omega)=\left(\varnothing,\dots,\varnothing\right)$. This is equivalent to say that all subwords $\pmod t$ are of the form $\ldots 0011\ldots$, which is, an infinite sequence of ``$0$'''s and then an infinite sequence of ``$1$'''s. For any $i\in\lbrace 0,\dots,t-1\rbrace$ let us define $n_i:=\min\lbrace k \in\bbbz\mid c_{i+kt}=1\rbrace$. Each $n_i$ corresponds to the index of the first ``$1$'' in the subword of $s(\omega)$ whose index is congruent to $i \pmod t$. Recall that the word $s(\omega)$ has as many ``$1$'''s of negative index as ``$0$'''s of positive index. This is equivalent to require $\sum_{i=0}^{t-1}n_i=0$, hence there is a natural bijective map between $\omega\in \ccp_{(t)}$ and $\left(n_i\right)_{i\in \lbrace 0,\dots,t-1\rbrace}\in\bbbz^t$ such that $\sum_{i=0}^{t-1}n_i=0$.


For example, if we take $\omega = (4,2) \text{ and } t=3$, then 
%$s(\lambda)=\ldots \color{red}{0} \color{blue}{0} \color{green}{1} \color{red}{1} \color{blue}{0}\color{green}{1} \color{black}| \color{red}{0} \color{blue}{1} \color{green}{0} \color{red}{0} \color{blue}{1} \color{green}{1}\color{black}\ldots$
\begin{align*}
\begin{array}{rc|lc}
& &s\left(w_{0}\right)=\ldots \color{red} 000 &\color{black}|\color{red}\underbrace{001}_{n_0=2}1\color{black}\ldots,\\
 s\left(\omega\right)=\ldots \color{red}{0} \color{blue}{0} \color{green}{0} \color{red}{0} \color{blue}{1}\color{green}{1} \color{black}| \color{red}{0} \color{blue}{1} \color{green}{1} \color{red}{0} \color{blue}{1} \color{green}{1}\color{red}{1}\color{blue}{1} \color{green}{1}\color{black} \ldots& \longmapsto& s\left(w_{1}\right)=\ldots \color{blue} 00\underbrace{1}_{n_1=-1}&\color{black}| \color{blue}111\color{black}\ldots , \\
  & & s\left(w_{2}\right)=\ldots \color{green} 00\underbrace{1}_{n_2=-1} &\color{black}| \color{green}111\color{black}\ldots .
\end{array}\\
\end{align*}
%Thus
%\begin{center}
%$s(\omega)=\ldots \color{red}{0} \color{blue}{0} \color{green}{0} \color{red}{0} \color{blue}{0}\color{green}{1} \color{black}| \color{red}{0} \color{blue}{1} \color{green}{1} \color{red}{1} \color{blue}{1} \color{green}{1}\color{black}\ldots $
%\end{center}

In \cite{Johnson}, Johnson uses the fermionic viewpoint of partitions (which is the same as the one described above) to prove that this bijection is actually the one used by Garvan--Kim--Stanton in \cite{GKS}. We reformulate here what is written in \cite[Section 2]{Johnson} in terms of index of words: let $\lambda$ be a partition and $t$ be a strictly positive integer. Abaci correspond exactly to $t$-subwords of $s(\lambda)$ with fixed residue $\pmod{t}$ while the $n_i$'s as defined in Bijection $2$ in \cite{GKS} correspond to the charge of the $i$-th runner on the abaci. This implies in particular that if $\omega$ is a $t$-core partition with corresponding word $s(\omega)=\left(c_i\right)_{i\in\bbbz}$ and if we set $\phi(\omega):=\left(n_0,\dots,n_{t-1}\right)$, then $n_i=\min\lbrace k\in\bbbz \mid c_{kt+i}=1\rbrace$. Moreover, we have:
\begin{equation}
|\omega|=\frac{t}{2}\sum_{i=0}^{t-1} n_i^2+\sum_{i=0}^{t-1} i n_i.
\label{eq:gks}
\end{equation}

%\begin{lem}(\cite[Theorem 2.10]{Johnson})\label{lem:johnson}
%\end{lem}


%
%Let us emphasize another interesting property of this bijection: by definition a hook in $\omega$ corresponds to a pair of indices $(k,l)\in\bbbz^2$ such that $k<l$, $c_k=1$ and $c_l=0$ $l-k=h$. Let $(q_k,i_k)$ and $(q_l,j_l)$ be respectively the quotient and the remainder of the euclidean division of $k$, respectively $l$, by $t$. As $\omega$ is a $t$-core, we have that $i_k\neq j_l$ (otherwise $h\equiv 0\pmod t$), $n_{i_k}\leq q_k$, $n_{j_l}-1\geq q_l$.
%Hence 
%\textcolor{red}{Revoir ce paragraphe et le changer}




\section{Hook length product of $t$-core partitions}\label{hooks}
The aim of this section is to introduce the material required to prove \cref{thm:DDpair} which is an enumerative result on hook length products. The latter is crucial to get \cref{NOC}, as explained in the introduction (see also \cref{sketch} for more details). This extended abstract only focuses on results for $\tilde{C}$ and $\tilde{C}^{\vee}$ that are stated independently in order to avoid characteristic functions (that complicate their statement). However note that analogous statements exist for all seven infinite families. As for technical results allowing to prove \cref{thm:DDpair}, they are stated only in the $\ccdd_{(2t+2)}$-case, once again to avoid characteristic functions and technicality they bring. The cases $\ccsc_{(2t)}$, $\ccsc_{(2t+1)}$ and $\ccdd_{(2t+1)}$ are obtained on the same way, uniform formulations and proofs can be found in \cite{Wmac}. \cref{thm:DDpair} involves $V_{g,t}$-codings: these are a particular kind of vector of integers (see \cref{def:vcoding}) that can be associated to any $\omega \in \ccdd_{(2t+2)}$ (see \cref{prop:core_vcoding}).

%
%Let $t \in \mathbb{N}^*$. \cref{thm:DDpair} above has an analogous statement for $\omega\in \ccdd_{(2t)}$, as stated below.
%
%\begin{thm}\label{thm:SCpair}
%Let $\omega\in \ccsc_{(2t)}$ and $(v_1,\dots,v_{t})\in\bbbz^{\lfloor t\rfloor}$ its associated $V_{g,t}$-coding, and set \linebreak $r_i = v_i -t+\frac{1}{2}$ for any $i\in \{1, \dots,t\}$. Then we have for any function $\tau$ defined on $\bbbz$:
%$$\lvert \omega\rvert = \frac{1}{2t}\sum_{i=1}^{t} r_i^2-\frac{(2t-1)(2t+1)}{24}$$
%
%%\textcolor{red}{adjust the weight}
%and if we set $\alpha_i(\omega)=\#\lbrace s\in\omega, h_s=2t-i, \varepsilon_s=1\rbrace$:
%\begin{multline}\label{eq:thmsc}
%\prod_{\substack{s\in\omega\\h_s\in\cch(\omega)}}\frac{\tau(h_s-\varepsilon_s g)}{\tau(h_s)}=\prod_{i=1}^{2t-1}\left(\frac{\tau(-i)}{\tau(i)}\right)^{\alpha_i(\omega)}
%\prod_{1\leq i<j\leq t} \frac{\tau(r_i-r_j)}{\tau(j-i)} \frac{\tau(r_i+r_j)}{\tau(i+j)}.
%\end{multline}
%\end{thm}


Let $t\in \mathbb{N}^*$. The end of \cref{lit} associates a vector of integers $\phi(\omega)$ to any $t$-core $\omega$. 
Following \cite{CRV}, this vector can be ordered ``naturally'' (meaning that this order is inherited by that of the residue $\pmod{t}$) by setting:
\begin{equation}\label{eqn:phi}
n_k:=\left\lfloor \frac{\lambda_i-i}{t}\right\rfloor+1, \quad i=\min\lbrace \nu \mid \lambda_\nu \equiv k \pmod t\rbrace.
\end{equation}

The Littlewood decomposition, when restricted to $\ccdd$, also has interesting properties and can be stated as follows (\cite{GKS,MP} for instance):
\begin{numcases}
{\lambda\in\ccdd \mapsto} \left(\omega,\underline{\tilde{\nu}}\right)\in\ccdd_{(t)}\times\ccdd\times\ccp^{(t-1)/2}\quad\text{if $t$ is odd,}\notag  \\
\left(\omega,\underline{\tilde{\nu}},\mu\right)\in\ccdd_{(t)}\times\ccdd\times\ccp^{(t-2)/2}\times\ccsc\quad\text{if $t$ is odd.}\label{todd}
\end{numcases}


This symmetrical behaviour of $\ccdd_{(2t+2)}$ partitions yields some additional conditions on the associated vector of integers. These have already been studied by Garvan--Kim--Stanton \cite{GKS} but are stated here in a slightly different way. Let $\omega\in \ccdd_{(2t+2)}$ and $\phi(\omega)=(n_0,n_1,\dots,n_{2t+1})$ be as above. Symmetries of $\ccdd$ ensure that $n_0=0$, and $n_i=-n_{2t+2-i}$ for all $i \in \lbrace 1,\dots,2t+1\rbrace$, which in particular implies that $n_t=0$. By \eqref{eq:gks}, we then have:
\begin{align}
\lvert \omega \rvert &=(t+1)\sum_{i=0}^{2t+1}n_i^2+\sum_{i=0}^{2t+1}in_i=2\left((t+1)\sum_{i=0}^{t-1}n_i^2+\sum_{i=0}^{t-1}(i-t-1)n_i)\right)\label{eq:gkddpair}
%&=\sum_{i=0}^{t-1}((t+1)n_i^2+in_i)+\sum_{i=t+1}^{2t+1}((t+1)n_i^2+in_i)\\
%&=\sum_{i=0}^{t-1}((t+1)n_i^2+in_i)+\sum_{i=0}^{t-1}((t+1)n_{2t+2-i}^2+(2t+2-i)n_{2t+2-i})\\
%&=\sum_{i=0}^{t-1}((t+1)n_i^2+in_i)+\sum_{i=0}^{t-1}((t+1)(-n_{i})^2-(2t+2-i)n_{i})\\
%&=2\left((t+1)\sum_{i=0}^{t-1}n_i^2+\sum_{i=0}^{t-1}(i-t-1)n_i)\right)
\end{align}
\noindent So with this formalism one can recover the vector of integers given in \cite{GKS} from the Littlewood decomposition together with the word decomposition. When developing \eqref{eq:macdostart} with respect of powers of $T$, half of the previous quadratic form appear to the exponent.
\begin{ex}
\cref{fig:wordddcore} above illustrates this for a $\ccdd$ when $t=6$. The arrows are sorted in six different colors, each of them corresponding to a fixed residue $\pmod 6$ of the index of the corresponding word of $\omega$.
The word corresponding to $\omega$ writes as follows:
\begin{center}
$s(\omega)=\cdots \color{blue}{0} \color{brown}{0}  \color{RoyalPurple}{0} \color{purple}{0} \color{teal}{0} \color{purple}{0}\color{blue}{1} \color{brown}{0}  \color{RoyalPurple}{0} \color{purple}{0} \color{teal}{0} \color{purple}{0} \color{blue}{1} \color{brown}{0}  \color{RoyalPurple}{0} \color{purple}{1} \color{black}|\color{teal}{1} \color{purple}{0} \color{blue}{1} \color{brown}{1}  \color{RoyalPurple}{0} \color{purple}{1} \color{teal}{1} \color{purple}{1} \color{blue}{1} \color{brown}{1}  \color{RoyalPurple}{0} \color{purple}{1} \color{teal}{1} \color{purple}{1}
\color{blue}{1} \color{brown}{1}  \color{RoyalPurple}{1} \color{purple}{1} \color{teal}{1} \color{purple}{1}\color{black}\cdots.$
\end{center}


By extracting the subwords of fixed residue $\pmod 6$, we obtain:
\begin{align*}
&s\left(w_{0}\right)=\cdots \color{teal} 000 \color{black}| \color{teal}111\color{black}\cdots, \quad
s\left(w_{1}\right)=\cdots \color{purple} 000 \color{black}| \color{purple}011\color{black}\cdots \quad
s\left(w_{2}\right)=\cdots \color{blue} 011 \color{black}| \color{blue}111\color{black}\cdots\\
&s\left(w_{3}\right)=\cdots \color{brown} 000 \color{black}| \color{brown}111\color{black}\cdots, \quad
s\left(w_{4}\right)=\cdots \color{RoyalPurple} 000 \color{black}| \color{RoyalPurple}001\color{black}\cdots, \quad
s\left(w_{5}\right)=\cdots \color{purple} 001 \color{black}| \color{purple}111\color{black}\cdots
\end{align*}

so that $(\color{teal}{n_0},\color{purple}{n_1},\color{blue}{n_2},\color{brown}{n_3},\color{RoyalPurple}{n_4},\color{purple}{n_5}\textcolor{black}{)=(}\color{teal}{0},\color{purple}{1},\color{blue}{-2},\color{brown}{0},\color{RoyalPurple}{2},\color{purple}{-1} \textcolor{black}{)\in\bbbz^6.}$

The properties of symmetry of any $\ccdd$ partition along its main diagonal yields the following restriction, as mentioned in \cite{GKS}:
\begin{center}
$ \omega \in\ccdd_{(6)}\longleftrightarrow (\color{purple}{1},\color{blue}{-2}\textcolor{black}{)\in\bbbz^2.}$
\end{center}
\end{ex}
Similarly, the symmetrical behaviour of $\ccsc_{(2t)}$ partitions yields additional conditions on the associated vector of integers: let $\omega\in \ccsc_{(2t)}$ and $\phi(\omega)=(n_0,n_1,\dots,n_{2t-1})$, then $n_i=-n_{2t-1-i}$ for all $i \in \lbrace 0,\dots,2t-1\rbrace$, in particular $n_t=0$. By \eqref{eq:gks}, we have:
\begin{align}
\lvert \omega \rvert &=(2t)/2\sum_{i=0}^{2t-1}n_i^2+\sum_{i=0}^{2t-1}in_i=\sum_{i=0}^{t-1}((2t)n_i^2+(2(i-t)-1)n_i).\label{eq:gkscimpair}
%&=\sum_{i=0}^{t-1}((2t+1)/2n_i^2+in_i)+\sum_{i=t+1}^{2t}((2t+1)/2n_i^2+in_i)\\
%&=\sum_{i=0}^{t-1}((2t+1)/2n_i^2+in_i)+\sum_{i=t+1}^{2t}((2t+1)/2n_{2t-i}^2+(2t-i)n_{2t+1-i})\\
%&=\sum_{i=0}^{t-1}((2t+1)/2n_i^2+in_i)+\sum_{i=t+1}^{2t+1}((2t+1)/2(-n_i))^2-(2t-i)n_{i})\\
%&=\sum_{i=0}^{t-1}((2t+1)n_i^2+(2i-2t)n_i)
\end{align}

%\subsection*{From Garvan--Kim--Stanton decomposition to Macdonald $V_{g,t}$-codings}
The ordered vector of integers $\phi(\omega)$ still lacks of some properties to be the combinatorial tool one needs to prove Theorems \ref{prop:MacdotypeC}, \ref{NOC} and \ref{thm:DDpair}. These properties are the one satisfied by vectors of integers called $V_{g,t}$-codings (defined in \cite{Wmac}). The goal here is to attach a $V_{g,t}$-coding to any $\phi(\omega)$. Before going any further, let us motivate \cref{def:vcoding} below, with the following observation: making use of properties of $q$-series together with the Weyl group action on the affine root systems, Stanton provides a new proof of the Macdonald identities \cite{Stanton}. Sums that appear in these identities involve vectors of integers, so that the existence of $\phi$, which is a bijection between partitions and vectors of integers, leads us to rephrase Stanton's work. From this perspective, \cite[Proposition 3.7]{Stanton} that focuses on type $\tilde{C}_t$ establishes why only partitions in $\ccdd_{(2t+2)}$ are to be considered on type $\tilde{C}_t$: they are actually the only ones whose associated vector of integers contribute in a non-trivial way to the sum in the Macdonald identity for the type $\tilde{C}_t$.  

As this extended abstract only focuses on results for types $\tilde{C}_t$ and $\tilde{C}_t^{\vee}$, \cref{def:vcoding} below restricts to these types. Both the general definition (of \cite{Wmac}) and \cref{def:vcoding} involve a parameter $g$ of the affine root system defined by the equality $M=g\Lambda$ in \cite[p.$134$]{Mac}. In particular $g = 2t+2$ if the affine algebra considered is of type $\tilde{C}_t$ and $ g = 2t$ if it is of type $\tilde{C}_t^{\vee}$. 
%Note that $g=\begin{cases} h^{\vee}

\begin{definition}\label{def:vcoding}
Let $t \in \mathbb{N}^*$ and $g \in \mathbb{N}^*$ such that $t\leq g$, then set $\lambda \in \ccp$ and let $s(\lambda)=(c_k)_{k\in\bbbz}$ be its corresponding binary word. For $i\in\lbrace 0,\dots,g-1\rbrace$, define $\beta_i:=\max\lbrace (k+1)g+i\mid c_{kg+i}=0\rbrace$. Let $\sigma:\lbrace 1,\dots,g\rbrace\rightarrow \lbrace 0,\dots,g-1\rbrace$ be the unique permutation such that $\beta_{\sigma(1)}>\dots>\beta_{\sigma(g)}$. The vector $\mathbf{v}:=(\beta_{\sigma(1)},\dots,\beta_{\sigma(t)})$ is called the $V_{g,t}$-coding corresponding to $\lambda$.
\end{definition}


%\begin{definition}\label{def:vcoding}
%Let $\omega\in\mathcal{A}_{(g)}$ be a $g$-core. Set $\phi(\omega)=(n_0,\dots,n_{g-1})$ as defined by \eqref{eqn:phi}. Then there exists a unique $\sigma:= \lbrace 1,\dots, t \rbrace  \to  \lbrace 0,\dots,g-1\rbrace$ such that $n_{\sigma(i)}g+\sigma(i)>n_{\sigma(j)}g+\sigma(j)$ for $i<j$ and $i\in \{1,\dots, t\}$. This defines $v_{i}=n_{\sigma(i)}t+\sigma(i)$ for any $i \in \{1,\dots, t\}$ and the resulting vector of integers $(v_1,\dots,v_{t})$ is a $V_{g,t}$-coding.
%\end{definition}

Note that a notion of $V_{t}$-coding introduced by Dehaye--Han in \cite{HD} to connect Macdonald identities for type $\tilde{A}_{t-1}$ and $t$-core partitions corresponds to the $V_{t,t}$-coding. We can now state \cref{thm:DDpair} and provide the key technical results required to prove it.
%The definition of $V_{g,t}$-coding generalises this notion: it reduces to their $V_t$-coding when applied to type $\tilde{A}_{t-1}$. 
 
 \begin{thm}\label{thm:DDpair}
Set $t$ a positive integer and $g=2t+2$. Let $\omega\in \ccdd_{(2t+2)}$ and $\mathbf{v}\in\bbbz^{t}$ its associated $V_{g,t}$-coding, and set $r_i=v_i-t-1$ for any $i\in\lbrace1,\dots,t\rbrace$. Then we have
\begin{equation*}\label{eq:ddpoids}
\lvert \omega\rvert = \frac{1}{g}\sum_{i=1}^{t} r_i^2-\frac{(g/2-1)(g-1)}{12},
\end{equation*}
%\textcolor{red}{adjust the weight}
and setting $\alpha_i(\omega):=\#\lbrace u\in\omega, h_u=g-i, \varepsilon_u=1\rbrace$, and for any function $\tau:\bbbz\rightarrow F^{\times}$, where $F$ is a field, we also have
\begin{equation*}\label{eq:thmdd}
%\prod_{\substack{s\in\omega\\h_s\in\cch(\omega)}}
\prod_{s\in\omega}\frac{\tau(h_s-\varepsilon_s g)}{\tau(h_s)}=\prod_{i=1}^{g-1}\left(\frac{\tau(-i)}{\tau(i)}\right)^{\alpha_i(\omega)}
\prod_{i=1}^{t}\frac{\tau(r_i)}{\tau(i)}\prod_{1\leq i<j\leq t} \frac{\tau(r_i-r_j)}{\tau(j-i)} \frac{\tau(r_i+r_j)}{\tau(g-i-j)}.
\end{equation*}

%Let $\omega\in \ccdd_{(2t+2)}$ and $(v_1,\dots,v_{t})\in\bbbz^{\lfloor t\rfloor}$ its associated $V_{g,t}$-coding, and set $r_i = v_i -t-1$ for any $i\in \{1, \dots,t\}$. Then for any function $\tau$ defined on $\bbbz$ we have:
%$$\lvert \omega\rvert = \frac{1}{2t+2}\sum_{i=1}^{t} r_i^2-\frac{t(2t+1)}{12}$$
%
%%\textcolor{red}{adjust the weight}
%and if we set $\alpha_i(\omega)=\#\lbrace s\in\omega, h_s=(2t+2)-i, \varepsilon_s=1\rbrace$:
%\begin{multline}\label{eq:thmdd}
%\prod_{\substack{s\in\omega\\h_s\in\cch(\omega)}}\frac{\tau(h_s-\varepsilon_s (2t+2))}{\tau(h_s)}=\prod_{i=1}^{2t+1}\left(\frac{\tau(-i)}{\tau(i)}\right)^{\alpha_i(\omega)}
%\prod_{i=1}^{t}\frac{\tau(r_i)}{\tau(i)}\prod_{1\leq i<j\leq t} \frac{\tau(r_i-r_j)}{\tau(j-i)} \frac{\tau(r_i+r_j)}{\tau(i+j)}.
%\end{multline}
\end{thm}

To show \cref{thm:DDpair} one needs to associate bijectively a vector of integers to any partition. For technical and conceptual reasons (respectively facilitating inductive proofs and interpreting the determinants as Schur functions) the involved vectors need to be ordered. The $V_{g,t}$-codings introduced so far are natural candidate, but one still need to show that they correspond bijectively to partitions. This is the purpose of \cref{prop:core_vcoding}.
\begin{proposition}\label{prop:core_vcoding}
Let $t$ be a positive integer. Any $\lambda$ in one of the sets $\ccdd_{(2t+2)}$ and $\ccsc_{(2t)}$ is in bijective correspondence with its $V_{g,t}$-coding, where $g$ is the index of the corresponding set.
\end{proposition}
 
Let $s(\omega)$ be the word corresponding to $\omega$ via the word decomposition. The $V_{g,t}$-coding associated to $\omega$ by \cref{prop:core_vcoding} is immediately given when reading $s(\omega)$ from right to left: for instance $v_1$ is the index of the first ``$1$'' in the subword $\pmod{g}$ of $s(\omega)$ that contains the last ``$0$''.
%\textcolor{red}{sans doute à virer
%Now \begin{definition}\label{def:t_inter}
%Let $t$ be a strictly positive integer. Let $\lambda\in \mathcal{A}$
%\end{definition}}
\cref{lem:maxhook} allows to prove \cref{thm:DDpair} by induction on the length of the Durfee square of the partition. Following the same philosophy as in \cref{lem:indices,lem:durf}, it enumerates the boxes in the largest hook by means of their pair of indices. Ultimately, and as illustrated in \cref{fig:wordddcore} in the red and blue shaded areas, let $s$ be a box of $\lambda$ in the first hook. Then $\varepsilon_s=1$, respectively $\varepsilon_s=-1$, implies $j_s=v_1-g$, respectively $i_s=-v_1+g$ and $j_s<v_1-g$. 

We define the \textit{$g$-intervals} as as $\I_{m,M}^{g,+}:=\lbrace k\in \bbbz \mid m\leq k<M, k\equiv m \pmod g\rbrace$ and $\I_{m,M}^{g,-}:=\lbrace l\in \bbbz \mid m<l\leq M, l\equiv M \pmod g\rbrace$. This notion is of particular interest in our case ever since we are trying to enumerate hook lengths with a fixed residue $\pmod g$. The key ingredient of the proof of \cref{thm:DDpair} is that $g$-intervals allow us to transform products involving generic functions $\tau$ into telescopic factors.


\begin{lem}\label{lem:maxhook}
Set $\omega\in \ccdd_{(2t+2)}$ so that $g = 2t+2$ and let $(v_1,\dots,v_{t})\in\bbbz^t$ be its associated $V_{g,t}$-coding. Then the biggest hook of $\omega$, denoted by $H_1$, corresponds to the collection of boxes $\cch_{1,+}\cup\cch_{1,-}$ where $\cch_{1,+}$ is the set of indices of boxes $s$ in the first hook such that $\varepsilon_s=1$:
\begin{equation*}
\cch_{1,+}=\I_{-v_1+g,v_1-g}^{g,+}  \cup \I_{0,v_1-g}^{g,+} \cup\I_{g/2,v_1-g}^{g,+}
 \displaystyle\bigcup_{i=2}^{t}\left(\I_{v_i,v_1-g}^{g,+}\cup \I_{-v_i+g,v_1-g}^{g,+}\right),
 \end{equation*}
and $\cch_{1,-}$ is the set of indices of boxes $s$ in the first hook such that $\varepsilon_s=-1$:
\begin{equation*}
\cch_{1,-}= \I_{-v_1+g,v_1-2g}^{g,-} \cup\I_{-v_1+g,-g}^{g,-}\cup\I_{-v_1+g, -g/2}^{g,-} \displaystyle\bigcup_{i=2}^{t}\left(\I_{-v_1+g,v_i-g}^{g,-}\cup \I_{v_1-g,-v_i}^{g,-}\right).
\end{equation*}

%$\lbrace(k,l) \mid k=-v_1+g \text{ and } \exists (i,a) \in\lbrace 2,\dots,t\rbrace\times \mid l=v_i \rbrace$
\end{lem}

Finally we will also need the following lemma to apply \cref{thm:DDpair} to Macdonald identities specializations.
\begin{lem}\label{lem:signschur}
Let $\omega\in\ccdd_{(2t+2)}$ and $s(\omega)=(c_k)_{k\in\bbbz}$. Let $u_i=(2t+2)n_i+i$ the index of the first letter ``$1$'' in the subword $(c_{i+(2t+2)k})_{k\in\bbbz}$, where $(n_i)=\phi^{-1}(\omega)$.
Let $(v_i)_{i\in\lbrace 1,\dots t\rbrace}$ be a $V_{g,t}$-coding. Then $\sigma: \lbrace 1,\dots,t\rbrace\rightarrow\lbrace 0,\dots 2t+1\rbrace$ of \cref{def:vcoding} is such that that $v_i=u_{\sigma(i)}$ for $i \in \{1, \dots t\}$. We have that:
\begin{equation*}
\lvert H_+\rvert=\#\lbrace s\in\omega, h_s<2t+2, \varepsilon_s=1\rbrace\equiv d_\omega+\sgn(\sigma)\pmod{2}.
\end{equation*}
\end{lem}
%We also have the following lemma for $\ccsc_{(2t)}$
%\begin{lem}\label{lem:signschursc}
%Let $\omega\in\ccsc_{(2t)}$ and $s(\omega)=(c_k)_{k\in\bbbz}$. Let $u_i=2tm_i+i-1$ the index of first letter ``$1$'' in the subword $(c_{i+2tk})_{k\in\bbbz}$.
%Let $(v_i)_{i\in\lbrace 1,\dots t\rbrace}$ be a $V_{g,t}$-coding. Then $\sigma: \lbrace 1,\dots,t\rbrace\rightarrow\lbrace 0,\dots 2t-1\rbrace$ of \cref{prop:core_vcoding} is such that $v_i=u_{\sigma(i)}$ for $i \in \{1, \dots, t\}$. 
%We have that:
%\begin{equation}
%\lvert H_+\rvert=\#\lbrace s\in\omega, h_s<2t, \varepsilon_s=1\rbrace\equiv \sgn(\sigma)\pmod{2}
%\end{equation}
%\end{lem}


\section{Sketch of proof of \cref{prop:MacdotypeC,NOC}}\label{sketch}
 We first expand the right-hand side of \eqref{eq:macdostart} and we extract the terms in $T$. The power of $T$ corresponds to \eqref{eq:gkddpair}. By changes of variables and \cref{lem:signschur}, we are able to take $\omega=\phi^{-1}({\bf m})$ and derive \cref{prop:MacdotypeC}.



%and $\textsf{so}$ is the classical odd orthogonal Schur function as defined for instance in \cite{FH,AyyerKumari}.
%\section{From Macdonald identities to $q$-Nekrasov--Okounkov formulas}

%\section{Sketch of proof of \cref{NOC}}

We now derive \cref{NOC} for $\ccdd$. In order to do so, we start by proving the equality setting $u=q^t$ for any strictly positive integer $t$ on both sides. By technical manipulations on products, the right hand side of \eqref{eq:noc} is exactly that of \cref{prop:MacdotypeC} when $x_i=q^i$ which completes the first step of the proof.

 The left-hand side of \eqref{eq:noc} can be obtained by setting $\tau(x)=1-q^x$ in \cref{thm:DDpair} and  multiplying the resulting expression by the product of hook lengths on the main diagonal $\Delta$. We prove that:
 \begin{equation}\label{eq:schurinter}
\sp_{\mu}(q,q^2,\dots,q^t)=(-1)^{\lvert H+\rvert} q^{(t+1)d_\lambda}\prod_{s\in\lambda}\frac{1-q^{h_s-(2t+2)\varepsilon_s}}{1-q^{h_s}}\prod_{s\in\Delta}\frac{1+q^{t+1+h_s/2}}{1+q^{-t-1+h_s/2}}.
 \end{equation}
The proof of the equality above is done by induction on $r_1=\max_{1\leq i\leq t}(r_i)$ as defined in \cref{thm:DDpair}. First remark that the product over elements of $\Delta$ on the right-hand side of \eqref{eq:schurinter} is a telescopic product equal to $\prod_{i=1}^t (1+q^{r_i})/(1+q^i)$. On the left-hand side of \eqref{eq:schurinter}, the quotient of determinants $\det(q^{ir_j}-q^{-ir_j})/\det(q^{i(t+1-j)}-q^{-i(t+1-j)})$ can be computed by noting that it is equal to $\det({(q^{r_j})}^{i}-{(q^{r_j})}^{-i})/\det({(q^{t+1-j})}^i-{(q^{t+1-j})}^{-i})$. Finally we conclude by checking that both sides of the equation verify the same initial value and the same induction property.

The last step of the proof is to check that both sides of \eqref{eq:noc} are Laurent polynomials in the variable $u$. An argument of polynomiality that \eqref{eq:noc} holds for any $u$ then allows one to conclude.

%\acknowledgements{\lipsum[20]}

%% 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



\end{document}