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\usepackage{amscd,amsfonts,epic,eepic,bbm,graphicx,ytableau}
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\newcommand{\la}{\lambda}

\Crefname{prop}{Proposition}{Propositions}
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%% define your title in the usual way
\title[A bijection between self-conjugate with same disparity and ordinary partitions]{A bijection between ordinary partitions and self-conjugate partitions with same disparity} 
%and counting simultaneous cores}

%% define your authors in the usual way
%% use \addressmark{1}, \addressmark{2} etc for the institutions, and use \thanks{} for contact details
\author[H. Cho, J. Huh, and J. Sohn]{Hyunsoo Cho\thanks{\href{coconut@yonsei.ac.kr}{coconut@yonsei.ac.kr}}\addressmark{1}, JiSun Huh\thanks{\href{hyunyjia@g.skku.edu}{hyunyjia@g.skku.edu} JiSun Huh was supported by NRF grant \#2015R1D1A1A01057476.}\addressmark{2}, \and Jaebum Sohn\thanks{\href{jsohn@yonsei.ac.kr}{jsohn@yonsei.ac.kr} Jaebum Sohn was supported by NRF grant \#2017R1A2B4009501.}\addressmark{1}}

%% then use \addressmark to match authors to institutions here
\address{\addressmark{1}Department of Mathematics, Yonsei University, Seoul 03722, Republic of Korea \\ \addressmark{2}Applied Algebra and Optimization Research Center, Sungkyunkwan University, Suwon 16420, Republic of Korea}

%% put the date of submission here
\received{Nov. 14th}

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\revised{Mar. 20th}

%% put your English abstract here, or comment this out if you don't have one yet
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\abstract{We give a bijection between the set of ordinary partitions and that of self-conjugate partitions with some restrictions. Also, we show the relation between hook lengths of a self conjugate partition and its corresponding partition via the bijection. As a corollary, we give new combinatorial interpretations for the Catalan number and the Motzkin number in terms of self-conjugate simultaneous core partitions.}

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


%% put your keywords here, or comment this out if you don't have them yet
\keywords{partition, self-conjugate partition, hook length, simultaneous core partition}

%% 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
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\begin{document}

\maketitle
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\section{Introduction}



Let $\la=(\la_1,\la_2,\dots,\la_{\ell})$ be a partition of $n$. The {\it Young diagram} of $\la$ is a collection of $n$ boxes in $\ell$ rows with $\la_i$ boxes in row $i$. We label the columns of the diagram from left to right starting with column 1. The box in row $i$ and column $j$ is said to be in position $(i,j)$. For example, the Young diagram for $\la=(5,4,2,1)$ is below.
\begin{center}
\scriptsize{
\begin{ytableau}
~&~&~&~&~ \\
~&~&~&~ \\
~&~ \\
~
\end{ytableau}}
\end{center}
For the Young diagram of $\la$, the partition $\la '=(\la_1 ',\la_2 ',\dots, \la_{\la_1} ')$ is called the {\it conjugate} of $\la$, where $\la'_j$ denotes the number of boxes in column $j$ of $\la$. For each box in its Young diagram, we define its {\it hook length} by counting the number of boxes directly to its right or below, including the box itself. Equivalently, for the box in position $(i,j)$, the hook length of $\la$ is defined by 
$$h_\la(i,j)=\la_i+\la'_j-i-j+1.$$ 
For example, the hook lengths in the first row above are $8$, $6$, $4$, $3$, and $1$, respectively. We denote $h_{\la}(i,j)$ by $h(i,j)$ when $\la$ is clear.

For a positive integer $t$, a partition $\la$ is called a {\it $t$-core} if none of its hook lengths are multiples of $t$. 
The number of $t$-core partitions of $n$ is denoted by $c_t(n)$. The study of core partitions arises from the representation theory of the symmetric group $S_n$. (See \cite{JK} for details.) Many researches on core partitions are being made through various ways, such as representation theory and analytic methods--see, for example,  \cite{BN2,BY,Chen2,GO,HS,Olsson,Olsson2}.

A partition whose conjugate is equal to itself is called {\it self-conjugate}. Let $sc_t(n)$ denote the number of $t$-core partitions of $n$ which are self-conjugate. 
A number of properties of self-conjugate core partitions have been found and proved. (See \cite{BDFKS,BN3}.)

Garvan, Kim, and Stanton \cite{GKS} found the generating functions of $c_t(n)$ and $sc_t(n)$;

\begin{equation}\label{eqn:core}
\sum_{n=0}^{\infty}c_t(n)q^n=\prod_{n=1}^{\infty}\frac{(1-q^{nt})^t}{1-q^n}.
\end{equation}

\begin{equation}\label{eqn:selfcore}
\sum_{n=0}^{\infty}sc_{2t}(n)q^n=\prod_{n=1}^{\infty} (1-q^{4nt})^t (1+q^{2n-1}).
\end{equation} 

Now by combining (\ref{eqn:core}), (\ref{eqn:selfcore}), and Gauss' well-known identity
$$ \sum_{n=0}^{\infty}q^{\frac{n(n+1)}{2}}=\prod_{n=1}^{\infty}\frac{1-q^{2n}}{1-q^{2n-1}}=\prod_{n=1}^{\infty}(1-q^{2n})(1+q^n),$$
%=\prod_{n=1}^{\infty}\frac{(1-q^{2n})^2}{1-q^n}
one can obtain the following proposition which shows a relation between two numbers $c_t(n)$ and $sc_t(n)$.

\begin{prop}\label{prop:core}
$$\sum_{n=0}^{\infty} sc_{2t}(n) q^n =\left(\sum_{n=0}^{\infty} c_t(n) q^{4n}\right) \left(\sum_{n=0}^{\infty} q^{\frac{n(n+1)}{2}}\right).$$
\end{prop}


Also, if we let $p(n)$ be the number of partitions of $n$ and let $sc(n)$ be the number of self-conjugate partitions of $n$, then $p(n)$ and $sc(n)$ have a similar relation. 

\begin{prop}\label{prop:nocore}
$$\sum_{n=0}^{\infty}  sc(n) q^n =\left(\sum_{n=0}^{\infty} p(n) q^{4n}\right) \left(\sum_{n=0}^{\infty} q^{\frac{n(n+1)}{2}}\right).$$
\end{prop}

In this paper, we construct a bijection between the set of ordinary partitions and the set of self-conjugate partitions with the same disparity. Our bijection can be obtained by combining Wright's bijection for proving Jacobi triple product identity and a bijection between self-conjugate partitions and diagonal sequence pairs. (See \cite{Pak, Wright}.) The bijection leads to a new combinatorial proof for \cref{prop:nocore}. Also, from the bijection, we find a relation between hook lengths of a self-conjugate partition and that of the corresponding partition. (See \cref{thm:second}.) As a result of this relation, we can also reprove \cref{prop:core}. Another result comes from \cref{thm:second} is \cref{prop:simul} which is a generalization of \cref{prop:core}. Here, we use the notation of a $(t_1,t_2,...,t_p)$-core partition if it is simultaneously a $t_1$-core,\dots, and a $t_p$-core.

\begin{prop}\label{prop:simul}
 Let $c_{(t_1,t_2,\dots,t_p)}(n)$ be the number of $(t_1,t_2,\dots,t_p)$-core partitions of $n$ and $sc_{(t_1,t_2,\dots,t_p)}(n)$ be the number of self-conjugate $(t_1,t_2,\dots,t_p)$-core partitions of $n$. Then we have
$$\sum_{n=0}^{\infty} sc_{(2t_1,2t_2,...,2t_p)}(n) q^n 
=\left(\sum_{n=0}^{\infty}c_{(t_1,t_2,\dots,t_p)}(n)q^{4n}\right)\left(\sum_{n=0}^{\infty}q^{\frac{n(n+1)}{2}}\right).$$
\end{prop}

At the end of this paper, new interpretations of the Catalan number and the Motzkin number in terms of self-conjugate simultaneous core partitions is given (see \cref{cor:catalanmotzkin}) as a corollary of \cref{prop:simul}.\\


This paper is organized as follows. In \cref{sec:pre}, we define the disparity and introduce new classification of the set of self-conjugate partitions. In \cref{sec:bijections}, we give a bijection between the set of ordinary partitions and that of self-conjugate partitions with the same disparity. In \cref{sec:hook}, we explain the relation between even hook lengths in a self-conjugate partition and hook lengths in the corresponding partition via the bijection. Furthermore, we give some new results on counting self-conjugate simultaneous cores.


%===================================================================================

\section{Self-conjugate partitions with same disparity}\label{sec:pre}

%===================================================================================

In this section, we give some basic notions and introduce a set partition of the set $\mathcal{SC}$ of self-conjugate partitions. 

Let $\la$ be a partition.  We often use the notation $\delta_i$ for the hook length $h(i,i)$ of the $i$th box on the main diagonal. The set $D(\la)=\{\delta_i~:~i=1,2,\dots\}$ is called the {\it set of main diagonal hook lengths of $\la$}. It is clear that if $\la$ is self-conjugate, then $D(\la)$ determines $\la$, and elements of $D(\la)$ are all distinct and odd. Hence, for a self-conjugate partition $\la$, $D(\la)$ can be partitioned into the following two subsets;
\begin{eqnarray*}
D_1(\la)&=&\{\delta_{i}\in D(\la)~:~\delta_{i}\equiv 1\pmod{4}\},\\
D_3(\la)&=&\{\delta_{i}\in D(\la)~:~\delta_{i}\equiv 3\pmod{4}\}.
\end{eqnarray*}

\begin{example}\label{ex:hook} 
Let $\la=(4,4,4,3)$ be a self-conjugate partition of $15$. \cref{fig:example} shows its Young diagram and the hook lengths. The set $D(\la)=\{7,5,3\}$ of main diagonal hook lengths is partitioned into $D_1(\la)=\{5\}$ and $D_3(\la)=\{7,3\}$.

\begin{figure}[ht!]
\centering
\small{\begin{ytableau}
{\bf 7}&6&5&3 \\ 6&{\bf 5}&4&2 \\ 5&4&{\bf 3}&1 \\ 3&2&1
\end{ytableau}}
\caption{The Young diagram of a self-conjugate partition and its hook lengths}\label{fig:example}
\end{figure}
\end{example}

The set of hook lengths of boxes in the first column of the Young diagram of $\la$ is called the {\it beta-set} of $\la$ and denoted by $\beta(\la)$. 

Let $\mathcal{SC}(n)$ be the set of self-conjugate partitions of $n$ and $\la\in\mathcal{SC}(n)$. Using the value $|D_1(\la)|-|D_3(\la)|$, we split $\mathcal{SC}(n)$ as follows: For $m,n\geq 0$, we define a set $\mathcal{SC}^{(m)}(n)$ by
$$\mathcal{SC}^{(m)}(n)=\{\la\in\mathcal{SC}(n)~:~|D_1(\la)|-|D_3(\la)|=(-1)^{m+1} \left\lceil \frac{m}{2} \right\rceil\}.$$
We note that for a self-conjugate partition $\la$, if $|D_1(\la)|-|D_3(\la)|=k$ for $k\geq1$, then $\la\in\mathcal{SC}^{(2k-1)}(n)$.
Otherwise, if $|D_1(\la)|-|D_3(\la)|=-k$ for $k\geq0$, then $\la\in\mathcal{SC}^{(2k)}(n)$.
Therefore, $\mathcal{SC}(n)=\bigcup\limits_{m=0}^{\infty}\mathcal{SC}^{(m)}(n)$. \\

We use the notation $sc^{(m)}(n)$ for $|\mathcal{SC}^{(m)}(n)|$ and $\mathcal{SC}^{(m)}$ for $\bigcup_{n\geq 0} \mathcal{SC}^{(m)}(n)$.\\

For a partition $\la$, we define the {\it disparity} of $\la$ by 
$${\rm dp}(\la)=|\{(i,j)\in\la~:~h(i,j) \text{~is odd}\}|-|\{(i,j)\in\la~:~h(i,j) \text{~is even}\}|.$$ 

For example, for $\la=(4,4,4,3)$ given in \cref{ex:hook}, $|D_1(\la)|-|D_3(\la)|=-1$, and $\la$ is an element of $\mathcal{SC}^{(2)}(15)$. 
Moreover, the disparity of $\la$ is ${\rm dp}(\la)=9-6=3$. \\

It is not hard to show that each element of $\mathcal{SC}^{(m)}(n)$ has the same disparity.

\begin{prop} \label{prop:oddeven}
For $m\geq 0$, if $\la$ is in the set $\mathcal{SC}^{(m)}$, then its disparity ${\rm dp}(\la)$ is $\frac{m(m+1)}{2}$. 
\end{prop}



By \cref{prop:oddeven}, one may notice that the disparity of a self-conjugate partition is a triangular number $\frac{m(m+1)}{2}$, and the set of self-conjugate partitions with the disparity $\frac{m(m+1)}{2}$ is $\mathcal{SC}^{(m)}$. In fact, the disparity of any ordinary partition is a triangular number. 


%===================================================================================

\section{Bijections between $\mathcal{SC}^{(m)}$ and $\mathcal{P}$} \label{sec:bijections}

%===================================================================================

The set of partitions of $n$ is denoted by $\mathcal{P}(n)$, and the set of partitions is denoted by $\mathcal{P}$. In this section we construct bijections between two sets $\mathcal{SC}^{(m)}(4n+m(m+1)/2)$ and $\mathcal{P}(n)$ which play a key role throughout the paper.\\ 

Before constructing bijections, we give a notation.
For a self-conjugate partition $\la$, if
\begin{eqnarray*}
D_1(\la)&=&\{4a_1+1, 4a_2+1,\dots, 4a_r+1\},\\
D_3(\la)&=&\{4b_1-1, 4b_2-1,\dots, 4b_s-1\},
\end{eqnarray*}
we say that $\la$ has the {\it diagonal sequence pair} $((a_1,a_2,\dots,a_{r}),(b_1,b_2,\dots,b_{s}))$, where $a_1>a_2>\cdots>a_{r}\geq0$ and $b_1>b_2>\cdots>b_{s}\geq 1$. For convenience, we allow an empty sequence if $r$ or $s$ is equal to $0$.

For $\la=(4,4,4,3)\in \mathcal{SC}^{(2)}(15)$, its diagonal sequence pair is $((1),(2,1))$. \\

We note that if $((a_1,\dots,a_{r}),(b_1,\dots,b_s))$ is the diagonal sequence pair of a self-conjugate partition $\la\in \mathcal{SC}^{(m)}(4n+m(m+1)/2)$, then 
$$r-s+(-1)^m \left\lceil \frac{m}{2} \right\rceil =0$$ 
and 
$$4\left(\sum_{i=1}^{r}a_i +\sum_{j=1}^{s}b_j\right)+r-s=4n+\frac{m(m+1)}{2}.$$
 
Now, we are ready to construct our mapping.\\

\textbf{Mapping $\phi^{(m)}_n$ : $\mathcal{SC}^{(m)}(4n+m(m+1)/2)~\rightarrow~\mathcal{P}(n)$}\\

Let $\la\in\mathcal{SC}^{(m)}(4n+m(m+1)/2)$ be a self-conjugate partition with the diagonal sequence pair $((a_1,\dots,a_r),(b_1,\dots,b_s))$. We define $\phi^{(m)}_n(\la)$ by the partition $\mu=(\mu_1, \dots, \mu_{\ell})$ such that 
$$\mu_i=a_{i}+i+s-r \qquad \text{for} \quad i \leq r,$$ 

and $(\mu_{r+1}, \dots, \mu_{\ell})$ is the conjugate of the partition $\gamma=(b_1-s, b_2-s+1, \dots, b_{s}-1)$.

(We allow that $\gamma$ has some zero parts.) \\

In \cref{fig:2}, the diagram after deleting the shaded area is the Young diagram of $\mu$.

\begin{figure}[ht]
\centering
\begin{tikzpicture}[scale=.43]
\filldraw[fill=gray!30,color=gray!30] (0,0) rectangle (3,7);

\foreach \i in {1,2,3,4,5,6}
 \draw[dashed] (\i,0) -- (\i,7-\i); 

\draw[thick] (0,0) -- (0,7)
                    (0,0) -- (7.03,0)
                    (1,7) -- (15,7) -- (15,6) --(2,6)
                    (14,6) -- (14,5) --(3,5)
                    (12,5) -- (12,4) --(4,4)
                    (12,4) -- (12,3) --(5,3)
                    (11,3) -- (11,2) --(6,2)
                    (10,2) -- (10,1) --(7,1)
                    (10,1) -- (10,0) --(6,0)
                    (3,0) -- (3,-6) -- (4,-6) --(4,0)
                    (4,-5) -- (5,-5) -- (5,0)
                    (5,-5) -- (6,-5) -- (6,0)
                    (6,-4) -- (7,-4) -- (7,0);

\draw[<->] (-0.5,0) -- (-0.5,4);
\draw[<->] (-0.5,4) -- (-0.5,7);
\draw[<->] (0,7.5) -- (3,7.5);
\draw[<->] (3,7.5) -- (7,7.5);

\node at (-1,2) {\begin{turn}{90} 
$s$
\end{turn}};
\node at (-1,5.5) {\begin{turn}{90} 
$r-s$
\end{turn}};
\node at (1.5, 8) {$r-s$};
\node at (5, 8) {$s$};
\node at (8.5, 0.45) {$a_r$};
\node at (8.5, 1.45) {$a_{r-1}$};
\node at (8.5, 2.45) {$a_{r-2}$};
\node at (8.5, 3.7) {$\vdots$};
\node at (8.5, 4.45) {$a_{3}$};
\node at (8.5, 5.45) {$a_{2}$};
\node at (8.5, 6.45) {$a_{1}$};

\node at (3.5, -2) {\begin{turn}{90} 
$b_1-s$
\end{turn}};
\node at (4.5, -2) {\begin{turn}{90} 
$b_2-s+1$
\end{turn}};
\node at (5.55,-2) {$\dots$};
\node at (6.5, -2) {\begin{turn}{90} 
$b_s-1$
\end{turn}};

\foreach \i in {0,1,2,3,4,5,6}
 \draw[thick] (0,\i+1) -- (7-\i,\i+1) --  (7-\i,\i); 

\node at (7.5,-7) {i) $r > s$}; 
\end{tikzpicture}
\quad\qquad\begin{tikzpicture}[scale=.43]

\foreach \i in {1,2,3}
 \draw[dashed] (\i,0) -- (\i,3); 

\draw[thick] (0,0) -- (0,3) -- (14,3) -- (14,2) -- (0,2); 
\draw[thick] (12,2) -- (12,1) -- (0,1); 
\draw[thick] (10,1) -- (10,0) -- (0,0); 
\draw[thick] (0,0) -- (0,-10) -- (1,-10) --(1,0); 
\draw[thick] (1,-8) -- (2,-8) -- (2,0); 
\draw[thick] (2,-7) -- (3,-7) -- (3,0); 
\draw[thick] (3,-7) -- (4,-7) -- (4,3);
\draw[thick] (4,-5) -- (5,-5) -- (5,0);  
\draw[thick] (5,-4) -- (6,-4) -- (6,0);
\draw[thick] (6,-4) -- (7,-4) -- (7,1);    
\draw[thick] (6,1) -- (6,2);    
\draw[thick] (5,2) -- (5,3);    
\draw[dashed] (5,2) -- (5,0);  
\draw[dashed] (6,1) -- (6,0);  

\node at (8.5,0.45) {$a_r$};
\node at (8.5,1.7) {$\vdots$};
\node at (8.5,2.45) {$a_1$};

\node at (0.5, -2) {\begin{turn}{90} 
$b_1-s$
\end{turn}};
\node at (1.5, -2) {\begin{turn}{90} 
$b_2-s+1$
\end{turn}};
\node at (2.5, -2) {\begin{turn}{90} 
$b_3-s+2$
\end{turn}};
\node at (3.55,-2) {$\dots$};
\node at (4.5, -2) {\begin{turn}{90} 
$b_{s-2}-3$
\end{turn}};
\node at (5.5, -2) {\begin{turn}{90} 
$b_{s-1}-2$
\end{turn}};
\node at (6.5, -2) {\begin{turn}{90} 
$b_s-1$
\end{turn}};

\draw[<->] (-0.5,0) -- (-0.5,3);
\draw[<->] (0,3.5) -- (4,3.5);
\draw[<->] (4,3.5) -- (7,3.5);

\node at (2, 4) {$s-r$};
\node at (5.5, 4) {$r$};
\node at (-1,1.5) {\begin{turn}{90} 
$r$
\end{turn}};

\node at (7,-11) {ii) $r \leq s$}; 
\end{tikzpicture}
\caption{Graphical interpretations of mapping $\phi^{(m)}_n$}\label{fig:2}
\end{figure}


\begin{thm}\label{thm:bijective}
For nonnegative integers $m$ and $n$, the mapping $\phi_n^{(m)}$ is bijective.
\end{thm}

We define the bijection $\phi^{(m)}:\mathcal{SC}^{(m)} \rightarrow \mathcal{P}$ by $\phi^{(m)}_n(\la)$, for a partition $\la\in\mathcal{SC}^{(m)}$ of  $4n+\frac{m(m+1)}{2}$.
We say that $\mu$ is the {\it corresponding partition} of $\la$ when $\phi^{(m)}(\la)=\mu$. \\ 

We give two examples of the bijection $\phi^{(m)}$.

\begin{example} \label{ex:bijection}
We consider two self-conjugate partitions $\la$ and $\tilde{\la}$ with the set of main diagonal hook lengths $D(\la)=\{21,15,13,9,3,1\}$ and $D(\tilde{\la})=\{31,19,11,5\}$, respectively. 

\begin{itemize}
\item Since $D_1(\la)=\{21,13,9,1\}$ and $D_3(\la)=\{15,3\}$, $\la\in\mathcal{SC}^{(3)}$ and $((5,3,2,0),(4,1))$ is the diagonal sequence pair of $\la$. If we let $\mu$ be the partition $\phi_{14}^{(3)}(\la)$, then 
$$\mu_1=5+1-2=4,\quad \mu_2=3+2-2=3, \quad \mu_3=2+3-2=3, \quad \mu_4=0+4-2=2$$ 
and $(\mu_{5},\mu_6,\dots)$ is the conjugate of the partition $(4-2,1-2+1)$.\\
Therefore, $\mu=(4,3,3,2,1,1)$.
\item Since $D_1(\tilde{\la})=\{5\}$ and $D_3(\tilde{\la})=\{31,19,11\}$, $\tilde{\la} \in \mathcal{SC}^{(4)}$ and $((1),(8,5,3))$ is the diagonal sequence pair of $\tilde{\la}$. If we let $\tilde{\mu}$ be the partition $\phi_{14}^{(4)}(\tilde{\la})$, then $\mu_1=1+1+2=4$ and 
$(\mu_{2},\mu_3,\dots)$ is the conjugate of the partition $(8-3,5-3+1,3-3+2)$.\\ 
Therefore, $\tilde{\mu}=(4,3,3,2,1,1)$.
\end{itemize}

For given $\mu\in\mathcal{P}$ and $m\geq0$, we consider the following diagram to find $\la$ such that $\phi^{(m)}(\la)=\mu$. For convenience, even if $i\leq 0$, we set the $i$th column is the column on the left side of the $(i+1)$st column and the $i$th row is on the above of the $(i+1)$st row.
\begin{itemize}
\item For $m=2k-1$, we consider the diagram $\nu$ obtained from the Young diagram of $\mu$ by attaching $\frac{k(k-1)}{2}$ boxes on the left side such that $\nu$ has $\mu_i+k-i$ boxes in row $i$ for $i<k$ and $\mu_i$ boxes in row $i$ for $i\geq k$. 
Then, the number of (white) boxes $(i,j)$ in row $i$ such that $i-j<k$ is equal to $a_i$ and the number of (gray) boxes $(i,j)$ in column $j$ such that $i-j\geq k$ is equal to $b_j$. See the first diagram in \cref{fig:graphic} for $\mu=(4,3,3,2,1,1)$  and $m=3$.  
\item For $m=2k$, we consider the diagram $\nu$ obtained from the Young diagram of $\mu$ by attaching $\frac{k(k+1)}{2}$ boxes on the above such that $\nu$ has $k-i$ boxes in row $-i$ for $i=0,1,\dots,k-1$ and $\mu_i$ boxes in row $i$ for $i>0$. 
Then, the number of (white) boxes $(i,j)$ in row $i$ such that $i-j<-k$ is equal to $a_i$ and the number of (gray) boxes $(i,j)$ in column $j$ such that $i-j\geq -k$ is equal to $b_j$. See the second diagram in \cref{fig:graphic} for $\mu=(4,3,3,2,1,1)$  and $m=4$.   
\end{itemize}

\begin{figure}[ht]
\centering
\begin{tikzpicture}[scale=.42]
\filldraw[fill=gray!50,color=gray!50] (0,-2) rectangle (1,-6)
                                                   (1,-3) rectangle (2,-4)
                                                   (10,2) rectangle (11,-6)
                                                   (11,1) rectangle (12,-4)
                                                   (12,0) rectangle (13,-3);

\draw[ultra thick,color=gray!50] (-0.7,-0.5) -- (3.7,-0.5);
\draw[ultra thick,color=gray!50] (0.3,-1.5) -- (2.7,-1.5);
\draw[ultra thick,color=gray!50] (1.3,-2.5) -- (2.7,-2.5);

\draw[ultra thick,color=gray] (0.5,-2.3) -- (0.5,-5.7);
\draw[ultra thick,color=gray] (1.5,-3.3) -- (1.5,-3.7);

\draw[thick] (0,0) -- (0,-6) --(1,-6) -- (1,0)
                    (0,-4) -- (2,-4) -- (2,0) 
                    (0,-3) -- (3,-3) -- (3,0)
                    (0,-1) -- (4,-1) -- (4,0) -- (0,0)
                    (0,-2) -- (3,-2)
                    (0,-5) -- (1,-5)
                    (0,0) -- (-1,0) -- (-1,-1) -- (0,-1);
 
\draw[ultra thick,color=gray] (10.5,1.7) -- (10.5,-5.7);
\draw[ultra thick,color=gray] (11.5,0.7) -- (11.5,-3.7);
\draw[ultra thick,color=gray] (12.5,-0.3) -- (12.5,-2.7);

\draw[ultra thick,color=gray!50] (13.3,-0.5) -- (13.7,-0.5);

                  
\draw[thick] (10,0) -- (10,-6) --(11,-6) -- (11,0)
                    (10,-4) -- (12,-4) -- (12,0) 
                    (10,-3) -- (13,-3) -- (13,0)
                    (10,-1) -- (14,-1) -- (14,0) -- (10,0)
                    (10,-2) -- (13,-2)
                    (10,-5) -- (11,-5)
                    (10,0) -- (10,2) -- (11,2) -- (11,0)
                    (10,1) -- (12,1) -- (12,0);
               
\node at (1.5,-7.5) {$\updownarrow$};
\node at (12,-7.5) {$\updownarrow$};
\node at (1.5,-9) {$((5,3,2,0),(4,1))$};
\node at (12,-9) {$((1),(8,5,3))$};
\end{tikzpicture}
\caption{Graphical interpretations for odd $m$ and even $m$ of the bijection $\phi^{(m)}$}\label{fig:graphic}
\end{figure}

\end{example}
   

\begin{prop}\label{prop:size}    
For $m\geq 0$, the number of self-conjugate partitions of $n$ with the disparity $\frac{m(m+1)}{2}$ is
$$sc^{(m)}(n)=\begin{cases}
               p(k) & \text{if}~n=4k+\frac{m(m+1)}{2},\\
               0    & \text{otherwise}.
               \end{cases}$$
\end{prop}

By \cref{thm:bijective} and \cref{prop:size}, we have the following corollary and as a consequence of \cref{cor:first}, we have \cref{prop:nocore}.

\begin{cor}\label{cor:first}
For a nonnegative integer $m$, we have 
$$\sum_{\la\in\mathcal{SC}^{(m)}} q^{|\la|}=q^{\frac{m(m+1)}{2}}\sum_{\mu\in\mathcal{P}}q^{4|\mu|}.$$
\end{cor}





%===============================================================================  
  
\section{Properties of hook lengths of $\mathcal{SC}^{(m)}$}\label{sec:hook}

%=============================================================================== 

In this section we provide some properties of hook lengths of $\la\in \mathcal{SC}^{(m)}$. 

%-------------------------------------------------------------------------------

\subsection{Hook lengths of the first row or column}        
                        
%-------------------------------------------------------------------------------

For the partitions $\la\in \mathcal{SC}^{(m)}$ and $\mu=\phi^{(m)}(\la)$, we give a relation between their hook lengths in the first row or the first column. 

For a self-conjugate partition $\la$, we define the {\it half-even beta set} of $\la$ by
$$\beta_{e/2}(\la)=\{h(i,1)/2~:~h(i,1)~\text{is even},~1\leq i \leq \la_1\}.$$             
                
\begin{prop} \label{prop:evenhook}  
Let $\la\in\mathcal{SC}^{(m)}$ with $D(\la)=\{\delta_1,\dots,\delta_d\}$ and $\mu=\phi^{(m)}(\la)=(\mu_1,\dots,\mu_{\ell})$. Then the half-even beta set of $\la$ is
$$\beta_{e/2}(\la)=\begin{cases}
                   \beta(\mu') &\text{if}\quad \delta_1\in D_1(\la),\\
                   \beta(\mu)  &\text{if}\quad \delta_1\in D_3(\la).
                   \end{cases}$$  
\end{prop}            

   
  
\begin{example} Let $\la$, $\tilde{\la}$ be self-conjugate partitions we considered in \cref{ex:bijection}. We remind that $\phi^{(3)}(\la)=\phi^{(4)}(\tilde{\la})=\mu=(4,3,3,2,1,1)$. We note that $h_{\la}(1,1)=21\in D_{1}(\la)$ and $h_{\tilde{\la}}(1,1)=31\in D_{3}(\tilde{\la})$. As in \cref{prop:evenhook}, $\beta_{e/2}(\la)=\beta(\mu')=\{9,6,4,1\}$ and $\beta_{e/2}(\tilde{\la})=\beta(\mu)=\{9,7,6,4,2,1\}$. See \cref{fig:hookrelation} for the Young diagrams of $\mu$, $\la$, $\tilde{\la}$, and their hook lengths. 
\end{example}
\begin{figure}[ht!]
\small
\qquad
$\qquad\mu=~$\begin{ytableau}      
*(gray!65)9&*(gray!30)6&*(gray!30)4&*(gray!30)1\cr
*(gray!80)7&4&2\cr
*(gray!80)6&3&1\cr
*(gray!80)4&1\cr
*(gray!80)2\cr
*(gray!80)1\cr
\end{ytableau}
\qquad\qquad\qquad $\leftrightarrow$\qquad $\la=~$\begin{ytableau}
21&18&17&15&12&11&8&7&5&2&1\cr
*(gray!30)18&15&14&12&9&8&5&4&2\cr
17&14&13&11&8&7&4&3&1\cr
15&12&11&9&6&5&2&1\cr
*(gray!30)12&9&8&6&3&2\cr
11&8&7&5&2&1\cr
*(gray!30)8&5&4&2\cr
7&4&3&1\cr
5&2&1\cr
*(gray!30)2\cr
1\cr
\end{ytableau}\\

$\qquad\qquad\updownarrow$\\

$\qquad\qquad\tilde{\la}=~$\begin{ytableau}
31&25&21&18&15&14&12&11&9&8&7&5&4&3&2&1\cr
25&19&15&12&9&8&6&5&3&2&1\cr
21&15&11&8&5&4&2&1\cr
*(gray!80)18&12&8&5&2&1\cr
15&9&5&2\cr
*(gray!80)14&8&4&1\cr
*(gray!80)12&6&2\cr
11&5&1\cr
9&3\cr
*(gray!80)8&2\cr
7&1\cr
5\cr
*(gray!80)4\cr
3\cr
*(gray!80)2\cr
1\cr
\end{ytableau}
\caption{Hook length relations between corresponding partitions}\label{fig:hookrelation}
\end{figure}
  
%-------------------------------------------------------------------------------

\subsection{The relation between hook lengths of $\mathcal{SC}^{(m)}$ and $\mathcal{P}$}        
                        
%-------------------------------------------------------------------------------
  
We start this subsection by stating a proposition.
     
\begin{prop} \label{prop:delete} 
For $\la\in\mathcal{SC}^{(m)}$ with $D(\la)=\{\delta_1,\delta_2,\dots,\delta_d\}$, let $\bar{\la}$ be the self-conjugate partition with $D(\bar{\la})=\{\delta_i\in D(\la)~:~2\leq i \leq d\}$, and let $\mu=(\mu_1,\mu_2,\dots,\mu_{\ell})$ and $\bar{\mu}$ be the corresponding partitions of $\la$ and $\bar{\la}$, respectively. If $\mu=(\mu_1,\mu_2,\dots,\mu_{\ell})$, then 
$$\bar{\mu}=\begin{cases}
            (\mu_2,\mu_3,\dots,\mu_{\ell}) & \text{if}\quad \delta_1\in D_1(\la)\\  (\mu_1-1,\mu_2-1,\dots,\mu_{\ell}-1) & \text{if}\quad \delta_1\in D_3(\la)
                \end{cases} $$ 
\end{prop}

One may notice that there are more relations between hook lengths of corresponding partitions from \cref{fig:hookrelation}. By using \cref{prop:evenhook,prop:delete}, we have the following theorem.
 
\begin{thm}\label{thm:second} Let $\la\in \mathcal{SC}^{(m)}$ be a self-conjugate partition with the disparity $m(m+1)/2$.\\ 
If $\phi(\la)=\mu$, then for each positive integer $k$, the number of boxes $(i,j)$ with $h_{\la}(i,j)=2k$ is equal to twice the number of boxes $(\tilde{i},\tilde{j})$ with $h_{\mu}(\tilde{i},\tilde{j})=k$.  
\end{thm}


The following corollary is obtained directly from \cref{thm:second}.

\begin{cor}
For a self-conjugate partition $\lambda$ with the disparity $m(m+1)/2$, let $\phi(\la)=\mu$.\\ Then $\lambda$ is a $(2t_1,2t_2,\dots,2t_p)$-core partition if and only if $\mu$ is a $(t_1,t_2,\dots,t_p)$-core partition.
\end{cor}

We denote the set of self-conjugate $(t_1,t_2,\dots,t_p)$-core partitions $\la\in\mathcal{SC}^{(m)}$ of $n$ by $\mathcal{SC}^{(m)}_{(t_1,...,t_p)}(n)$, and use notation $sc^{(m)}_{(t_1,...,t_p)}(n)$ for 
$|\mathcal{SC}^{(m)}_{(t_1,...,t_p)}(n)|$.\\

By using \cref{thm:bijective,thm:second}, we obtain the cardinality of $\mathcal{SC}^{(m)}_{(2t_1,...,2t_p)}(n)$.

\begin{prop}\label{prop:coresize} For a nonnegative integer $m$, the number of self-conjugate $(2t_1,2t_2,\dots,2t_p)$-core partitions of $n$ with the disparity $\frac{m(m+1)}{2}$ is
$$sc_{(2t_1,...,2t_p)}^{(m)}(n)=\begin{cases}
                c_{(t_1,...,t_p)}(k) & {\text if} \quad n=4k+\frac{m(m+1)}{2},\\
                               0     & {\text otherwise}.
                               \end{cases}$$
\end{prop}

Let $\mathcal{SC}_{(t_1,...,t_p)}=\bigcup_{m\geq0}\mathcal{SC}^{(m)}_{(t_1,...,t_p)}$, where $\mathcal{SC}^{(m)}_{(t_1,...,t_p)}$ denote the set of self-conjugate $(t_1,t_2,\dots,t_p)$-core partitions $\la$ with the disparity $\frac{m(m+1)}{2}$. From the previous proposition, we have the following corollary and \cref{prop:simul}.


\begin{cor}\label{cor:corere}
$$\sum_{\la \in \mathcal{SC}_{(2t_1,...,2t_p)}^{(m)}} q^{|\la|} 
=q^{\frac{m(m+1)}{2}}\sum_{\mu\in \mathcal{P}_{(t_1,\dots,t_p)}}q^{4|\mu|}.$$
\end{cor}


%-------------------------------------------------------------------------------

\subsection{Counting self-conjugate $(2t_1,\dots,2t_p)$-cores with same disparity}        
                        
%-------------------------------------------------------------------------------

In this subsection, we give some sets of self-conjugate partitions each of them is counted by known special numbers.

It is well-known that there are finitely many $(t_1,...,t_p)$-core partitions when  $t_1,...,t_p$ are relatively prime positive integers. From \cref{prop:coresize}, we have the following result.  

\begin{cor}\label{cor:sim}
For relatively prime positive integers $t_1,...,t_p$, the number of self-conjugate $(2t_1,...,2t_p)$-core partitions with the disparity $\frac{m(m+1)}{2}$ is equal to the number of $(t_1,...,t_p)$-core partitions.
\end{cor}

Anderson \cite{Anderson} gives an interpretation for the Catalan number in terms of simultaneous core partitions, and Amderberhan and Leven \cite{AL}, Yang, Zhong, and Zhou \cite{YZZ}, Wang \cite{Wang}, respectively, gives an identity for the Motzkin number. 


\begin{thm}[\cite{Anderson}]\label{thm:anderson}
For relatively prime integers $t_1,t_2\geq1$, the number of $(t_1,t_2)$-core partitions is
$$c_{(t_1,t_2)}=\frac{1}{t_1+t_2}\binom{t_1+t_2}{t_1}.$$ 
In particular, $c_{(n,n+1)}=C_n$, where $C_n=\frac{1}{n+1}\binom{2n}{n}$ is the $n$th Catalan number.
\end{thm}

\begin{thm}[\cite{Wang}]\label{thm:Wang}
For relatively prime integers $n,d\geq1$, the number of $(n,n+d,n+2d)$-core partitions is
$$c_{(n,n+d,n+2d)}=\frac{1}{n+d}\sum_{i=0}^{\lfloor n/2\rfloor}\binom{n+d}{i,i+d,n-2i}.$$
In particular, $c_{(n,n+1,n+2)}$ is the $n$th Motzkin number $M_n=\sum_{i\geq 0} \frac{1}{i+1}\binom{n}{2i}\binom{2i}{i}$.
\end{thm}  
    
By using \cref{cor:sim} and the above known results, we have the following corollary.

\begin{cor}\label{cor:catalanmotzkin}
Let $m\geq0$ be an integer.
\begin{enumerate} 
\item[\rm{(a)}] For relatively prime integers $t_1,t_2\geq1$, the number of self-conjugate $(2t_1,2t_2)$-core partitions with the disparity $\frac{m(m+1)}{2}$ is
$$sc^{(m)}_{(2t_1,2t_2)}=\frac{1}{t_1+t_2}\binom{t_1+t_2}{t_1}.$$
In particular, $sc^{(m)}_{(2n,2n+2)}=C_n$, where $C_n$ is the $n$th Catalan number.
%$$sc^{(m)}_{(2n,2n+2)}=\frac{1}{n+1}\binom{2n}{n}.$$
\item[\rm{(b)}] For relatively prime integers $n,d\geq1$, the number of self-conjugate $(2n,2n+2d,2n+4d)$-core partitions with the disparity $\frac{m(m+1)}{2}$ is 
$$sc^{(m)}_{(2n,2n+2d,2n+4d)}=\frac{1}{n+d}\sum_{i=0}^{\lfloor n/2\rfloor}\binom{n+d}{i,i+d,n-2i}.$$
In particular, $sc^{(m)}_{(2n,2n+2,2n+4)}$ is the $n$th Motzkin number.
\end{enumerate}
\end{cor}
\acknowledgements{
We thank Byungchan Kim for helpful comments and anonymous referees for their suggestions which improved the presentation of this extended abstract.
}




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