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\newtheorem{theorem}{Theorem}[section]
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\newtheorem{remark}[theorem]{Remark}

\usepackage{mathtools, ytableau, tikz}

%% define your title in the usual way
\title[GBG-rank generating functions for ordinary, self-conjugate, and doubled distinct partitions]{GBG-rank generating functions for ordinary, self-conjugate, and doubled distinct partitions}

%% define your authors in the usual way
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\author[H. Cho, E. Kim, H. Lee, K. Lee, A. J. Yee and J. Yoo]{\texorpdfstring{Hyunsoo Cho\thanks{\href{mailto:hyunsoo@ewha.ac.kr}{hyunsoo@ewha.ac.kr}. Hyunsoo Cho was supported by NRF grant No. 2021R1C1C2007589 and No. 2019R1A6A1A11051177.}\addressmark{1},
Eunmi Kim\thanks{\href{mailto:ekim67@ewha.ac.kr}{ekim67@ewha.ac.kr}. Eunmi Kim was supported by the National Research Foundation of Korea (NRF) grant No. RS--2023-00244423 and NRF--2019R1A6A1A11051177.}\addressmark{1},
Ho-Hyeong Lee\thanks{\href{mailto:two-fifty@yonsei.ac.kr}{two-fifty@yonsei.ac.kr}. }\addressmark{2},
Kyeongjun Lee\thanks{\href{mailto:gk53719@yonsei.ac.kr}{gk53719@yonsei.ac.kr}. Kyeongjun Lee was supported by the National Research Foundation of Korea (NRF) grant No. RS--2024-00347040.}\addressmark{2},
Ae Ja Yee\thanks{\href{mailto:yee@psu.edu}{yee@psu.edu}. Ae Ja Yee was partially supported by a grant ($\#$633963) from the Simons Foundation.}\addressmark{3}, \and
Jaeyeong Yoo\thanks{\href{mailto:anamory726@kangwon.ac.kr}{anamory726@kangwon.ac.kr}. }\addressmark{4} 
}{Hyunsoo Cho,
Eunmi Kim,
Ho-Hyeong Lee,
Kyeongjun Lee,
Ae Ja Yee, and
Jaeyeong Yoo 
}}

%% then use \addressmark to match authors to institutions here
\address{\addressmark{1}Institute of Mathematical Sciences, Ewha Womans University, Seoul 03760, Republic of Korea \\ 
\addressmark{2}Department of Mathematics, Yonsei University, Seoul 03722, Republic of Korea \\
\addressmark{3}Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, USA\\
\addressmark{4}Department of Mathematics, Kangwon National University, Chuncheon, Gangwon 24341, Republic of Korea}




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\abstract{BG-ranks and GBG-ranks, introduced by Berkovich and Garvan, are combinatorial statistics that have been used to prove some partition congruences and refinements. In this paper, we give a thorough study on these statistics and obtain formulas for the GBG-rank generating functions for ordinary partitions, self-conjugate partitions, and doubled distinct partitions.}

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\keywords{partition ranks, cranks, BG-ranks, GBG-ranks, self-conjugate partitions, doubled distinct partitions}

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\begin{document}

\maketitle
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%          Section.     %   Introduction
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}

In the theory of partitions,  the most well-known combinatorial statistics are Dyson's rank and crank. In 1944,  Dyson \cite{dyson44} defined the rank of a partition claiming that this statistic combinatorially explains Ramanujan’s mod 5 and 7 partition congruences. Dyson’s claim was confirmed by Atkin and Swinnerton-Dyer \cite{atkin-swinnerton-dyer} in 1955. In the same paper, Dyson \cite{dyson44} also conjectured the existence of another statistic for Ramanujan's mod 11 congruence, namely crank, and this conjecture was settled by Andrews and Garvan \cite{andrews-garvan88} in 1988. 

In 1990, Garvan, Kim and Stanton \cite{garvan-kim-stanton} found another crank, which splits the set of partitions into $t$ equinumerous classes for $t=5,7,11$, and thus gives a combinatorial account for the three congruences of Ramanujan. In their proof, they constructed two bijections, in which $t$-core partitions are an essential component.  Since then, their crank along with Dyson's rank and crank has been adopted to prove other partition congruences and refinements. 

In \cite{BG0,BG1}, utilizing the idea of Garvan, Kim and Stanton, Berkovich and Garvan introduced another partition statistic, namely BG-rank, and used it to prove some generalizations and refinements of Ramanujan's mod 5 partition congruence. The BG-rank involves $2$-core partitions. Later, Berkovich and Garvan \cite{BG2} further generalized this statistic by considering $t$-core partitions. 

Recently, the BG-rank has played an important role in the recent works of Chern, Li, Stanton, Xue and Yee \cite{yee1} and Li, Seo, Stanton and Yee \cite{yee2} on the enumeration of Kleshchev bipartitions. Motivated by their works, in this paper, we study the BG-rank and its generalization GBG-rank more in depth. 

An integer partition $\lambda=(\lambda_1,\lambda_2,\dots,\lambda_\ell)$ is a weakly decreasing sequence of positive integers. These positive integers are called 
the parts of $\lambda$. The sum of all parts of $\lambda$ and the number of parts are denoted by $|\lambda|$ and $\ell(\lambda)$, respectively. If $|\lambda|=n$, then $\lambda$ is called a partition of $n$ and denoted by $\lambda\vdash n$.  It is a convention that the empty sequence $\varnothing$ is considered a partition of $0$.  We denote by $\mathcal{P}$ the set of all partitions. 

For a partition $\lambda$, its {\it Young diagram} $Y_\lambda$, also known as the {\it Ferrers diagram}, is a graphical representation of $\lambda$, which consists of $n$ boxes placed left justified in rows with $\lambda_i$ boxes in the $i$-th row. In \Cref{fig1},  the Young diagram of $\lambda=(5,4, 2, 1)$ is illustrated. 
\begin{figure}[ht]
    %\bigskip
    \centering
    	\begin{tikzpicture}[scale=1]
		\draw[black] (2, 1) -- (4.5, 1);
		\draw[black] (2, 0.5) -- (4.5, 0.5);
		\draw[black] (2, 0) -- (4, 0);
		\draw[black] (2, -0.5) -- (3, -0.5);
		\draw[black] (2, -1) -- (2.5, -1);
		
		\draw[black] (2, 1) -- (2, -1);
		\draw[black] (2.5, 1) -- (2.5, -1);
		\draw[black] (3, 1) -- (3, -0.5);
		\draw[black] (3.5, 1) -- (3.5, 0);
		\draw[black] (4, 1) -- (4, 0);
		\draw[black] (4.5, 1) -- (4.5, 0.5);
			
	\end{tikzpicture}
	  \caption{ The Young diagram $Y_{(5, 4, 2, 1)}$ of the partition $(5,4,2,1)$}
    \label{fig1}
\end{figure} 

We label the box in the $i$-th row and the $j$-th column of the Young diagram $Y_{\lambda}$ by $(i,j)$.  For a positive integer $t$, the {\it $t$-residue of the box $(i,j)$ } is defined as follows \cite{JK}:
$$
Res(i,j):=j-i \mod{t}.
$$

The Young diagram with each box filled with its $t$-residue is called the {\it $t$-residue diagram of $\lambda$}.
The $3$-residue diagram of the partition $(5,4,2,1)$ is shown in \Cref{fig:residue}.
\begin{figure}[ht!] 
\centering
\begin{tikzpicture}[scale=1]
		\draw[black] (2, 1) -- (4.5, 1);
		\draw[black] (2, 0.5) -- (4.5, 0.5);
		\draw[black] (2, 0) -- (4, 0);
		\draw[black] (2, -0.5) -- (3, -0.5);
		\draw[black] (2, -1) -- (2.5, -1);
		
		\draw[black] (2, 1) -- (2, -1);
		\draw[black] (2.5, 1) -- (2.5, -1);
		\draw[black] (3, 1) -- (3, -0.5);
		\draw[black] (3.5, 1) -- (3.5, 0);
		\draw[black] (4, 1) -- (4, 0);
		\draw[black] (4.5, 1) -- (4.5, 0.5);
		
		\draw (2.25,0.7) node{\footnotesize{$0$}};		
		\draw (2.75,0.7) node{\footnotesize{$1$}};		
		\draw (3.25,0.7) node{\footnotesize{$2$}};		
		\draw (3.75,0.7) node{\footnotesize{$0$}};	
		\draw (4.25,0.7) node{\footnotesize{$1$}};	
		
		\draw (2.25,0.2) node{\footnotesize{$2$}};	
		\draw (2.75,0.2) node{\footnotesize{$0$}};		
		\draw (3.25,0.2) node{\footnotesize{$1$}};	
		\draw (3.75,0.2) node{\footnotesize{$2$}};	
		\draw (2.25,-0.3) node{\footnotesize{$1$}};	
		\draw (2.75,-0.3) node{\footnotesize{$2$}};
		\draw (2.25,-0.8) node{\footnotesize{$0$}};	
			
	\end{tikzpicture}

%\begin{ytableau} %[scale=1]
%	0&1&2&0 &1\\
%	2&0&1&2\\
%	1&2\\
%	0
%\end{ytableau}
\caption{The $3$-residue diagram of the partition $(5,4,2,1)$}\label{fig:residue}
\end{figure}


For a positive integer $t$, the {\it GBG-rank of $\lambda$ mod $t$} is the sum of $\zeta_t^{j-i}$ for $(i,j)\in Y_{\lambda}$, where $\zeta_t$ is a $t$-th root of unity. In this paper, we denote the GBG-rank mod $t$ by $\omega_{t}(\lambda)$, namely,
\begin{equation*}
\omega_t(\lambda):=\sum_{(i,j)\in Y_{ \lambda} } \zeta_{t}^{j-i}.
\end{equation*}
 We also define the {\it $t$-GBG diagram of $\lambda$} by assigning $\zeta_t^{j-i}$ to each box $(i,j)$ in the Young diagram.  The $3$-GBG diagram of the partition $(5, 4, 2, 1)$ is given in \Cref{fig:GBG}, where $\zeta:=\zeta_3$. Since  $1+\zeta_3+\zeta_3^2=0$, the GBG-rank of $(5,4, 2, 1)$ mod $3$ is $0$.
   
\begin{figure}[ht!]
	\centering
	\begin{tikzpicture}[scale=1]
		\draw[black] (2, 1) -- (4.5, 1);
		\draw[black] (2, 0.5) -- (4.5, 0.5);
		\draw[black] (2, 0) -- (4, 0);
		\draw[black] (2, -0.5) -- (3, -0.5);
		\draw[black] (2, -1) -- (2.5, -1);
		
		\draw[black] (2, 1) -- (2, -1);
		\draw[black] (2.5, 1) -- (2.5, -1);
		\draw[black] (3, 1) -- (3, -0.5);
		\draw[black] (3.5, 1) -- (3.5, 0);
		\draw[black] (4, 1) -- (4, 0);
		\draw[black] (4.5, 1) -- (4.5, 0.5);
		
		\draw (2.25,0.7) node{\footnotesize{$1$}};		
		\draw (2.75,0.7) node{\footnotesize{$\zeta$}};		
		\draw (3.25,0.7) node{\footnotesize{$\zeta^2$}};		
		\draw (3.75,0.7) node{\footnotesize{$1$}};	
		\draw (4.25,0.7) node{\footnotesize{$\zeta$}};	
		
		\draw (2.25,0.2) node{\footnotesize{$\zeta^2$}};	
		\draw (2.75,0.2) node{\footnotesize{$1$}};		
		\draw (3.25,0.2) node{\footnotesize{$\zeta$}};	
		\draw (3.75,0.2) node{\footnotesize{$\zeta^2$}};	
		\draw (2.25,-0.3) node{\footnotesize{$\zeta$}};	
		\draw (2.75,-0.3) node{\footnotesize{$\zeta^2$}};
		\draw (2.25,-0.8) node{\footnotesize{$1$}};	
			
	\end{tikzpicture}

	%\begin{ytableau}
	%	*(gray!30)1&\zeta&\zeta^2&1 & \zeta \\
          %  	\zeta^2&*(gray!30)1&\zeta&\zeta^2 \\
           % 	\zeta&\zeta^2\\
           % 	1
	%\end{ytableau}
	\caption{The $3$-GBG diagram of the partition $(5,4,2,1)$}\label{fig:GBG} 
\end{figure}


For $t=2$, the GBG-rank is called the BG-rank \cite{BG0}. Berkovich and Garvan used the BG-rank to prove some generalizations and refinements of Ramanujan's mod $5$ partition congruence \cite{BG0, BG1}. They \cite{BG2} also studied the number of distinct values of GBG-ranks of $s$-core partitions mod $t$. In \cite{BU}, Berkovich and Uncu gave the generating functions for ordinary and strict partitions with the largest part bounded and a fixed integral BG-rank. These results were generalized by Berkovich and Dhar \cite{BD} to integral GBG-ranks mod $t$ for any prime $t$ for ordinary partitions and partitions into parts repeating no more than $t-1$ times. The proofs of these results rely on the characterization of $t$-core partitions in the work of Garvan, Kim and Stanton \cite{garvan-kim-stanton}.

Our main objective of this paper is to give a more direct way to compute the generating function for partitions with a fixed GBG-rank mod $t$, which enables us to get the generating function for non-integral values of GBG-ranks mod $t$. This approach can also be adopted to other types of partitions such as self-conjugate partitions and doubled distinct partitions. We only state our results on ordinary partitions in this section. More results on the other partitions will be discussed in later sections.  

We fix integers $t$ and $\ell$ with $t\ge 2$.  For $\omega \in \zeta_t^{\ell} \mathbb{Z}$, let %$G_t(\omega, q)$ be the generating function for partitions with the GBG-rank mod $t$ equal to $\omega$, i.e.,
\begin{align*}
	G_t(\omega,q):=\sum_{\substack{\lambda \in \mathcal{P} \\ \omega_t(\lambda)=\omega}} q^{|\lambda|} .
\end{align*}

\begin{theorem} \label{thm:ord}
	For any prime $t$ and $k,\ell\in \mathbb{Z}$, we have
	\begin{equation*}
		G_t(k \zeta_t^{\ell},q)= \begin{dcases}  
			\frac{q^{tk^2 -(t-1)k} }{(q^t;q^t)_{\infty}^t} & \text{if}  \hspace{10mm} \ell=0,\\
 			\frac{q^{t k^2+k}}{(q^t;q^t)_{\infty}^t } & \text{if}  \hspace{5mm} 1\le \ell \le t-1.
		\end{dcases}
	\end{equation*}
\end{theorem}
Here, we use the following standard $q$-series notation: For $a,q\in \mathbb{C}$ and a positive integer $n$,
\[
	(a;q)_0 := 1, \quad (a;q)_n := \prod_{j=0}^{n-1}(1-aq^j), \quad 
	(a;q)_\infty := \lim_{n\to \infty} (a;q)_n, \quad |q|<1.
\]

%Berkovich and Garvan \cite{BG0} defined the BG-rank of a partition, which is the GBG-rank mod $2$, and used it in \cite{BG0, BG1} to generalize and refine the Ramanujan mod $5$ partition congruences. 

\begin{remark}
Letting $N \to \infty$ in \cite[Theorem 1.1]{BD} and \cite[Theorem 4.1]{BD} gives \Cref{thm:ord} for $\ell=0$ and $1 \le \ell \le t-1$, respectively.
\end{remark}

By restricting the number of parts and the size of parts of partitions, we can also obtain the finite form of the generating function $G_t(\omega,q)$.  Let 
\begin{align*}
G_{M, N, t}(\omega, q):=\sum_{\substack{\lambda \in \mathcal{P}, \lambda_1\le N, \ell(\lambda)\le M \\ \omega_t(\lambda)=\omega}} q^{|\lambda|}.
\end{align*}
% be the generating function for the number of partitions into at most $M$ parts each less than or equal to $N$ with the GBG-rank mod $t$ equal to $\omega$.  %Then we have the following result.



\begin{theorem}\label{thm:ord_finite}
	For any prime $t$, any $k,\ell \in\mathbb{Z}$, and nonnegative integers $M$ and $N$, we have
	\begin{align*}
		G_{M,N,t}(k\zeta_t^\ell,q)  &= \begin{cases} q^{tk^2-(t-1)k} \qbinom{N_{t,1}+M_{t,1}}{N_{t,1}-k }_{q^t} \qbinom{N_{t,t}+M_{t,t}}{N_{t,t}+k }_{q^t} ~{\displaystyle\prod_{j=2}^{t-1}}\qbinom{N_{t,j}+M_{t,j}}{N_{t,j}}_{q^t}  & \text{if}  \hspace{10mm} \ell=0, \\ \\
		 q^{tk^2+k} \qbinom{N_{t,\ell+1}+M_{t,\ell+1}}{N_{t,\ell+1}-k }_{q^t} \qbinom{N_{t,\ell}+M_{t,\ell}}{N_{t,\ell}+k }_{q^t}
		~{\displaystyle\prod_{\substack{1\leq j\leq t \\ j\not=\ell,\ell+1}}} \qbinom{N_{t,j}+M_{t,j}}{ N_{t,j}}_{q^t} & \text{if}  \hspace{5mm} 1\le \ell \le t-1,
		\end{cases}
	\end{align*}
	where $M_{t,j}:=\left\lfloor\frac{M+j-1}{t}\right\rfloor$ and $N_{t,j}:=\left\lfloor\frac{N-j}{t}\right\rfloor+1$.
\end{theorem}
Here, $\qbinom{m}{n}_q$ is the $q$-binomial coefficient defined by
\[
	\qbinom{m}{n}_q = \begin{dcases}
	\frac{(q;q)_m}{(q;q)_n (q;q)_{m-n}} & \text{ for } m \geq n \geq 0,\\
	0 & \text{otherwise}.
	\end{dcases}
\]

Since 
\[
	\lim_{m\to \infty} \qbinom{m}{n}_q=\frac{1}{(q;q)_{n}},
\]
letting $M \to \infty$ for $\ell=0$ in \Cref{thm:ord_finite} yields the following identity, which was given by Berkovich and Dhar \cite[Theorem 1.1]{BD}.


\begin{corollary}
	For $\nu \in \{0,1\ldots, t-1\}$, 
	\begin{align*}
		G_{\infty, \, tN+\nu, t}(k,q) = \frac{q^{tk^2-(t-1)k}}{(q^t;q^t)_{N+k}(q^t;q^t)_{N+\lceil\frac{\nu}{t}\rceil-k}(q^t;q^t)_{N+\lceil\frac{\nu-1}{t}\rceil} \cdots (q^t;q^t)_{N+\lceil\frac{\nu-(t-2)}{t}\rceil}}.
	\end{align*}
\end{corollary}

Similarly, letting $M \to \infty$ in \Cref{thm:ord_finite} for $1 \le \ell \le t-1$ gives \cite[Theorem 4.1]{BD}.

\begin{corollary}
For $\nu \in \{0,1,\dots,t-1\}$ and $\ell \in \{1,2,\dots,t-1\}$,
\[
	G_{\infty,tN+\nu,t}(k\zeta_t^\ell,q)=\frac{q^{tk^2+k}}{(q^t;q^t)_{N+\lceil\frac{\nu -\ell +1}{t}\rceil+k}(q^t;q^t)_{N+\lceil\frac{\nu -\ell}{t}\rceil-k} \displaystyle\prod_{\substack{1\leq j\leq t \\ j\not=\ell,\ell+1}}(q^t;q^t)_{N+\lceil\frac{\nu -j +1}{t}\rceil}}.
\]
\end{corollary}

%The $\ell=0$ case of \Cref{thm:ord_finite} was ... \textcolor{red}{Add more}

%The GBG-rank mod $2$, namely the BG-rank, was introduced earlier by Berkovich and Garvan  in \cite{BG0} and used in \cite{BG0, BG1} for congruences related to the mod 5 partition congruence of Ramanujan  Berkovich and Garvan \cite{BG2} also studied the number of distinct values of the GBG-rank mod $t$ of $s$-core partitions. 
%In \cite{BU}, Berkovich and Uncu  gave the generating functions for partitions having the bounded largest part and a fixed integral BG-rank. These were generalized in \cite{BD} with a fixed integral GBG-rank mod $t$ for any prime $t$ and there are similar results for partitions into parts repeating no more $t-1$ times.

The rest of this paper is organized as follows. In \Cref{sec:pre}, we recall some necessary definitions.  In \Cref{sec:ord}, we give a general formula for the GBG-rank generating function for ordinary partitions, from which Theorems \ref{thm:ord} and \ref{thm:ord_finite} can be derived. We also give an asymptotic formula for the number of partitions with integral GBG-ranks mod $t$. Sections~\ref{sec:sc} and \ref{sec:dd} are devoted to GBG-ranks for self-conjugate partitions and doubled distinct partitions, respectively. Adapting the method used for ordinary partitions, we give the GBG-rank generating functions and refinements for self-conjugate partitions and doubled distinct partitions.

\iffalse

Let 
\begin{align*}
	F_t(q):=\sum_{n\ge 0} a_t(n) q^n =\frac{1}{(q^t;q^t)_{\infty}^t} \sum_{k=-\infty}^{\infty} q^{tk^2-(t-1)k}.
\end{align*}
\begin{theorem}
	For a fixed $t\ge 2$, 
	\begin{align*}
	a_t(n)=\frac{t^{\frac{t-1}{2}}}{2(24)^{t/4} } n^{-(t+2)/4} e^{\pi \sqrt{2n /3}} +O\bigg(n^{-\frac{t+3}{4}} e^{\pi \sqrt{ 2n/3}} \bigg) \text{ as $n\to \infty$}. 
	\end{align*}
Moreover, the asymptotic ratio for ordinary partitions to have integral ranks is
\begin{align*}
	\frac{a_t(n)}{p(n)} =\frac{t^{(t-1)/2}}{\sqrt{2} (24)^{(t-2)/4}} n^{-(t-2)/4} \bigg(1+ O\left(n^{-1/4}\right)\bigg) \text{ as $n\to \infty$}. 
\end{align*}
\end{theorem}

\fi



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%          Section.     %   Preliminaries
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Preliminaries}\label{sec:pre}
%In this section, we 

%For a partition $\lambda$, a partition $\lambda'=(\lambda'_1,\lambda_2,\dots,\lambda'_m)$ is called the conjugate of $\lambda$ if $\lambda'_j$ is the number of boxes in column $j$ of the Young diagram of $\lambda$.

The hook of the box $(i,j)$ in the Young diagram of a partition $\lambda$  is the following set of boxes:
$$
H_{(i,j)}:=\{ (k,m) \in Y_{\lambda} \;|\; k\ge i, m\ge j\}.
$$
The size of $H_{(i,j)}$ is called the hook length of the box $(i,j)$ and denoted by $h_{(i,j)}$.   
\iffalse
In \Cref{fig2}, the number in each box indicates its hook length. 
\begin{figure}[ht]
   % \bigskip
    \centering
    \begin{ytableau} 
	8&6&4&3 & 1\\
	6&4&2&1\\
	3&1\\
	1
\end{ytableau}
	  \caption{ The hook lengths of the partition ${(5,4, 2, 1)}$}
    \label{fig2}
\end{figure} 
\fi
For a positive integer $t$, $\lambda$ is called a $t$-core partition if no hooks have length divisible by $t$. 
%As seen in \Cref{fig2}, the partition $(5,4, 2,1)$ is a $5$-core partition.  

%There is a well-known algorithm for getting a $t$-core partition from an arbitrary partition $\lambda$ \cite{JK}. The rim hook of a box $x$ consists of the boxes of the rim between two ends of the hook of $x$. We successively remove rim hooks of length $t$ until no hooks of length $t$ remain. 
%We call this resulting partition the $t$-core of $\lambda$ and denote it  by $\lambda_{\text{$t$-core}}$. 



A Frobenius symbol for a positive integer $n$ is a two-rowed array 
\begin{equation*}
	\left( \begin{matrix} a_1 & a_2 & \cdots & a_s \\ 
	b_1 & b_2 & \cdots & b_s \end{matrix} \right)
\end{equation*}
such that  $\sum_{j=1}^{s}(a_j +b_j+1)=n$,  $a_1>a_2>\cdots>a_s \ge 0$, and
$b_1>b_2>\cdots>b_s \ge 0$.  There is a natural one-to-one correspondence between partitions and Frobenius symbols.  For a partition $\lambda$, let $s$ be the largest integer such that $\lambda_{s}-s \ge 0$, i.e., $s$ is the side of the largest square of $\lambda$ that fits inside the Young Diagram of $\lambda$. This largest square is unique and called the {\it Durfee square} of $\lambda$. We now consider the following two-rowed array:
\begin{equation*}
	\left( \begin{matrix} \lambda_1-1& \lambda_2-2 & \cdots & \lambda_{s} -s \\
	 \lambda'_1-1& \lambda'_2-2 & \cdots & \lambda'_{s} - s 	\end{matrix}\right),
\end{equation*}
where $\lambda_j'$ is the number of boxes in the $j$-th column of the Young diagram of $\lambda$.  %denotes the conjugate of $\lambda$.
Clearly, this satisfies the conditions for Frobenius symbols, and this process is reversible. Thus, there is a unique Frobenius symbol associated with $\lambda$. 
\iffalse 
For instance,
the associated Frobenius symbol of the partition $(11,11,8,4,4,2)$  is
\begin{equation*}
	\left(\begin{matrix} 10 & 9& 5 & 0\\ 5 & 4 & 2& 1\end{matrix} \right). 
\end{equation*}
\fi
In this paper, however, we will denote the Frobenius symbol of $\lambda$ by 
\[
	\mathfrak{F}(\lambda):=(\lambda_1, \lambda_2-1\dots, \lambda_s -s+1 \mid \lambda'_1-1,\lambda'_2-2,\dots, \lambda'_s -s ).
\]
Note that each number on the left of the bar in $\mathfrak{F}(\lambda)$ is one larger than the corresponding entry in the above two-rowed array notation. 
 In \Cref{fig:GBG}, the Frobenius symbol of the partition $(5,4,2,1)$ is
 \[
 (5, 3  \mid 3, 1).
 \] 

 The {\it conjugate of $\lambda$} is the partition resulting from reflecting the Young diagram of $\lambda$ about the main diagonal, and we denote the conjugate partition by $\lambda'$. We call $\lambda$ a {\it self-conjugate partition} if $\lambda=\lambda'$.
 
 Let $MD(\lambda)$ be the set of hook lengths in the main diagonal of $\lambda$. Then, for 
$$
\mathfrak{F}(\lambda)=(a_1,\dots,a_s \mid b_1,\dots,b_s),
$$
we can easily see that 
$$
MD(\lambda)=\{a_1+b_1,a_2+b_2,\dots,a_s+b_s\}.
$$ 

If $\lambda$ is self-conjugate, then 
\begin{equation}
\mathfrak{F}(\lambda)=(\lambda_1, \lambda_2-1 \ldots, \lambda_s -s+1 \mid \lambda_1-1,\lambda_2 -2, \ldots, \lambda_{s}-s), \label{frob:sc}
\end{equation}
so
$$
MD(\lambda)=\{2\lambda_1-1, 2\lambda_2-3, \ldots, 2\lambda_{s}-(2s-1) \},
$$
i.e., all elements of $MD(\lambda)$ are distinct and odd, and $MD(\lambda)$ determines $\lambda$. Thus, 
\[
	\sum_{\lambda \in \mathcal{SC}}q^{|\lambda|}=(-q;q^2)_\infty,  
\]
where $\mathcal{SC}$ denotes the set of self-conjugate partitions. 
%\begin{proposition}\label{pro2.1}
%The set of self-conjugate partitions are in bijection with the set of partitions into distinct odd parts.
%\end{proposition}

For a partition $\lambda$ into distinct parts, the shifted Young diagram of $\lambda$ is the diagram resulting from shifting the $i$-th row to the right by $i-1$ boxes for each $i$ in its Young diagram. The doubled distinct partition of $\lambda$, denoted by $\lambda\lambda$, is the partition whose Young diagram is defined by adding $\lambda_i$ boxes to the $(i-1)$st column of the shifted Young diagram of $\lambda$. That is, for a partition $\lambda=(\lambda_1,\lambda_2,\dots,\lambda_\ell)$ with $\lambda_1>\lambda_2>\cdots>\lambda_\ell$,
\begin{equation}
	\mathfrak{F}(\lambda\lambda)=(\lambda_1+1,\lambda_2+1,\dots,\lambda_\ell+1 \mid \lambda_1-1,\lambda_2-1,\dots,\lambda_\ell-1)\label{frob:dd}
\end{equation}
and
$$MD(\lambda\lambda)= \{ 2\lambda_1, 2\lambda_2, \dots,\allowbreak  2\lambda_\ell \}.$$  
Thus,
\[
	\sum_{\lambda\lambda \in \mathcal{DD}} q^{|\lambda\lambda|}=(-q^2;q^2)_\infty,
\]
where $\mathcal{DD}$ denotes the set of doubled distinct partitions. 

%\begin{proposition}\label{pro2.2}
%The set of self-conjugate partitions are in bijection with the set of partitions into distinct even parts.
%\end{proposition}




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%          Section.     %   Partitions
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Results on ordinary partitions}\label{sec:ord}

In this section and throughout the rest of this paper, we assume that $t$ is a positive integer with $t\ge 2$ unless it is stated otherwise. We also assume that $\zeta$ is a $t$-th root $\zeta_t$ of unity.    

%\subsection{Generating functions}
We first note that it follows from Frobenius symbols that the generating function for partitions is
$$
\sum_{\lambda \in \mathcal{P}} q^{|\lambda|}= [z^0] (-zq;q)_{\infty} (-z^{-1};q)_{\infty},
$$
where $[z^0] G(z)$ means the constant term of $G(z)$. Next, we will consider the following GBG-rank generating function:
$$
\sum_{\lambda \in \mathcal{P}} x^{\omega_t(\lambda)} q^{|\lambda|}.
$$

For a nonnegative integer $j$, let 
$
r_j:=r_j(\zeta):=\sum_{i=0}^{j-1}\zeta^i.
$ 
Note that $r_0=0$ because the sum is empty. Also, for $0 \le j\le t-1$, 
\begin{equation}
\sum_{i=1}^{t-j} \zeta^{-i} =\sum_{i=1}^{t-j}  \zeta^{t-i}=\sum_{i=j}^{t-1} \zeta^i = -r_j \label{rj}
\end{equation}
since $r_{t}=1+\zeta+\cdots+\zeta^{t-1}=0$.

Let $\lambda$ be a partition with the following Frobenius symbol:
$$\mathfrak{F}(\lambda)=(a_1,\dots,a_s \mid b_1,\dots,b_s)$$ 
for some positive integer $s$. Then,
$$
\omega_t(\lambda)=\sum_{k=1}^s \left( \sum_{i=0}^{a_k-1}\zeta^{i} + \sum_{i=1}^{b_k} \zeta^{-i} \right) =\sum_{k=1}^s \left( \sum_{i=0}^{\bar{a}_k-1}\zeta^{i} + \sum_{i=1}^{\bar{b}_k} \zeta^{-i} \right)  =\sum_{k=1}^s ( r_{\bar{a}_k} - r_{t-\bar{b}_k} ),
$$
where $\bar{a}_k$ and $\bar{b}_k$ are the least nonnegative residues of $a_k$ and $b_k$ mod $t$, respectively, and the last equality follows from \eqref{rj}. 
Thus, 
\begin{align}
	\sum_{\lambda\in\mathcal{P}} x^{\omega_t(\lambda)} q^{|\lambda|} =& [z^0] \prod_{j=1}^{t} \left(-x^{r_{j}} z q^j ;q^t\right)_\infty \left(-x^{-r_{j}}z^{-1}q^{t-j};q^t\right)_\infty  \nonumber\\
	=&\frac{1}{(q^t;q^t)^t_\infty}[z^0]\prod_{j=1}^t\sum_{n_j = -\infty}^{\infty}z^{n_j} x^{n_j r_j}q^{\frac{tn_j(n_j-1)}{2}+jn_j} \notag\\
	=& \frac{1}{(q^t;q^t)^t_{\infty}} \sum_{\substack{n_1,\ldots, n_t=-\infty \\ n_1+\cdots +n_t=0}}^{\infty} x^{n_1r_1+\cdots + n_tr_t} q^{\sum\limits_{j=1}^t \left(\frac{t n_j(n_j-1)}{2}+ j n_j\right)}  \label{ord_gen},
\end{align}
where the second equality follows from the Jacobi triple product identity \cite{gea1}:
\begin{equation}\label{JTP}
	 (-x;q)_\infty (-q/x;q)_\infty (q;q)_\infty = \sum_{n=-\infty}^{\infty} x^n q^{n(n-1)/2}.
\end{equation}

We are now ready to prove \Cref{thm:ord}.

\begin{proof}[Proof of \Cref{thm:ord}]
For $\ell =0$, we have $\omega_t(\lambda)=k \in \mathbb{Z}$. When $t$ is a prime, the minimal polynomial of $\zeta$ over $\mathbb{Z}$ is $1+x+\cdots +x^{t-1}$, so $1,\zeta, \ldots, \zeta^{t-2}$ are linearly independent over $\mathbb{Z}$. Hence, the power of $x$ in \eqref{ord_gen}  is an integer if and only if $n_1+n_t=0$ and $n_2=\cdots =n_{t-1}=0$. Thus, 
%we only need two sums when $j=1$ and $t$ in the product part of \eqref{ord_gen} so that
\begin{align}
	\sum_{\substack{\lambda \in \mathcal{P}\\ \omega_t(\lambda) \in \mathbb{Z}}} x^{\omega_t(\lambda)} q^{|\lambda|} 
	%&= \frac{1}{(q^t;q^t)_\infty^t} [z^0] \left(\sum_{n =- \infty}^{\infty} z^n x^{n} q^{tn(n-1)/2+n}\right) \left(\sum_{n =- \infty}^{\infty} z^n x^0  q^{tn(n-1)/2+tn}\right) \nonumber\\
	&= \frac{1}{(q^t;q^t)_\infty^t} \sum_{n_1 =- \infty}^{\infty} x^{n_1} q^{\frac{ t n_1(n_1-1)}{2} +\frac{t n_1(n_1+1)}{2} + n_1- tn_1}. \label{ord_int} 
\end{align}
The theorem  immediately follows from  comparing the coefficients of $x^{n_1}$ on both sides.
\iffalse
\[
	G_t (k, q) = \frac{q^{tk^2-(t-1)k}}{(q^t; q^t)_\infty^t}.
\]
\fi

%Note that $t$ has to be a prime. Suppose $t$ is not a prime, i.e., $t=t_1t_2$ for some $t_1,t_2>1$. Then $(\zeta^{t_1})^{t_2}=1$, so the evaluation is invalid.
	
Next we consider the case when $1 \leq \ell \leq t-1$. We first note that
$
\zeta^\ell = r_{\ell+1}(\zeta) - r_\ell(\zeta).
$ 
Again, due to the minimal polynomial of $\zeta$ and the linear independence of $1, \zeta, \ldots, \zeta^{t-2}$ over $\mathbb{Z}$,   
we see that the power of $x$  in \eqref{ord_gen} is  in $\zeta^\ell \mathbb{Z}$ if and only if $n_{\ell}+n_{\ell+1}=0$ and $n_1=\ldots =n_{\ell-1}=n_{\ell+2}=\cdots =n_{t}=0$. Thus,
\begin{align*}
	\sum_{\substack{\lambda\in\mathcal{P} \\ \omega_t(\lambda)\in\zeta^\ell\mathbb{Z}}}x^{\omega_t(\lambda)}q^{|\lambda|}
	%=&\frac{1}{(q^t;q^t)^t_\infty}[z^0]\left(\sum_{n=-\infty}^{\infty} z^nx^{n r_\ell}q^{\frac{tn(n-1)}{2}+\ell n}\right)
	%\left(\sum_{n=-\infty}^{\infty}z^nx^{n r_{\ell+1}}q^{\frac{tn(n-1)}{2}+(\ell+1)n}\right)\\
	=&\frac{1}{(q^t;q^t)^t_\infty} \sum_{n_{\ell+1}=-\infty}^{\infty}  x^{n_{\ell+1} \,\zeta^\ell }q^{t n_{\ell+1}^2+n_{\ell+1}},
\end{align*}
which completes the proof.
\end{proof}

The finite form of $G_t(k\zeta^{\ell}, q)$ in \Cref{thm:ord_finite} can be proved in a similar way by applying the following finite form of the Jacobi triple product identity \cite{FH98}:
\begin{equation}\label{JTP_finite}
	(-1/x;q)_M (-xq;q)_N = \sum_{n=-M}^N \qbinom{N+M}{M+n}_{q} x^n q^{n(n+1)/2}.
\end{equation}


%\subsection{Asymptotics}
Next, we will give the asymptotic ratio for partitions with integral GBG-ranks mod $t$ among all ordinary partitions by applying Wright's circle method.
Let 
\begin{align*}
F_t(q):=\sum_{n\ge 0} a_t(n) q^n =\frac{1}{(q^t;q^t)_{\infty}^t} \sum_{k=-\infty}^{\infty} q^{tk^2-(t-1)k}.
\end{align*}

\begin{theorem}
For a fixed $t\ge 2$, 
\begin{align*}
a_t(n)=\frac{t^{\frac{t-1}{2}}}{2(24)^{t/4} } n^{-(t+2)/4} e^{\pi \sqrt{2n /3}} +O\bigg(n^{-\frac{t+3}{4}} e^{\pi \sqrt{ 2n/3}} \bigg) \text{ as $n\to \infty$}. 
\end{align*}
Moreover, the asymptotic ratio for ordinary partitions to have integral GBG-ranks mod $t$ is
\begin{align*}
\frac{a_t(n)}{p(n)} =\frac{t^{(t-1)/2}}{\sqrt{2} (24)^{(t-2)/4}} n^{-(t-2)/4} \bigg(1+ O\left(n^{-1/4}\right)\bigg) \text{ as $n\to \infty$},
\end{align*}
where $p(n)$ is the partition function of $n$.
\end{theorem}





%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%          Section.     %  Self-conjugate partitions
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Results on self-conjugate partitions}\label{sec:sc}

For a nonnegative integer $j$, let 
$
r^*_j:=r_j^*(\zeta):=\sum_{i=-j}^{j}\zeta^i.
$  

As seen in \Cref{sec:pre}, self-conjugate partitions are completely determined by their hook lengths in the main diagonal, which are distinct odd numbers.

 Let $\lambda$ be a self-conjugate partition with 
$
MD(\lambda)=\{2d_1+1,2d_2+1,\dots,2d_s+1\}.
$
\iffalse
Then, $\omega_{t}(\lambda)=\sum_{j=1}^s r^*_{d_j}$.
Thus, for $t=2\ell$ for some positive integer $\ell$, 
$$
\sum_{\lambda\in \mathcal{SC}} x^{\omega_t(\lambda)} q^{|\lambda|} = \prod_{j=0}^{2\ell-1} (-x^{r_j^*}q^{2j+1};q^{4\ell})_{\infty} \\= \frac{1}{(q^{4\ell};q^{4\ell})_\infty^\ell} \prod_{j=0}^{\ell-1} \sum_{n=-\infty}^{\infty} x^{n r_j^*} q^{2\ell n^2-(2\ell-2j-1)n}.
$$
Utilizing this formula, we can obtain the following theorem. 
\fi
Then, $\omega_{t}(\lambda)=\sum_{j=1}^s r^*_{d_j}$ and $r^*_j=r^*_{j+2t}$. 
Thus, 
$$
\sum_{\lambda\in \mathcal{SC}} x^{\omega_t(\lambda)} q^{|\lambda|} = \prod_{j=0}^{t-1} (-x^{r_j^*}q^{2j+1};q^{2 t })_{\infty} . %\\= \frac{1}{(q^{4\ell};q^{4\ell})_\infty^\ell} \prod_{j=0}^{\ell-1} \sum_{n=-\infty}^{\infty} x^{n r_j^*} q^{2\ell n^2-(2\ell-2j-1)n}.
$$
Utilizing this formula, we can obtain the following theorem.

\begin{theorem}\label{thm:sc}
	We have
	\[
		\sum_{\substack{\lambda \in \mathcal{SC}}} x^{\omega_t(\lambda)} q^{|\lambda|} =
		\begin{dcases}
			\frac{ {(-q^{t};q^{2t})_\infty}}{(q^{2t};q^{2t}){_\infty^{(t-1)/2}}} \prod_{j=0}^{(t-3)/2} \sum_{n=-\infty}^{\infty} x^{nr_j^*} q^{tn^2-(t-2j-1)n}  &\text{if $t$ is odd},\\ 
			\frac{1}{(q^{2t};q^{2t}){_\infty^{t/2}}} \prod_{j=0}^{t/2-1} \sum_{n=-\infty}^{\infty} x^{nr_j^* } q^{tn^2-(t-2j-1)n}  &\text{if $t$ is even}.
		\end{dcases}
	\]
\end{theorem}


For a nonnegative integer $N$, let $\mathcal{SC}_N$ be the set of self-conjugate partitions into parts less than or equal to $N$. Clearly, each partition in $\mathcal{SC}_N$ has at most $N$ parts.
%For a positive integer $t$ and a nonnegative integer $j$, let $r_j^*(\zeta_t):=\sum_{i=-j}^{j}\zeta_t^i$. We have the following generating function identity.
\begin{theorem}\label{thm:sc_main}
For a nonnegative integer $N$, we have
	\begin{align*}
		&\sum_{\lambda \in \mathcal{SC}_N} x^{\omega_t(\lambda)} q^{|\lambda|}\\ &=
		\begin{cases}
			{(-q^{t};q^{2t})_{N_{t,\frac{t-1}{2}}}}
			{\displaystyle\prod_{j=0}^{(t-3)/2} \sum_{n=-N_{t,t-j-1}}^{N_{t,j}}} \qbinom{N_{t,j} + N_{t,t-j-1}}{N_{t,t-j-1} - n}_{q^{2t}} x^{nr_j^*} q^{tn^2 - (t-2j-1)n}  & \text{if $t$ is odd},\\
			{\displaystyle\prod_{j=0}^{t/2-1} \sum_{n=-N_{t,t-j-1}}^{N_{t,j}}} \qbinom{N_{t,j} + N_{t,t-j-1}}{N_{t,t-j-1} - n}_{q^{2t}} x^{nr_j^*} q^{tn^2 - (t-2j-1)n} & \text{if $t$ is even},
		\end{cases}
	\end{align*}
	where $N_{t,j} := \lfloor \frac{N-j-1}{t} \rfloor +1$.
\end{theorem}


Note that when $t=2,3,4,6$, we have $r_j^* \in \mathbb{Z}$ for all positive integers $j$, which yields that $\omega_t(\lambda) \in \mathbb{Z}$.

Next, we consider integral GBG-ranks mod $t$ for an odd prime $t$. Let $GSC_t(k,q)$ be the generating function for the number of self-conjugate partitions with the GBG-rank mod $t$ equal to $k$.

\begin{theorem}\label{thm:sc_gen}
	For an odd prime $t$ and an integer $k$,
	\[
		GSC_t(k,q) = \frac{ {(-q^{t};q^{2t})_\infty} q^{tk^2-(t-1)k} }{(q^{2t}; q^{2t} ){_\infty^{(t-1)/2}}}.
	\]
\end{theorem}

%Let $GSC_{N,t}(k,q)$ be the generating function for the number of self-conjugate partitions into parts less than or equal to $N$ with the GBG-rank mod $t$. Clearly, each partitions in $GSC_{N,t}(k,q)$ has at most $N$ parts.

For a nonnegative integer $N$, let $GSC_{N,t}(k,q)$ be the generating function for the number of self-conjugate partitions into parts less than or equal to $N$ with the GBG-rank mod $t$ equal to $k$. %Clearly, each partitions in $GSC_{N,t}(k,q)$ has at most $N$ parts.
\begin{theorem}\label{thm:sc_finite}
	For an odd prime $t$ and integers $k,N$ with $N\ge 0$,
	\begin{align*}
		GSC_{N,t}(k,q)&= q^{tk^2 -(t-1)k} (-q^t;q^{2t})_{N_{t, \frac{t-1}2}}\qbinom{N_{t,0} + N_{t,t-1}}{N_{t,t-1} - k}_{q^{2t}} \prod_{j=1}^{(t-3)/2} \qbinom{N_{t,j} +N_{t,t-j-1}}{N_{t,t-j-1}}_{q^{2t}},%   \\
		%&\times \qbinom{M_{t,1} + M_{t,t-2}}{M_{t,1}}_{q^{2t}}\qbinom{M_{t,2} + M_{t,t-3}}{M_{t,2}}_{q^{2t}}\cdots \qbinom{M_{t,\ell -1} + M_{t,\ell}}{M_{t,\ell -1}}_{q^{2t}},
	\end{align*}
	where $N_{t,j}=\lfloor \frac{N-j-1}{t} \rfloor +1$.
\end{theorem}


\begin{remark}
\Cref{thm:sc_finite} is the same as \cite[Theorem 1.3]{BD} when $N$ is replaced by $tN+\nu$.
\end{remark}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%          Section.     %  Doubled-distinct partitions
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Results on doubled distinct partitions}\label{sec:dd}

For a nonnegative integer $j$, let 
$
r'_j:=r'_j(\zeta):=\sum_{i=-j+1}^{j}\zeta^i.
$  

As seen in \Cref{sec:pre}, doubled distinct partitions are completely determined by their hook lengths in the main diagonal, which are distinct even numbers. Thus, for a doubled distinct partition $\lambda\lambda$ with
$
MD(\lambda\lambda)=\{2d_1,2d_2,\dots,2d_s \},
$
we see that $\omega_{t}(\lambda\lambda)=\sum\limits_{j=1}^s r'_{d_j}$. 

\iffalse
For example in \Cref{fig:dd-gbg}, when $\lambda\lambda=(8,8,6,6,4,2,2)$ is a doubled distinct partition with $MD(\lambda\lambda)=\{14,12,6,4\}$, we check that $\omega_t(D_1)=r'_7$, $\omega_t(D_2)=r'_6$, $\omega_t(D_3)=r'_3$, and $\omega_t(D_4)=r'_2$.


\begin{figure}[ht!]
        \centering
        \begin{ytableau}
                *(gray!50)1&\zeta&\zeta^2&\zeta^3&\zeta^4&\zeta^5&\zeta^6&\zeta^7\\
                \zeta^{-1}&*(gray!50)1&\zeta&\zeta^2&\zeta^3&\zeta^4&\zeta^5&\zeta^6\\
                \zeta^{-2}&\zeta^{-1}&*(gray!50)1&\zeta&\zeta^2&\zeta^3\\
                \zeta^{-3}&\zeta^{-2}&\zeta^{-1}&*(gray!50)1&\zeta&\zeta^2\\
                \zeta^{-4}&\zeta^{-3}&\zeta^{-2}&\zeta^{-1}\\
                \zeta^{-5}&\zeta^{-4}\\
                \zeta^{-6}&\zeta^{-5}
        \end{ytableau}
        \caption{The $t$-GBG diagram of the doubled distinct partition $\lambda\lambda=(8,8,6,6,4,2,2)$ with $MD(\lambda)=\{14,12,6,4\}$, where $\zeta=\zeta_t$}\label{fig:dd-gbg}
\end{figure}
\fi

In a similar way to the self-conjugate partition case, we get the following generating function for doubled distinct partitions with its GBG-rank mod $t$.


\begin{theorem}\label{thm:gen}
	We have
	\begin{align*}
		\sum_{\lambda\lambda\in\mathcal{DD}}x^{\omega_t(\lambda\lambda)}q^{|\lambda\lambda|}=
		\begin{dcases}
			\frac{(-q^{2t};q^{2t})_{\infty}}{(q^{2t};q^{2t})_{\infty}^{(t-1)/2}}\prod_{j=1}^{{(t-1)/2}}\sum_{n=-\infty}^{\infty}x^{nr'_j}q^{tn^2-(t-2j)n}&\text{if $t$ is odd},\\
			\frac{(-q^{t};q^{2t})_{\infty}(-q^{2t};q^{2t})_{\infty}}{(q^{2t};q^{2t})_{\infty}^{t/2-1}}\prod_{j=1}^{t/2-1}\sum_{n=-\infty}^{\infty}x^{nr'_j}q^{tn^2-(t-2j)n}&\text{if $t$ is even}.
		\end{dcases}
	\end{align*}
\end{theorem}

We also get the finite form of the results in \Cref{thm:gen}. 
For a nonnegative integer $N$, let $\mathcal{DD}_N$ be the set of doubled distinct partitions into parts less than or equal to $N$. By the definition, each partitions in $\mathcal{DD}_N$ has at most $N-1$ parts.

\begin{theorem}\label{thm:dd_main}
	For a nonnegative integer $N$, we have 
	\begin{align*}
		&\sum_{\lambda\lambda\in\mathcal{DD}_N}x^{\omega_t(\lambda\lambda)}q^{|\lambda\lambda |}\\
		&=\begin{cases}
			(-q^{2t};q^{2t})_{N_{t,t}}{\displaystyle\prod_{j=1}^{(t-1)/2}\sum_{n=-N_{t,t-j}}^{N_{t,j}}}\qbinom{N_{t,j}+N_{t,t-j}}{ N_{t,j}-n}_{q^{2t}} x^{nr'_j}q^{tn^2-(t-2j)n} & \text{if $t$ is odd}, \\
			(-q^{t};q^{2t})_{N_{t, \frac{t}2}}(-q^{2t};q^{2t})_{N_{t,t}}{\displaystyle\prod_{j=1}^{t/2-1}\sum_{n=-N_{t,t-j}}^{N_{t,j}}} \qbinom{N_{t,j}+N_{t,t-j}}{ N_{t,j}-n}_{q^{2t}} x^{nr'_j}q^{tn^2-(t-2j)n} &\text{ if $t$ is even},
		\end{cases}
	\end{align*}
	where $N_{t,j}=\lfloor\frac{N-j-1}{t}\rfloor +1$.
\end{theorem}




Let $GDD_t (k,q)$ be the generating function for the number of doubled distinct partitions with the GBG-rank mod $t$ equal to $k$. We have the following results.
\begin{theorem}\label{thm:dd_int}
	For an integer $k$, we have
	\begin{align*}
		GDD_{t}(0,q)=\begin{dcases}
		\frac{(-q^{2t};q^{2t})_{\infty}}{(q^{2t};q^{2t})_{\infty}^{(t-1)/2}}\quad&\text{if $t$ is odd},\\
		\frac{(-q^{t};q^{2t})_{\infty} (-q^{2t};q^{2t})_{\infty}}{(q^{2t};q^{2t})_{\infty}^{t/2-1}}\quad&\text{if $t$ is even},
		\end{dcases}
	\end{align*}
	and $GDD_{t}(k,q)=0$ for $k\neq 0$. Moreover, for a doubled distinct partition $\lambda\lambda$, if $\omega_t(\lambda\lambda)\in \mathbb{Z}$, then $\omega_t(\lambda\lambda)=0$.
\end{theorem}

\begin{theorem}
	For an odd prime $t$ and an integer $k$, we have
	\[
		GDD_{t}(k+k\zeta_t,q)=\frac{(-q^{2t};q^{2t})_{\infty}q^{tk^2-(t-2)k}}{(q^{2t};q^{2t})_{\infty}^{(t-1)/2}}.
	\]
	%where $t=2\ell +1$.
\end{theorem}

For a nonnegative integer $N$, let $GDD_{N,t}(k,q)$ be the generating function for the number of doubled distinct partitions into parts less than or equal to $N$ with the GBG-rank mod $t$ equal to $k$. %By the definition, each partitions in $GDD_{N,t}(k,q)$  has at most $N-1$ parts.

\begin{theorem}
Let $t\ge 3$. For integers $k,N$ with $N\ge 0$, $GDD_{N,t}(k,q)=0$ when $k\neq 0$ and 
	\begin{align*}
		GDD_{N,t}(0,q)=\begin{cases}
			(-q^{2t};q^{2t})_{N_{t,t}} {\displaystyle\prod_{j=1}^{(t-1)/2}}\qbinom{N_{t,j}+N_{t,t-j}}{ N_{t,j}}_{q^{2t}} &\text{if $t$ is odd},\\
			(-q^{t};q^{2t})_{N_{t,\frac{t}{2}}}(-q^{2t};q^{2t})_{N_{t,t}}{\displaystyle\prod_{j=1}^{t/2-1}} \qbinom{N_{t,j}+N_{t,t-j}}{ N_{t,j}}_{q^{2t}} &\text{if $t$ is even},
		\end{cases}
	\end{align*}
	where $N_{t,j}=\left\lfloor\frac{N-j-1}{t}\right\rfloor +1$.
\end{theorem}

\iffalse
For now, we have only considered the case when the GBG-rank is an integer so far. Recall that $MD(\lambda\lambda)=\{2\lambda_1,2\lambda_2,\dots,2\lambda_\ell\}$ when $\lambda=(\lambda_1,\lambda_2,\dots,\lambda_\ell)$ is a partition into distinct parts, we obtain $\omega_t(\lambda\lambda)=\sum_{i=1}^{\ell}r'_{\lambda_i}$. We obtain that $\omega_t(\lambda\lambda)=x$ for some $\lambda\lambda\in \mathcal{D}\mathcal{D}$ means that
\[
	x=(1+\zeta_t)f(\zeta_t+\zeta_t^{-1})
\]
for some integer coefficient polynomial $f$. Furthermore, we get the following theorem.
\fi





\acknowledgements{The authors are grateful to the referees for their careful reading and helpful comments. This work was initiated at the KIAS-KRIMS-KWMS Summer School on Modular Forms and Partition Theory.}

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