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\title{\bf Group Actions on Partitions}

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\author{Byungchan Kim\thanks{This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2016R1D1A1A09917344).}\\
\small School of Liberal Arts\\[-0.8ex]
\small Seoul National University of Science and Technology\\[-0.8ex] 
\small Seoul, Republic of Korea\\
\small\tt bkim4@seoultech.ac.kr
}

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\date{\dateline{Dec 9, 2016}{Sep 1, 2017}{Sep 22, 2017}\\
\small Mathematics Subject Classifications: 05A17, 11P81}

\begin{document}

\maketitle



\begin{abstract}
We introduce group actions on the integer partitions and their variances. Using generating functions and Burnside's lemma, we study arithmetic properties of the counting functions arising from group actions. In particular, we find a modulo 4 congruence involving the number of ordinary partitions and the number of partitions into distinct parts.


  % keywords are optional
  \bigskip\noindent \textbf{Keywords:} Partitions, unimodal sequences, group action, partition congruence, Bailey pairs, Bailey Lemma
\end{abstract}

\section{Introduction}

  In the theory of integer partitions, the conjugation is one of the most important operations on partitions. The Durfee square of size $d \times d$ in the partition $\lambda$ is defined by the largest integer $d$ such that the partition $\lambda$ has at least $d$ parts $\geq d$. Using the Durfee square, we can represent the partition $\lambda$ by $(d, \pi_1, \pi_2)$, where $d$ is the size of the Durfee square and $\pi_1$ and $\pi_2$ are partitions into parts $\leq d$. Let $D$ be the set of partitions represented by the Durfee square decomposition
\[
D := \{ (d, \pi_1, \pi_2 ) : d \geq 0, \text{ $\pi_1$ and $\pi_2$ are partitions into parts $\leq d$} \}.
\]

For the symmetric group $G=S_2$, we define a group action on $D$ by
\[
\sigma (d, \pi_1, \pi_2) = (d, \pi_{\sigma(1)}, \pi_{\sigma(2)} ),
\]
for $\sigma \in G$. When $\sigma= (1 2)$, this group action corresponds to the conjugation of the partition. In the light of that we can understand the conjugation of partitions as a group action, it is natural to ask whether there are interesting consequences on partitions in the view of group actions. Moreover, once we define a group action on the set $X$, then it  is also natural to examine how many orbits there are. For an element $x$ in an orbit, we define the weight of $x$ as an element of $X$ and we define $|X/G|(n)$ is the number of orbits of weight $n$. For example, $|D/S_2|(n)$ is the number of partitions of $n$, where if two partitions are conjugate to each other, then they are considered to be the same partition. 



The goal of this paper is investigating properties of counting functions arising from group actions on various partition sets $X$ and groups $G$.  To this end, we will use Burnside's lemma frequently, which says that  the number of orbits $| X / G|$ is given by
\[
| X / G| = \frac{1}{|G|} \sum_{g \in G} |X^{g}|,
\]
where $X^{g}$ is the invariant subset of $X$ under the action of $g$. %We will use Burnside's lemma frequently to deduce the generating function for $|X/G|(n)$.


One of the main results is a modulo 4 congruence involving the number of ordinary partitions and the number of partitions into distinct parts.  Before stating the result, we introduce some notation. Let $p(n)$ be the number of ordinary partitions of $n$ and let $d(n)$ be the number of partitions of $n$ into distinct parts. For a given partition $\lambda$, $\ell(\lambda)$ is defined by the largest part of $\lambda$. If $\lambda$ is an empty partition, then we define $\ell(\lambda)=0$.


\begin{theorem}\label{main1}
Let $s(n)$ be the number of partitions of $n$ of which Durfee decompositions $(d, \pi_1, \pi_2)$ satisfy $\ell(\pi_1 ) = \ell(\pi_2)$. Then, for all non-negative integers $n$,
\[
p(n) + s(n) \equiv 2d(n) \pmod{4}.
\]
\end{theorem}


\begin{remark}
By Euler's pentagonal number theorem, we know that
\[
2d(n) \equiv 
\begin{cases}
2 \pmod{4}, &\text{if $n = \frac{k(3k-1)}{2}$ for an integer $k$,}\\
0 \pmod{4}, &\text{otherwise.}
\end{cases}
\]
\end{remark}



In the next section we prove Theorems \ref{main1}, and we consider multi-partitions and unimodal sequences under group actions in the remainder of the paper.




\section{Proof of Theorem \ref{main1}}
Let $C_2$ be the set defined by
\[
C_2:=\{ (d, \nu_1, \nu_2 ) : d \geq 0, \text{ $\nu_1$ and $\nu_2$ are compositions into non-negative integers $\leq d$} \}.
\]
Note that we allow $\nu_i$'s can have zeros as parts and recall that two sequences that differ in the order of their terms define different compositions, while they are considered to be the same partition. The weight of an element $(d,\nu_1,\nu_2) \in C_2$ is defined by $d^2$ plus the sum of parts in $\nu_1$ and the first part of $\nu_2$. For the convenience, the empty composition is considered to be $\{ 0 \}$ so that we can define the first part of an empty composition is zero.  For the symmetric group $G=S_2$, we define a group action on $C_2$ by exchanging the first part of $\nu_i$ and $\nu_{\sigma(i)}$ for $i=1$ or $2$. By employing Burnside's lemma, we find that
\[
|C_2/S_2| (n) = \frac12 \left(  |C_{2}^{(1)}|(n) + |C_{2}^{(1 2)}| (n)    \right).
\]
Here and in the sequel, for a given set $X$, $|X|(n)$ denotes the number of elements in the set $X$ with the weight $n$. Now we will count how many partitions there are among the elements in the invariant subsets. To this end, we define
\[
|C_2/S_2|_D (n) :=  \frac12 \left(  |C_{2}^{(1)} \cap D |(n) + |C_{2}^{(1 2)} \cap D| (n)    \right),
\]
where $D$ is the set of partitions represented by Durfee square decompositions.  By employing general combinatorial arguments \cite{And:Book}, we obtain that
\[
\begin{aligned}
\sum_{n=0}^{\infty} |C_{2}^{(1)} \cap D |(n) q^n &= \sum_{n=0}^{\infty} \frac{q^{n^2}}{(q)_{n}^{2}} \\
\sum_{n=0}^{\infty} |C_{2}^{(1 2)} \cap D |(n) q^n &=  \sum_{n= 0}^{\infty} q^{n^2} \sum_{k=0}^{n} \frac{q^{2k}}{(q)_{k}^2},
\end{aligned}
\]
which implies that
\begin{equation}
\label{psteq1}
\sum_{n=0}^{\infty} |C_{2}/S_2|_D (n) q^n = \frac12 \left(  \frac{1}{(q)_{\infty}}  +  \sum_{n= 0}^{\infty} q^{n^2} \sum_{k=0}^{n} \frac{q^{2k}}{(q)_{k}^2} \right). 
\end{equation}

Here we have  used the usual $q$-series notation,
\[
(a)_n = (a;q)_n := \prod_{j=0}^{n-1} (1-aq^{j})
\]
for $n \in \mathbb{N}_{0} \cup \{ \infty \}$. 

By using the inclusion-exclusion principle, we derive that
\begin{equation}\label{qn2}
\frac{1}{(q)_{n}^{2}} = 2 \sum_{k=0}^{n} \frac{q^k}{(q)_{k}^{2}} - \sum_{k=0}^{n} \frac{q^{2k}}{(q)_{k}^{2}}.
\end{equation}

Therefore, we find that 
\[
f(q):=\sum_{n=0}^{\infty} |C_{2}/S_2|_D (n)q^n = \sum_{n =0}^{\infty} q^{n^2}\sum_{k=0}^{n} \frac{q^k}{(q)_{k}^{2}}.
\]

It turns out that we can reformulate $f(q)$ by using a false theta function.


\begin{lemma}\label{keylemma}
We have
\[
f(q) = \frac{1}{(q)_{\infty}} \left( 1 - \sum_{n =1}^{\infty} q^{n(3n-1)} (1-q^{2n}) \right).
\]
\end{lemma}

Before proving the above lemma, we show that Lemma \ref{keylemma} implies Theorem \ref{main1}.

\begin{proof}[Proof of Theorem \ref{main1}]
We first note that \eqref{psteq1} implies that
\begin{equation}\label{psteq2}
f(q) =\sum_{n=0}^{\infty} |C_{2}/S_2 |_D (n) q^n = \frac12 \left( \sum_{n=0}^{\infty} p(n) q^n + \sum_{n=0}^{\infty} s(n) q^n \right).
\end{equation}
From Euler's pentagonal number theorem \cite[Cor 1.7]{And:Book}, we find that 
\[
(q^2; q^2)_{\infty} = 1 + \sum_{n=1}^{\infty} (-1)^n q^{n(3n-1)} (1 + q^{2n} ).
\]
Therefore, by Lemma \ref{keylemma}, we obtain that
\[
f(q) \equiv \frac{ (q^2; q^2)_{\infty}}{(q)_{\infty}} \equiv (-q)_{\infty} \pmod{2}.
\]
From \eqref{psteq2} and the fact that 
\[
(-q)_{\infty} = \sum_{n=0}^{\infty} d(n),
\]
we conclude Theorem \ref{main1}.
\end{proof}

Now we turn to prove Lemma \ref{keylemma}.

\begin{proof}[Proof of Lemma \ref{keylemma}]
First we recall that a Bailey pair relative to $a$ is a pair of sequences $(\alpha_n,\beta_n)_{n \geq 0}$ satisfying
\begin{equation} \label{pairdef}
\beta_n = \sum_{k=0}^n \frac{\alpha_k}{(q)_{n-k}(aq)_{n+k}}.
\end{equation}
If $(\alpha_n, \beta_n)$ is a Bailey pair relative to 1, then Bailey Lemma (see \cite{WarSur} for example) implies that
\begin{equation}\label{BaileyL}
\frac{1}{(q)_{\infty}} \sum_{n =0}^{\infty} q^{n^2} \alpha_n = \sum_{n=0}^{\infty} q^{n^2} \beta_n.
\end{equation}
From \cite{Sl1}, we find that
\begin{equation} \label{bpair1}
\begin{aligned}
\alpha_0 &=1, \\
\alpha_n &= -q^{n(n-1)} (1-q^{2n} ) \text{ for $n >0$,} \\
\beta_n &= \frac{q^n}{(q)_{n}^2}
\end{aligned}
\end{equation}
consists of a Bailey pair relative to $1$.  Recall (see \cite{An-Hi1}, for example) that if $(\alpha_n, \beta_n)$ is a Bailey pair relative to $a$, then 
\begin{align*}
\alpha_{n}' &= a^n q^{n^2} \alpha_n,  \\
\beta_{n}' &= \sum_{k=0}^{n} \frac{a^k q^{k^2}}{(q)_{n-k}} \beta_k 
\end{align*}
is also a Bailey pair relative to $a$. Using this with \eqref{bpair1}, we obtain a new Bailey pair relative to $1$, 
\begin{equation} \label{bpair2}
\begin{aligned}
\alpha_0' &=1, \\
\alpha_n' &= -q^{n(2n-1)} (1-q^{2n} ) \text{ for $n >0$,} \\
\beta_n' &= \sum_{k=0}^{n} \frac{q^{k^2+k}}{(q)_{n-k} (q)_{k}^2}.
\end{aligned}
\end{equation}
Using Andrews' finite Heine transformation \cite[Corollary 4]{An.1}
\[
\sum_{k=0}^{n} \frac{ (q^{-n})_k (\alpha)_{k} (\beta)_{k} q^{k}}{(q)_{k} (\gamma)_{k} (q^{1-n} /\tau)_{k}} = \frac{ (\alpha \beta \tau / \gamma )_{n}}{(\tau)_{n}} \sum_{k=0}^{n} \frac{ (q^{-n})_{k} (\gamma/\alpha)_{k} (\gamma/\beta)_{k} q^k}{ (q)_{k} (\gamma)_{k} (\gamma q^{1-n} / (\alpha \beta \tau) )_{k}}
\]
with $\alpha, \beta \to 0$ and $\gamma=\tau=q$, we derive that
\begin{equation}\label{bpair3}
\beta_{n}' = \sum_{k=0}^{n} \frac{ q^{k^2 +k}}{(q)_{n-k}(q)_{k}^{2}} = \sum_{k=0}^{n} \frac{q^{k}}{(q)_{k}^{2}}.
\end{equation}
From \eqref{bpair2}, \eqref{bpair3}, and \eqref{BaileyL}, we obtain the desirable identity.
\end{proof}



Note that
\begin{equation} \label{xxx}
\sum_{n=0}^{\infty} |C_2/S_2|_D (n) q^n = \sum_{n=0}^{\infty} \frac{q^{n^2}}{(q)_{n}} \sum_{k=0}^{n} \frac{ q^{k^2 +k}}{(q)_{k}} \begin{bmatrix} n \\ k \end{bmatrix},
\end{equation}
where we use the $q$-binomial coefficient defined by 
\begin{equation} 
\begin{bmatrix} n \\ k \end{bmatrix} = \frac{(q)_n}{(q)_k(q)_{n-k}}.
\end{equation}
Recall that the $q$-binomial coefficient $\begin{bmatrix} n \\ k \end{bmatrix}$ generates the partitions which fit inside a $(n-k) \times k$ rectangle. From \eqref{xxx}, we can see that $|C_2/S_2|_D (n)$ is the number of partitions of $n$ with certain Durfee dissections (See \cite{An.2} for more details on Durfee dissections.).


\begin{corollary}
Let $D_2 (n)$ be the number of partitions of $n$ with 2 successive Durfee squares of sizes $d_1 \times d_1$ and $d_2 \times d_2$ such that there is a part of size $d_2$ below the second Durfee square.  Then, for all non-negative integers $n$,
\[
|C_2/S_2|_D (n) = D_2 (n) = \frac{1}{2} \left(p(n) + s(n)\right).
\]
\end{corollary}





\section{Other Results} 

 In this section, we study the number of orbits in the sets of multi-partitions and in the sets of unimodal sequences under group actions.
 

\subsection{Multi-partitions}

Let $P_2$ be the set of bi-partitions
\[
P_2 := \{ (\lambda_1, \lambda_2 ) : \text{ $\lambda_1$ and $\lambda_2$ are ordinary partitions} \}.
\]
For the group $G=S_2$, we define a  group action on $P_2$ by 
\[
\sigma (\lambda_1, \lambda_2 ) = (\lambda_{\sigma(1)}, \lambda_{\sigma(2)} ).
\]

Let $|P_2 / S_2| (n)$ be the number of orbits with weight $n$. Then, $|P_2 /S_2|(n)$ is the number of bi-partitions of $n$, where  bi-partitions in the same orbit define the same bi-partition. By simple combinatorial arguments and Burnside's formula, we find that 
\[
\sum_{n=0}^{\infty} |P_2 / S_2| (n) q^n = \frac{1}{2} \left( \frac{1}{(q)_{\infty}^2} + \frac{1}{(q^2 ;q^2)_{\infty}} \right).
\]

Let $p_{-2} (n)$ be the number of bi-partitions of $n$. Then, we have
\[
\sum_{n=0}^{\infty} p_{-2} (n) q^n = \frac{1}{(q)_{\infty}^2}.
\]
From the famous Ramanujan congruence $p(5n+4)\equiv 0 \pmod{5}$ and a bi-partition congruence \cite{HL} $p_{-2} (5n+3) \equiv 0 \pmod{5}$, we can easily see that 
\[
|P_2 / S_2 | (5n+3) \equiv 0 \pmod{5}.
\]

We define the set of tri-partitions   
\[
P_3 := \{ (\lambda_1, \lambda_2, \lambda_3) : \text{ $\lambda_1$, $\lambda_2$, and $\lambda_3$ are ordinary partitions} \}.
\]

For $G= S_3$, we define a group action on $P_3$ by 
\[
\sigma (\lambda_1, \lambda_2, \lambda_3) = (\lambda_{\sigma(1)}, \lambda_{\sigma(2)}, \lambda_{\sigma(3)} ).
\]

We may think each partition $\lambda_i$ is on a vertex of a regular triangle. Then, we can understand $|P_3 /S_3| (n)$ is the number of tri-partitions of $n$, where the partitions are considered to be the same if they are invariant under a reflection or a rotation fixing the triangle. From Burnside's lemma, we observe that
\[
\sum_{n=0}^{\infty} |P_3 / S_3 | (n) q^n = \frac{1}{6} \left( \frac{1}{(q)_{\infty}^3} +  \frac{3}{(q)_{\infty} (q^2;q^2)_{\infty}} + \frac{2}{(q^3 ; q^3)_{\infty}} \right).
\]

\begin{proposition}
For all non-negative integers $n$, 
\[
 |P_3 / S_3 | (3n+2) \equiv 0 \pmod{3}.
\] 
Moreover, 
\[
\sum_{n=0}^{\infty} |P_3 /S_3| (3n+2) q^n = \frac{3}{2} \left( \frac{(q^3 ;q^3)_{\infty}^{9}}{(q)_{\infty}^{12}}  + \frac{ (q^3;q^3)_{\infty}^{3} (q^6 ; q^6)_{\infty}^{3}}{ (q)_{\infty}^{4} (q^2 ;q^2)_{\infty}^{4}} \right).
\]
\end{proposition}

\begin{proof}
H.-C. Chan \cite{Chan} defined the cubic partition function $c(n)$ by
\[
\sum_{n=0}^{\infty} c(n) q^n = \frac{1}{(q)_{\infty} (q^2 ; q^2)_{\infty}},
\]
and proved that $c(3n+2) \equiv 0 \pmod{3}$. Therefore, it suffices to show that 
\begin{equation} \label{p3con}
p_{-3} (3n+2) \equiv 0 \pmod{9},
\end{equation}
where $p_{-3} (n)$ is defined by
\[
\sum_{n=0}^{\infty} p_{-3} (n) q^n = \frac{1}{(q)_{\infty}^{3}}.
\]
We define the Dedekind eta function $\eta(z)$ by
\[
\eta(z) = q^{1/24} (q)_{\infty} \qquad \text{with $q=\exp (2\pi i z)$.}
\]
Since $\frac{\eta^9 (3z)}{\eta^3(z)}$ is a modular form of weight 3 and level 3 with character $\left( \frac{-3}{\cdot} \right)$ and the dimension of such modular forms space is 2, we find that
\begin{equation}\label{mod1eq}
\frac{\eta^9 (3z)}{\eta^3(z)} |U_3 = 9 \frac{\eta^9 (3z)}{\eta^3(z)}
\end{equation}
by checking the first two coefficients. Here, $U_3$ is an operator defined by $\left( \sum a(n) q^n \right)|U_3 := \sum a(3n)q^n$. (For more details on modular forms, see \cite{Ono} for example.) 
From \eqref{mod1eq}, we find that
\begin{equation}\label{ccc1}
\begin{aligned}
\frac{\eta^9 (3z)}{\eta^3(z)} |U_3 &= \left( (q^3; q^3)_{\infty}^{9} \sum_{n=0}^{\infty} p_{-3} (n) q^{n+1} \right)\big|U_3 \\
&= (q)_{\infty}^{9} \sum_{n=0}^{\infty} p_{-3} (3n+2) q^{n+1} \\
&=9 q \frac{(q^3 ; q^3)_{\infty}^{9}}{(q)_{\infty}^{3}},
\end{aligned}
\end{equation}
which proves $p_{-3} (3n+2) \equiv 0 \pmod{9}$ as desired. We also note that $\frac{\eta^3 (3z) \eta^{3} (6z)}{\eta(z) \eta(2z)}$ is a modular form of weight 2 and level 6. Proceeding as before (for this time, we need to check the first three coefficients), we deduce that
\[
\frac{\eta^3 (3z) \eta^{3} (6z)}{\eta(z) \eta(2z)}|U_3 = 3 \frac{\eta^3 (3z) \eta^{3} (6z)}{\eta(z) \eta(2z)}.
\]
In terms of cubic partitions, the above modular identity is equivalent to
\begin{equation}\label{ccc2}
(q)_{\infty}^{3} (q^3;q^3)_{\infty} \sum_{n=0}^{\infty} c(3n+2) q^n = 3\frac{(q^3;q^3)_{\infty}^{3} (q^6;q^6)_{\infty}^{3}}{(q)_{\infty} (q^2;q^2)_{\infty}}.
\end{equation}
From \eqref{ccc1} and \eqref{ccc2}, we can obtain the desirable generating function for $|P_3 /S_3|(3n+2)$.
\end{proof}


\subsection{Unimodal sequences}

 Recall that a unimodal sequence is a sequence which is weakly increasing up to a point (called the peak), and then weakly decreasing thereafter. The weight of such a sequence is the sum of all of its terms.  We define the set of unimodal sequeces  
\[
U := \{ (p, \pi_1, \pi_2 ) : p \geq 0, \text{ $\pi_1$ and $\pi_2$ are partitions into parts $\leq p$} \}.
\]
For an element $(p, \pi_1, \pi_2)$, its weight is defined by $p$ plus the sum of parts in $\pi_1$ and $\pi_2$. Recall that $C_2$ is the set defined by
\[
C_2:=\{ (d, \nu_1, \nu_2 ) : d \geq 0, \text{ $\nu_1$ and $\nu_2$ are compositions into non-negative integers $\leq d$} \}.
\]
To match the weights, in this subsection, the weight of an element $(d,\nu_1,\nu_2) \in C_2$ is redefined by $d$ plus the sum of parts in $\nu_1$ and $\nu_2$.  For the symmetric group $G=S_2$, we define a group action on $C_2$ by exchanging the first part of $\nu_1$ and the first part of $\nu_2$.  By employing Burnside's lemma, we find that
\[
|C_2/S_2| (n) = \frac12 \left(  |C_{2}^{(1)}|(n) + |C_{2}^{(1 2)}| (n)    \right).
\]
 Here we will count how many unimodal sequences there are among the elements in the invariant subsets. To this end, we define
\[
|C_2/S_2|_U (n) :=  \frac12 \left(  |C_{2}^{(1)} \cap U |(n) + |C_{2}^{(1 2)} \cap U | (n)    \right).
\]


Then, we can see that 
\[
\begin{aligned}
\sum_{n =0}^{\infty} |C_2/S_2 |_U (n) q^n &= \frac12 \left(  \sum_{n=0}^{\infty} \frac{q^{n}}{(q)_{n}^{2}}   +  \sum_{n=0}^{\infty}  q^n \sum_{k=0}^{n} \frac{q^{2k}}{(q)_{k}^2} \right) \\
&=\sum_{n \geq k \geq 0} \frac{q^{n+k}}{(q)_{k}^{2}},
\end{aligned}
\]
where we have used \eqref{qn2} for the last equality.  From this, we find that $|C_2 /S_2|_U (n)$ is the number of unimodal sequences of weight $n$ such that the largest part in the partition before the peak is larger than or equal to the largest part in the partition after the peak.



From \cite[Theorem 1.1]{JL}, we find that
\begin{equation}\label{JL_thm}
\sum_{n=0}^{\infty} q^n \beta_n = \frac{1}{(aq)_{\infty} (q)_{\infty}} \sum_{n,r \geq 0} (-a)^{n} q^{n(n+1)/2 + (2n+1)r} \alpha_r,
\end{equation}
where $(\alpha_n, \beta_n)$ is a Bailey pair relative to $a$.  Therefore, by plugging the Bailey pairs \eqref{bpair2} and \eqref{bpair3} into \eqref{JL_thm}, we derive that
\[
\begin{aligned}
\sum_{n \geq k \geq 0} \frac{q^{n+k}}{(q)_{k}^{2}} &= \frac{1}{(q)_{\infty}^{2}} \sum_{n=0}^{\infty} (-1)^n q^{n(n+1)/2} \left( 1 - \sum_{r=1}^{\infty}  q^{(2n+1)r +r(2r-1)} (1-q^{2r} ) \right) \\
&=\frac{1}{(q)_{\infty}^{2}}\sum_{n, r \geq 0} (-1)^n q^{n(n+1)/2 + (2n+1)r + r(2r+1)} \\
&\qquad- \frac{1}{(q)_{\infty}^2} \sum_{n\geq 0, r>0} (-1)^n q^{n(n+1)/2 + (2n+1)r + r(2r-1)} \\
&=\frac{1}{(q)_{\infty}^{2}}  \left( \sum_{n, r \geq 0}  + \sum_{n, r < 0} \right)  (-1)^n q^{n(n+1)/2 + (2n+1)r + r(2r+1)},
\end{aligned}
\]
where we replace $n$ and $r$ by $-n-1$ and $-r$ in the second summation for the last equality. In summary, we have obtained a Hecke-type double sum expression for the $|C_2 / S_2|_U (n)$ generating function.
 
\begin{theorem} 
We have
\begin{equation} \label{falseindef}
\sum_{n =0}^{\infty} |C_2/S_2 |_U (n) q^n = \frac{1}{(q)_{\infty}^{2}}  \left( \sum_{n, r \geq 0}  + \sum_{n, r < 0} \right)  (-1)^n q^{n(n+1)/2 + 2nr + 2r(r+1)}.
\end{equation}
\end{theorem} 

We may think the double sum in the right hand side of \eqref{falseindef} as a false theta series in the sense of that it has a wrong sign compare to the theta series
\begin{equation} \label{fdef}
f_{a,b,c}(x,y,q) = \left(\sum_{r,s \geq 0} - \sum_{r,s < 0}\right) (-1)^{r+s}x^ry^sq^{a\binom{r}{2} + brs + c\binom{s}{2}},
\end{equation}
which is closely related to mock theta functions \cite{Hi-Mo1}.
 
 
In a recent paper \cite{KL}, the author and J. Lovejoy introduced a unimodal T-sequence. In a unimodal T-sequence, we have a peak and there are partitions into parts $\leq$ the peak in the right, in the left, and below of the peak.  We define the set of unimodal T-sequences 
\[
T := \{ (p, \pi_1, \pi_2, \pi_3 ) : p \geq 0, \text{ $\pi_1$, $\pi_2$, and $\pi_3$ are partitions into parts $\leq p$} \}.
\]
We also define a corresponding composition set $C_3$ by
\begin{align*}
C_3:=\{ (p, \nu_1, \nu_2, \nu_3 ) : p \geq 0, & \,\,\, \nu_1, \nu_2 \text{ and } \nu_3 \text{ are compositions } \\ &\text{ into non-negative integers }\leq d \}.
\end{align*}
For an element in $T$ (resp. $C_3$), its weight is defined by the size of the peak (i.e. $p$) plus the sum of parts in the partitions (resp. compositions).

For $G=S_3$, we define a group action on $C_3$ by switching the first part of $\nu_i$ and  the first part of $\nu_{\sigma(i)}$. As before, we investigate 
\[
| C_3 / S_3 |_T (n) := \frac{1}{6} \sum_{g \in S_3} | C_{3}^{g} \cap T| (n).
\]

Then, we find that
\[
\sum_{n=0}^{\infty} |C_3/S_3 |_T (n) q^n = \frac{1}{6} \sum_{n=0}^{\infty} \left( \frac{q^{n}}{(q)_{n}^{3}} + \frac{3q^n}{(q)_{n}} \sum_{k=0}^{n}  \frac{q^{2k}}{(q)_{k}^{2}} + 2q^n \sum_{k=0}^{n} \frac{q^{3k}}{(q;q)_{k}^{3}}  \right).
\]

By employing the inclusion-exclusion principle on the largest parts, we find that
\[
\frac{1}{(q)_{n}^{3}} = 3\sum_{k=0}^{n} \frac{q^k(1-q^k)}{(q)_{k}^{3}} + \sum_{k=0}^{n} \frac{q^{3k}}{(q)_{k}^{3}}.
\]
From \eqref{qn2}, \eqref{bpair3}, and the above, we derive that 
\[
\begin{aligned}
\sum_{n=0}^{\infty} |C_3/S_3 |_T (n) q^n &= \sum_{n=0}^{\infty} \frac{q^{n}}{(q)_{n}} \sum_{k=0}^{n} \frac{q^k}{(q)_{k}^{2}} - \sum_{n=0}^{\infty} q^{n} \sum_{k=0}^{n} \frac{q^k (1-q^k) }{(q)_{k}^{3}} \\
&=\sum_{n=0}^{\infty} \frac{q^{n}}{(q)_{n}^{2}} \sum_{k=0}^{n} \frac{q^{k^2+k}}{(q)_{k}} \begin{bmatrix} n \\ k \end{bmatrix}- \sum_{n \geq k \geq 0} \frac{ q^{n+k} (1-q^k)}{(q)_{k}^{3}}.
\end{aligned}
\]

\begin{proposition}
Let $T_1 (n)$  be the number of unimodal T-sequences of weight $n$ such that the partition below the peak has the Durfee rectangle of size $(k+1) \times k$. Let $T_2 (n)$ be the number of unimodal $T$-sequences of weight $n$ such that if the largest part of the partition in the right of the peak is $k$, then the largest part of the partition in the left (resp. below) of the peak is less than or equal to (resp. less than) $k$. Then, for all non-negative integers $n$,
\[
|C_3/S_3|_T (n) = T_1 (n) - T_2 (n).
\]
In particular, $T_1 (n) > T_2 (n)$ for all positive integers $n$.
\end{proposition}











%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection*{Acknowledgements}
The author would like to thank Bruce Berndt, Jeremy Lovejoy, and the referee for their valuable comments on an earlier version of this paper. The author is grateful to the referee for pointing out mistakes on the original definitions for group actions.  The author thanks the hospitality from the University of Illinois at Urbana-Champaign, where this work has been accomplished. 

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

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\end{document}
