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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\title{\bf A Combinatorial Proof of an Identity\\ 
from Ramanujan's Lost Notebook}

\author{Bernard L.S. Lin\\
\small School of Sciences\\[-0.8ex]
\small Jimei University\\[-0.8ex]
\small Xiamen, 361021, P.R. China\\
\small\tt linlsjmu@163.com\\
\and
Jian Liu\\
\small School of Banking and Finance\\[-0.8ex]
\small University of International Business and Economics\\[-0.8ex]
\small Beijing, 100029, P.R. China\\
\small\tt liujian8210@gmail.com
\and
Andrew Y.Z. Wang\thanks{Corresponding author}\\
\small School of Mathematical Sciences\\[-0.8ex]
\small University of Electronic Science and Technology of China\\[-0.8ex]
\small Chengdu, 611731, P.R. China\\
\small\tt yzwang@uestc.edu.cn}

\date{\dateline{Dec 15, 2012}{Jun 14, 2013}{Jun 21, 2013}\\
\small Mathematics Subject Classifications: 11P81, 05A19}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{document}

\maketitle

\begin{abstract}
In this note, we present a new combinatorial proof of an identity
originally stated in Ramanujan's lost notebook by keeping track of
an extra statistic in Yee's bijection for a special case of this
identity.

\bigskip\noindent \textbf{Keywords:} partition identities, bijective proof, Ramanujan's lost notebook
\end{abstract}

%--------------------------------------------------------------------
\section{Introduction}\allowdisplaybreaks
In his introduction to Ramanujan's ``lost" notebook, applying an
identity of Rogers, Andrews\cite{Andrews79} proved the following
wonderful identity:
\begin{eqnarray}
\sum_{n=0}^\infty
\frac{q^n}{(aq;q)_n(bq;q)_n}&=&(1-a^{-1})\sum_{n=0}^\infty
\frac{(-1)^nq^{n(n+1)/2}b^na^{-n}}{(bq;q)_n}\nonumber \\
&&{}+a^{-1}\sum_{n=0}^\infty
(-1)^nq^{n(n+1)/2}b^na^{-n}\frac{1}{(aq;q)_\infty(bq;q)_\infty}.\label{eq2}
\end{eqnarray}

Throughout the article, we assume that $q$ is a complex number with
$|q| < 1$ and we adopt the following customary $q$-series notation:
\begin{align*}
&(a;q)_0=1,\\[5pt]
&(a;q)_n=\prod_{j=1}^n (1-aq^{j-1}),\,\mbox{for positive integer $n$}\\
&(a;q)_\infty=\prod_{n=1}^\infty (1-aq^{n-1}).
\end{align*}

In his survey of partition bijections, Pak~\cite{Pak06} pointed out
that finding a combinatorial proof of \eqref{eq2} is an open
problem. Recently, Kim \cite{Kim10} and Levande \cite{Levande10}
independently solved Pak's problem. Kim \cite{Kim10} rewrote  the
identity only by multiplying both sides of \eqref{eq2} by a factor
$a$ and gave a combinatorial proof to show how the cancellations
work in his rewritten identity. Levande \cite{Levande10} not only
gave an involution proof of \eqref{eq2} but also presented a
combinatorial proof of its rewritten form with a little imperfect as
having to deal with zero parts in partitions.

For the case $a=1$, identity \eqref{eq2} reduces to
\[
\sum_{n=0}^\infty
\frac{q^n}{(bq;q)_n(q;q)_n}=\frac{1}{(bq;q)_\infty(q;q)_\infty}\sum_{n=0}^\infty
(-b)^nq^{n(n+1)/2},\] which is an identity derived by Fine
\cite{Fine88}.

Yee~\cite{Yee09} presented a combinatorial proof of the above
identity by rewriting it in the following form
\begin{equation}\label{eq3}
\sum_{n=0}^\infty
\frac{q^n(-bq^{n+1};q)_\infty}{(q;q)_n}=\frac{1}{(q;q)_\infty}\sum_{n=0}^\infty
b^nq^{n(n+1)/2}.
\end{equation}

In this note, we aim to give an alternative combinatorial
interpretation of identity \eqref{eq2}, based on Yee's combinatorial
proof of identity \eqref{eq3}, by keeping track of an extra
statistic. In Section 2, we will rewrite identity \eqref{eq2} in an
appealing form whose both sides are positive. The main goal of
Section 3 is to present a combinatorial proof of our Ramanujan's
rewritten identity.

%--------------------------------------------------------------------
\section{Ramanujan's Rewritten Identity}
Multiplying identity \eqref{eq2} by $(bq;q)_\infty$ and then
replacing $b$ by $-b$ yields
\begin{eqnarray}
\sum_{n=0}^\infty \frac{q^n(-bq^{n+1};q)_\infty}{(aq;q)_n} &=&
(1-a^{-1})\sum_{n=0}^\infty
q^{n(n+1)/2}b^na^{-n}(-bq^{n+1};q)_\infty \nonumber\\
&&{}+a^{-1}\sum_{n=0}^\infty
q^{n(n+1)/2}b^na^{-n}\frac{1}{(aq;q)_\infty}.\label{eq4}
\end{eqnarray}

As the fact that
$(-bq^{n+1};q)_\infty=(1+bq^{n+1})(-bq^{n+2};q)_\infty$ and
\[
\sum\limits_{n=0}^\infty
q^{\frac{n(n+1)}{2}}b^na^{-n}(-bq^{n+1};q)_\infty=(-bq;q)_\infty
+\sum\limits_{n=0}^\infty
q^{\frac{n(n+1)}{2}}b^{n}a^{-(n+1)}bq^{n+1}(-bq^{n+2};q)_\infty,
\]
we deduce
\[(1-a^{-1})\sum_{n=0}^\infty
q^{\frac{n(n+1)}{2}}b^na^{-n}(-bq^{n+1};q)_\infty=(-bq;q)_\infty-\sum_{n=0}^\infty
q^{\frac{n(n+1)}{2}}b^na^{-(n+1)}(-bq^{n+2};q)_\infty.\]


Adding the term $\sum\limits_{n=0}^\infty
q^{n(n+1)/2}b^na^{-(n+1)}(-bq^{n+2};q)_\infty$ to both sides of
identity \eqref{eq4}, we obtain that
\begin{align*}
\sum_{n=0}^\infty \frac{q^n(-bq^{n+1};q)_\infty}{(aq;q)_n}
+\sum_{n=0}^\infty&
q^{\frac{n(n+1)}{2}}b^na^{-(n+1)}(-bq^{n+2};q)_\infty\\
&=(-bq;q)_\infty+a^{-1}\sum_{n=0}^\infty \frac{
q^{\frac{n(n+1)}{2}}b^na^{-n}}{(aq;q)_\infty},
\end{align*}
which is equivalent to
\begin{eqnarray}
\sum_{n=1}^\infty \frac{aq^n(-bq^{n+1};q)_\infty}{(aq;q)_n}
+\sum_{n=0}^\infty q^{\frac{n(n+1)}{2}}b^na^{-n}(-bq^{n+2};q)_\infty
=\sum_{n=0}^\infty\frac{
q^{\frac{n(n+1)}{2}}b^na^{-n}}{(aq;q)_\infty}.\label{eq5}
\end{eqnarray}

\noindent \textsc{Remarks}.
\begin{enumerate}
\item In fact, (1.2) as introduced by Levande in \cite{Levande10}
is not an identity. To see this, we compare the constant term on each
side. The constant term on the left-hand side is $\frac{a}{1-a}$,
but it is $\frac{1}{1-a}$ on the right-hand side.

\item The rewritten Ramanujan's identity obtained by Levande is
essentially
\begin{align*}
&\frac{\prod\limits_{k=1}^{\infty}(1+acq^k)}{1-a}+\sum\limits_{n=1}^{\infty}
\frac{aq^n\prod\limits_{k=n+1}^{\infty}(1+acq^k)}{(1-a)(1-aq)\cdots(1-aq^n)}
+\sum\limits_{n=1}^\infty q^{{n+1}\choose
2}c^n\prod\limits_{k=n+1}^{\infty}(1+acq^k)\\
&=\left(\sum\limits_{n=0}^\infty q^{{n+1}\choose
2}c^n\right)\left(\prod\limits_{k=0}^{\infty}\frac{1}{(1-aq^k)}\right),
\end{align*}
which we name Levande's rewritten identity for concision.

\item The goal of Levande's bijection $\phi$ in \cite[Section
5]{Levande10} is to present a combinatorial proof of Levande's
rewritten identity. Levande's approach is basically correct, but
when processing with the empty partition, Levande's bijection $\phi$
cannot work properly. See $\phi(\lambda=1,\mu=\emptyset)$,
$\phi^{-1}(X=1,Y=\emptyset)$ and $\phi^{-1}(X=\emptyset,Y=1)$ for an
illustration.

\item Comparing with Levande's rewritten identity, our rewritten
identity \eqref{eq5} has a more neat form and does not involve the
factor $\frac{1}{1-a}$, so we do not need to introduce the zero
parts in a partition and can avoid the shortage of Levande's
bijection $\phi$ when we present the combinatorial interpretation of
identity \eqref{eq5} in Section 3.
\end{enumerate}

Now let
\[
P_{n}(a)=\sum_{\ell(\lambda)>
n}a^{\ell(\lambda)}q^{|\lambda|}\,\,\mbox{and}\,\,Q_{n}(a)=\sum_{\ell(\lambda)\leq
n}a^{\ell(\lambda)}q^{|\lambda|},
\]
where $\ell(\lambda)$ denotes the number of parts of a partition
$\lambda$, and $|\lambda|$ is the sum of parts of $\lambda$ and the
summation is taken over all  partitions satisfying the desired
condition.

It is easy to see that
\[
\frac{1}{(aq;q)_\infty}=P_{n}(a)+Q_{n}(a).
\]

Hence, identity \eqref{eq5} can be rewritten as
\begin{align}
\sum_{n=1}^\infty& \frac{aq^n(-bq^{n+1};q)_\infty}{(aq;q)_n}
+\sum_{n=0}^\infty
q^{n(n+1)/2}b^na^{-n}(-bq^{n+2};q)_\infty\nonumber \\
&=\sum_{n=0}^\infty q^{n(n+1)/2}b^na^{-n}P_{n}(a)+\sum_{n=0}^\infty
q^{n(n+1)/2}b^na^{-n}Q_{n}(a),\label{eq6}
\end{align}
which we will christen ``Ramanujan's rewritten identity''.

%--------------------------------------------------------------------
\section{Proof of Ramanujan's Rewritten Identity}
In this section, we aim to give a combinatorial interpretation of
Ramanujan's rewritten identity \eqref{eq6} to present a
combinatorial proof of identity \eqref{eq2}.

First we recall Yee's simple bijection in Lemma \ref{Yee}, which is
useful to the proof of our main result. We follow the notations in
\cite{Yee09} and introduce the necessary notions now.

Given a partition $\lambda$, we denote $s(\lambda)$ (resp.
$m(\lambda)$) the smallest (resp. largest) part of $\lambda$. For
any two partitions $\lambda=(\lambda_1,\lambda_2,\ldots)$ and
$\mu=(\mu_1,\mu_2,\ldots)$, we define the sum $\lambda+\mu$ to be
the partition $(\lambda_1+\mu_1,\lambda_2+\mu_2,\ldots)$ and their
union $\lambda\cup\mu$ to be the partition with parts
$\lambda_1,\lambda_2,\ldots,\mu_1,\mu_2,\ldots$ arranged in weakly
decreasing order.

The partition with no parts is denoted by $\epsilon$ and we define
$\tau_n$ to be  partition of $n(n+1)/2$ into $(n,n-1,\ldots,1)$,
which is called a triangular partition. The set of all triangular
partitions is denoted by $\Delta$.

Define $\mathcal{P}$ to be the set of all partitions and
$\mathcal{D}$ to be the set consisting of all partitions with
distinct parts.

For $n\geq 0$, let
\[
\mathcal{P}_n=\{\mbox{partitions $\pi$ with
$m(\pi)=n$}\}\,\,\mbox{and}\,\, \mathcal{P}_{\geq
n}=\{\mbox{partitions $\pi$ with $s(\pi)\geq n$}\}
\]
and denote $\mathcal{D}_{\geq n}$ the set $\mathcal{P}_{\geq
n}\cap\mathcal{D}$, i.e., the set of all partitions with distinct
parts and with the smallest part being greater than or equal to $n$.

\begin{lemma}[\mdseries{\cite[Thm 2.1]{Yee09}}]\label{Yee} We have
\[
\bigcup_{n=0}^\infty  \mathcal{D}_{\geq n+1}\times
\mathcal{P}_n=\Delta\times \mathcal{P}.
\]
\end{lemma}
\begin{proof}
Suppose $\lambda$ is a partition in $\mathcal{D}_{\geq n+1}$, since
\[ \lambda_1>\lambda_2>\cdots>\lambda_{\ell(\lambda)}>n,
\]
we can subtract $\ell(\lambda)-i+1$ from each part $\lambda_i$ of
$\lambda$ to obtain a partition $\lambda^{*}$ in $\mathcal{P}_{\geq
n}$. Thus we can write $\lambda$ as
$\tau_{\ell(\lambda)}+\lambda^{*}$ where
$\lambda^{*}\in\mathcal{P}_{\geq n}$.

Now we establish a bijection $\varphi$ between the sets
$\bigcup_{n=0}^\infty  \mathcal{D}_{\geq n+1}\times \mathcal{P}_n$
and $\Delta\times \mathcal{P}$.

Given a pair of partitions $(\lambda,\mu)\in\mathcal{D}_{\geq
n+1}\times \mathcal{P}_n$, we first write $\lambda$ as
$\tau_{\ell(\lambda)}+\lambda^{*}$ and define
$\varphi((\lambda,\mu))$ to be $(\tau_{\ell(\lambda)},
\lambda^{*}\cup \mu)$ where $\lambda^{*}\cup \mu$ is the union of
$\lambda^{*}$ and $\mu$.
It is easy to see that $\varphi$ is a one-to-one correspondence
between $(\lambda,\mu)$ and
$(\tau_{\ell(\lambda)},\lambda^{*}\cup\mu)$.
\end{proof}

In the above proof, for $n\geq 1$, we have $\ell(\mu)\geq 1$ and
$\ell(\lambda^{*})=\ell(\lambda)$, thus $\ell(\lambda^{*}\cup
\mu)>\ell(\lambda)$. As
$\ell(\Delta_{\ell(\lambda)})=\ell(\lambda)$, we have
\begin{corollary}
\begin{equation}\label{eqofyee}
\bigcup_{n=1}^\infty  \mathcal{D}_{\geq n+1}\times
\mathcal{P}_n=\{(\lambda,\mu)|(\lambda,\mu)\in\Delta\times
\mathcal{P}, \ell(\mu)>\ell(\lambda)\}:=S.
\end{equation}
\end{corollary}

Now we state our main results: Theorem \ref{thm3-1} and Theorem
\ref{thm3-2}, which provide a combinatorial proof of Ramanujan's
rewritten identity \eqref{eq6}.

\begin{theorem}\label{thm3-1}
\begin{equation}
\sum_{n=1}^\infty
\frac{aq^n(-bq^{n+1};q)_\infty}{(aq;q)_n}=\sum_{n=0}^\infty
q^{n(n+1)/2}b^na^{-n}P_{n}(a).
\end{equation}
\end{theorem}
\begin{proof}
It is straightforward to see the following partition identity
\begin{equation}\label{eqtemp1} \sum_{n=1}^\infty
\frac{aq^n(-bq^{n+1};q)_\infty}{(aq;q)_n}=\sum_{(\lambda,\mu)\in
\bigcup_{n=1}^\infty \mathcal{D}_{\geq n+1}\times
\mathcal{P}_n}a^{\ell(\mu)}b^{\ell(\lambda)}q^{|\lambda|+|\mu|}.
\end{equation}
From \eqref{eqofyee}, we deduce that
\begin{equation}\label{eqtemp2}
\sum_{(\lambda,\mu)\in \bigcup_{n=1}^\infty \mathcal{D}_{\geq
n+1}\times
\mathcal{P}_n}a^{\ell(\mu)}b^{\ell(\lambda)}q^{|\lambda|+|\mu|}=\sum_{(\lambda,\mu)\in
S} a^{\ell(\mu)-\ell(\lambda)}b^{\ell(\lambda)}q^{|\lambda|+|\mu|}.
\end{equation}
It is easy to show that
\begin{eqnarray}
\sum_{(\lambda,\mu)\in S}
a^{\ell(\mu)-\ell(\lambda)}b^{\ell(\lambda)}q^{|\lambda|+|\mu|}&=&
\sum_{n=0}^\infty
q^{n(n+1)/2}b^na^{-n}\sum_{\ell(\mu)>n}a^{\ell(\mu)}q^{|\mu|}\nonumber\\[5pt]
&=&\sum_{n=0}^\infty q^{n(n+1)/2}b^na^{-n}P_{n}(a).\label{eqtemp3}
\end{eqnarray}
Combining \eqref{eqtemp1}, \eqref{eqtemp2} and \eqref{eqtemp3}
together completes the proof.
\end{proof}

\begin{theorem}\label{thm3-2}
\begin{equation}
\sum_{n=0}^\infty
q^{n(n+1)/2}b^na^{-n}(-bq^{n+2};q)_\infty=\sum_{n=0}^\infty
q^{n(n+1)/2}b^na^{-n}Q_{n}(a).
\end{equation}
\end{theorem}

\begin{proof}
For a partition $\lambda=(\lambda_1,\lambda_2,\ldots,\lambda_m)$
with distinct parts where $\lambda_1>\lambda_2>\cdots>\lambda_m$,
let $\ell_{\triangle}(\lambda)$ be the largest $k$ such that
$\lambda_{m-i+1}=i$ holds for $1\leq i\leq k$. If $s(\lambda)>1$, we
set $\ell_{\triangle}(\lambda)=0$.

Then we have
\begin{equation}\label{eqtemp4}
\sum_{n=0}^\infty
q^{n(n+1)/2}b^na^{-n}(-bq^{n+2};q)_\infty=\sum_{\lambda\in
\mathcal{D}}
a^{-\ell_{\triangle}(\lambda)}b^{\ell(\lambda)}q^{|\lambda|}.
\end{equation}

For a partition $\lambda$ with distinct parts, we can split it into
a triangular partition $\tau_{\ell(\lambda)}$ and  an ordinary
partition $\mu$ whose length is not greater than $\ell(\lambda)$.
For example, $(7,5,4,1)=(4,3,2,1)+(3,2,2)$.  Thus,
$\ell_{\triangle}(\lambda)=\ell(\lambda)-\ell(\mu)$, which yields
\begin{equation}\label{eqtemp5}
\sum_{\lambda\in \mathcal{D}}
a^{-\ell_{\triangle}(\lambda)}b^{\ell(\lambda)}q^{|\lambda|}=\sum_{(\lambda,\mu)\in
(\mathcal{D},\mathcal{P})\atop {\ell(\mu)\leq\ell(\lambda)}}
a^{\ell(\mu)-\ell(\lambda)}b^{\ell(\lambda)}q^{\ell(\lambda)(\ell(\lambda)+1)/2+|\mu|}.\\[5pt]
\end{equation}
It is not hard to see that
\begin{equation}\label{eqtemp6}
\sum_{(\lambda,\mu)\in (\mathcal{D},\mathcal{P})\atop
{\ell(\mu)\leq\ell(\lambda)}}
a^{\ell(\mu)-\ell(\lambda)}b^{\ell(\lambda)}q^{\ell(\lambda)(\ell(\lambda)+1)/2+|\mu|}=\sum_{n=0}^\infty
a^{-n}b^nq^{n(n+1)/2}Q_n(a).
\end{equation}
The proof is completed by combining \eqref{eqtemp4}, \eqref{eqtemp5}
and \eqref{eqtemp6} together.
\end{proof}

%--------------------------------------------------------------------
\subsection*{Acknowledgements}
We wish to thank the referees for helpful comments leading to an
improvement of an earlier version. The first author was supported by
the National Science Foundation of China (Tianyuan Fund for
Mathematics, No.11226299), and the Scientific Research Foundation of
Jimei University, China. The second author was supported by China
Postdoctoral Science Foundation (No.2012M510377) and the third
author was supported by the Fundamental Research Funds for Central
Universities, ZYGX2012J116.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\bibitem{Kim10}
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\bibitem{Levande10}
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\bibitem{Pak06}
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\bibitem{Yee09}
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\end{thebibliography}

\end{document}
