Some Observations on Dyson's\\ New Symmetries of Partitions
Alexander Berkovich
We utilize Dyson's concept of the adjoint of a partition to derive an infinite family of new polynomial analogues of Euler's Pentagonal Number Theorem. We streamline Dyson's bijection relating partitions with crank $\le k$ and those with $k$ in the Rank-Set of partitions. Also, we extend Dyson's adjoint of a partition to MacMahon's ``modular'' partitions with modulus $2$. This way we find a new combinatorial proof of Gauss's famous identity. We give a direct combinatorial proof that for $n>1$ the partitions of $n$ with crank $k$ are equinumerous with partitions of $n$ with crank $-k$.
math0203111.tex
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%journal: JCT-A
%ref no: 53
%
% Sent to Bev Lantrip at: jcta@asu.edu
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% Previous versions:
% (1) version  Submitted to JCT(A) March 12, 2002 by alexb
%     to Bruce Rothschild <blr@math.ucla.edu>
% (2) edited version of (1) which includes appendix
%     file: paper+appendix.tex
% (3) (04/02/02):
%     file: paper+appendix-v2.tex
%     Edited version of (2) which contains generalizations found by Alex
%     and a note added in proof. Sent to Warnaar for comments.
% (4) (04/08/02):
%     file: paper+appendix+note_added.tex
%     Edited version of (3). 
%     * Changed 1 sentence on p.7
%     * Note added expanded to include derivation of Warnaar cubic
%       summation from cubic bibasic transformation of Gasper-Rahman.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% (5) Current version (04/23/02):
%     file: final.tex 
%     Edited version of (3).
%     * Fixed p.5 (added n=1, removed n=oo).
%     * p.25: changed "n->oo" to "n=1".
%     * p.5 Insert (#1) after (1.34). Gives mention of earlier proof by Andrews
%     * p.6 Insert (#2) at end of section 1. Mentions Appendix B. 
%     * p.21 Insert (#3) Mentions Appendix B for alt proof of (5.17). 
%     * p.21 Insert (#4) Mentions earlier paper of Andrews for proof of Cauchy
%     * p.22 Appendix --> Appendix A
%     * Appendix B -- Andrews proof of (5.17).
%     * Three new refs.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

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\begin{document}

\renewcommand{\theequation}{\arabic{section}.\arabic{equation}}

\title{Some Observations on Dyson's\\
New Symmetries of Partitions}

\author{Alexander Berkovich}
\email{alexb\char"40math.ufl.edu} 
\address{Department of Mathematics, University of Florida,
Gainesville, FL 32611}

\author{Frank G. Garvan}
\email{frank\char"40math.ufl.edu} 
\address{Department of Mathematics, University of Florida,
Gainesville, FL 32611}

\thanks{Research of the second author supported in part by the NSF under grant 
number DMS-9870052.}
\keywords{partitions, congruences, Dyson's rank, cranks, Euler's 
pentagonal number theorem, modular partitions,
$q$-series, polynomial analogues}
\subjclass{Primary: 11P81, 11P83, 05A17, 33D15}
\date{April 17, 2002}
%%\date{April 11, 2002}
%%\date{April 9, 2002}
%%\date{April 8, 2002}
%%\date{March 24, 2002}
%%\date{March 12, 2002}

\begin{abstract}
We utilize Dyson's concept of the adjoint of a partition to
derive an infinite family of new polynomial analogues of Euler's Pentagonal 
Number Theorem.
We streamline Dyson's bijection relating partitions with crank $\le k$
and those with $k$ in the Rank-Set of partitions. Also, we extend
Dyson's adjoint of a partition to MacMahon's ``modular'' partitions
with modulus $2$. This way we find a new combinatorial proof of
Gauss's famous identity.
We give a direct combinatorial proof that for $n>1$ the partitions of $n$
with crank $k$ are equinumerous with partitions of $n$ with
crank $-k$.
\end{abstract}


\maketitle
\pagestyle{myheadings}
\markboth{A. Berkovich and F.G. Garvan}{Dyson's symmetries of partitions}




\setcounter{equation}{0}
\section{Introduction}
\label{sec:intro}

%%%%%%SECTION 1
Let $p(n)$ denote the number of unrestricted partitions of $n$.
Ramanujan discovered three beautiful arithmetic properties of $p(n)$,
namely:
\begin{align}
p(5n+4) &\equiv 0 \pmod{5},\tag{1.1}\\
p(7n+5) &\equiv 0 \pmod{7},\tag{1.2}\\
p(11n+6) &\equiv 0 \pmod{11}.\tag{1.3}
\end{align}
The partition congruences modulo $5$ and $7$ were proved by Ramanujan 
in \cite{Ra1}. In \cite{Ra2} he proved \tagref{1.3} by a different method.
The most elementary proof of \tagref{1.3} similar to the one in \cite{Ra1}
is due to Winquist \cite{Win}.

Dyson \cite{D1} discovered empirically remarkable combinatorial
interpretations of \tagref{1.1} and \tagref{1.2}.
Defining the {\it rank} of a partition as the {\it largest part minus
the number of parts}, he observed that
\begin{align}
N(k,5,5n+4) &= \frac{p(5n+4)}{5},\quad 0 \le k \le 4, \tag{1.4}\\
N(k,7,7n+5) &= \frac{p(7n+5)}{7},\quad 0 \le k \le 6, \tag{1.5}
\end{align}
where $N(k,m,n)$ denotes the number of partitions of $n$ with
rank congruent to $k$ modulo $m$.
Identities \tagref{1.4} and \tagref{1.5} were later proved by
Atkin and Swinnerton-Dyer \cite{ASD}.
However, the rank failed to explain \tagref{1.3}, and so Dyson conjectured
the existence of some analogue of the rank that would explicate the
Ramanujan congruence modulo $11$. He named his hypothetical statistic
the {\it crank}.

Forty four years later, Andrews and Garvan \cite{AG}, building on the
work of Garvan \cite{Ga1}, finally unveiled
Dyson's crank of a partition $\pi$:
\begin{equation}
\mbox{crank}(\pi) =
\begin{cases}
  \lambda(\pi), &\mbox{if $\mu(\pi)=0$}, \\
  \nutwid(\pi) - \mu(\pi), &\mbox{if $\mu(\pi)>0$},
\end{cases}
\tag{1.6}
\end{equation}
where $\lambda(\pi)$ denotes the largest part of $\pi$,
$\mu(\pi)$ denotes the number of ones in $\pi$ and $\nutwid(\pi)$
denotes the number of parts of $\pi$ larger than $\mu(\pi)$.

Remarkably, the crank provides combinatorial interpretations
of all three Ramanujan congruences \tagref{1.1}--\tagref{1.3}.
Namely,
\begin{align}
M(k,5,5n+4) &= \frac{p(5n+4)}{5},\quad 0 \le k \le 4, \tag{1.7}\\
M(k,7,7n+5) &= \frac{p(7n+5)}{7},\quad 0 \le k \le 6, \tag{1.8}\\
M(k,11,11n+6) &= \frac{p(11n+6)}{11},\quad 0 \le k \le 10, \tag{1.9}  
\end{align}
where $M(k,m,n)$ denotes the number of partitions of $n$ with
crank congruent to $k$ modulo $m$.

Let $P_m(q)$ denote the generating function
\begin{equation}
P_m(q) = \sum_{n=1}^\infty p_m(n) q^n,
\tag{1.10}
\end{equation}
where $p_m(n)$ is the number of partitions of $n$ with rank $m$.
Here we are using the convention that $p_m(0)=0$.
As a practical tool for his empirical calculations Dyson used the 
following formula for $P_m(q)$:
\begin{equation}
P_m(q) = \frac{1}{(q)_\infty}
\sum_{j\ge1} (-1)^{j-1} (1-q^j) q^{\frac{j(3j-1)}{2}+ |m|j},
\tag{1.11}
\end{equation}
with
\begin{equation}
(q)_\infty = \prod_{j\ge1}(1-q^j),\quad \mbox{for $|q|<1$.}
\tag{1.12}
\end{equation}
For later use we also define
\begin{equation}
(a;q)_n = (a)_n = \prod_{j=0}^{n-1} (1 - a q^j), \quad n\ge 0
\tag{1.13}
\end{equation}
and note that $\frac{1}{(q)_\infty}$ is the generating function
for unrestricted partitions.

Dyson knew how to prove \tagref{1.11} in 1942 \cite{D2}.
However the first published proof of \tagref{1.11} was given
by Atkin and Swinnerton-Dyer \cite{ASD} in 1954.
In 1968, Dyson \cite{D3} found a simple combinatorial argument
which not only explained \tagref{1.11} but also led to a new proof of Euler's
celebrated pentagonal number theorem:
\begin{equation}
1 = \frac{1}{(q)_\infty} \sum_{j=-\infty}^{\infty} (-1)^j q^{\frac{j(3j-1)}{2}}.
\tag{1.14}
\end{equation}

To paraphrase Dyson's argument in  \cite{D3} we introduce the
generating function
\begin{equation}
Q_m(q) = \sum_{n\ge1} \qtwid_m(n) q^n,
\tag{1.15}
\end{equation}
where $\qtwid_m(n)$ is the number of partitions of $n$
with rank $\ge m$. We adopt the convention that
\begin{equation}
\qtwid_m(0)=0.
\tag{1.16}
\end{equation}
Clearly,
\begin{equation}
P_m(q) = Q_m(q) - Q_{m+1}(q).
\tag{1.17}
\end{equation}
Next, following the treatment in \cite{D3} we will show that
\begin{equation}
Q_m(q) + Q_{1-m}(q) + 1 = \frac{1}{(q)_\infty},
\tag{1.18}
\end{equation}
and for $m\ge0$
\begin{equation}
Q_m(q) = q^{m+1}\left(Q_{-2-m}(q) + 1\right).
\tag{1.19}
\end{equation}

To prove \tagref{1.18} we note that any given nonempty partition $\pi$
counted by $\frac{1}{(q)_\infty}$ has either rank $\ge m$ or
rank $< m$. If rank $\ge m$, then $\pi$ is counted by $Q_m(q)$.
If rank $< m$, then we conjugate $\pi$ to get $\pi^{*}$ as
illustrated in Fig.\ 1.

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%%\caption{{Fig. 1}\enskip Conjugation of a partition $\pi$}
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%%\mycap{1}{\enskip Conjugation of a partition $\pi$}
\textsc{Fig 1}.\ \parbox[t]{5in}{
Conjugation of a partition $\pi$ with largest part $\lambda(\pi)$
and the number of parts $\nu(\pi)$, $\lambda(\pi)-\nu(\pi)<m$.}
\end{figure}

It is obvious in this case
that $\mbox{rank}(\pi^{*})\ge1-m$. Hence, $\pi^{*}$ is
counted by $Q_{1-m}(q)$. Finally, the empty partition is counted by $1$
on the left side of \tagref{1.18} and on the right side
by $\frac{1}{(q)_\infty}$.
The proof of \tagref{1.19} is more subtle. Here we will use a different
conjugation transformation (Dyson's adjoint) as follows.
Consider some partition $\pi$ with $\mbox{rank}(\pi)\ge m\ge0$.
This partition is counted by $Q_m(q)$ in \tagref{1.19}.
Clearly,
\begin{equation}
\lambda(\pi) - \nu(\pi) \ge m \ge 0.
\tag{1.20}
\end{equation}
Let us now remove the largest part of $\pi$ to end up with $\pitwid$.
Next, we conjugate $\pitwid$ to get $\pitwid^{*}$.
Finally, we attach to $\pitwid^{*}$ a new largest part of
size $\lambda(\pi) - m - 1$.
These transformations are illustrated in Fig.\ 2.

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\end{figure}

Note that the map $\pi \to \pi'$ is reversible. It is obvious that
\begin{align}
\lambda(\pi') &= \lambda(\pi) - m - 1,\tag{1.21}\\
\nu(\pi') &\le \lambda(\pi)  + 1,\tag{1.22}\\
|\pi'| &= |\pi|  -m - 1,\tag{1.23}
\end{align}
where $|\pi|$ denotes the sum of parts of $\pi$.

Since $\mbox{rank}(\pi')\ge -2-m$, we see that $\pi'$ is counted
by $Q_{-2-m}(q)$ in \tagref{1.19}, provided $|\pi'|\ne0$.
If $|\pi|=m+1$ and $\mbox{rank}(\pi)\ge m$, then
$\nu(\pi)=1$ and $\lambda(\pi)=1+m$. So in this case $\pi'$ represents
the empty partition, which is counted by $1$ in \tagref{1.19}. This
concludes the proof of \tagref{1.19}.

Combining \tagref{1.18} and \tagref{1.19}, we see that for $m\ge 0$
\begin{equation}
Q_m(q) + q^{m+1} Q_{m+3} = \frac{q^{m+1}}{(q)_\infty}.
\tag{1.24}
\end{equation}
Iterating \tagref{1.24} we obtain for $m\ge0$
\begin{align}
Q_m(q) &= \frac{q^{m+1}}{(q)_\infty} - q^{m+1} Q_{m+3} \nonumber\\
&= \frac{q^{m+1} - q^{2m+5}}{(q)_\infty} + q^{2m+5} Q_{m+6}= \cdots
\tag{1.25}\\
& = 
 \frac{1}{(q)_\infty} \sum_{j\ge1} (-1)^{j-1} q^{\frac{j(3j-1)}{2} + mj}.
\nonumber
\end{align}

We observe that \tagref{1.25} together with \tagref{1.18} with $m=0$
yields Euler's theorem \tagref{1.14}. On the other hand, \tagref{1.25}
together with \tagref{1.17} yields \tagref{1.11} with $m\ge0$.
To treat the $m<0$ case in \tagref{1.17} we make use of
\begin{equation}
P_m(q) = P_{-m}(q),
\tag{1.26}
\end{equation}
which is a straightforward consequence of the conjugation transformation.

In \cite{A1} Andrews utilized Dyson's adjoint to give
a new proof of a partition theorem due to Fine.
This seems to be the only known application of the Dyson transformation.

In the next section we will show that Dyson's formulas \tagref{1.18}, 
\tagref{1.19} can be generalized to yield a binary tree of
polynomial analogues of \tagref{1.14}. This tree contains 
Schur's well-known formula
\begin{equation}
1=\sum_{j=-\infty}^{\infty} (-1)^j q^{\frac{j(3j-1)}{2}}
\left[
\matrix 2L \\
L + \lfloor \tfrac{3j}{2} \rfloor
\endmatrix
\right]_q,
\tag{1.27}
\end{equation}
as well as a new polynomial version of \tagref{1.14}
\begin{equation}
1=\sum_{j=-\infty}^{\infty} (-1)^j q^{\frac{j(3j+1)}{2}}
\left[
\matrix 2L - j\\
L + j
\endmatrix
\right]_q,
\tag{1.28}
\end{equation}
where $\lfloor x \rfloor$ denotes the integer part of $x$ and
$q$-binomial coefficients are defined as
\begin{equation}
\left[
\matrix n + m \\
n
\endmatrix
\right]_q
=
\begin{cases}
  \frac{(q)_{n+m}}{(q)_n (q)_m}, &\mbox{if $n$, $m\ge0$}, \\
  0, &\mbox{otherwise.}       
\end{cases}
\tag{1.29}
\end{equation}

Actually, \tagref{1.27} and \tagref{1.28} are special cases
of the following more general formula
\begin{equation}
1 - \delta_{\sigma,-1} =
\sum_{j=-\infty}^{\infty} (-1)^j q^{\frac{j(3j-1)}{2} + \sigma j}
\qbin{2L + \sigma - \au(n,j)}{L + \sigma + \lfloor \frac{n+1}{n}j\rfloor},
\tag{1.30}
\end{equation}
where
$n=1$, $2$, $3$, $4$, $5$, \dots; $\sigma=-1$, $0$, $1$, and
\begin{equation}
\au(n,j) = 
\begin{cases}
   -j, &\mbox{if $n=1$},\\
   0, &\mbox{if $n=2$},\\
   \displaystyle\sum_{k=1}^{n-2}\left\lfloor \frac{j+k}{n} \right\rfloor, &\mbox{if $n>2$.}
\end{cases}
\tag{1.31}
\end{equation}
It is easy to verify that
\begin{equation}
\lim_{n\to\infty} \au(n,j) = j-1,\quad \text{if $j>0$,}
\tag{1.32}
\end{equation}
and
\begin{equation}
\au(n,-j) = -\au(n,j),\quad \text{for $n\ge1$.}
\tag{1.33}
\end{equation}

Note that \tagref{1.27} is \tagref{1.30} with $n=2$, $\sigma=0$
and \tagref{1.28} is \tagref{1.30} with $n=1$, $\sigma=0$.


In \S\S 3 and 4 we will streamline and generalize Dyson's treatment of 
partitions
with crank $\le k$. In \S5 we will use modular representations
with modulus $2$
of partitions 
in which odd parts do not repeat,
and an appropriate modification of Dyson's
adjoint transformation, to obtain a new proof
of the Gauss formula
\begin{equation}
\frac{(q^2;q^2)_\infty}{(q;q^2)_\infty}
= \sum_{j\ge0}  q^{\frac{j(j+1)}{2}}.
\tag{1.34}
\end{equation}
%%We conclude in \S6 with a brief description of some open questions for
%%future investigation.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%INSERT #1
We remark that the first combinatorial proof of \tagref{1.34}
was given by Andrews in \cite{X}. This early proof uses a
Franklin-type involution and is quite different from the one given
in \S5.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\S6 contains a brief description of some open questions for
future research.  In Appendix A we introduce
a new type of partition transformation, termed pseudo-conjugation,
in order to prove 
directly  that for $n>1$ the partitions of $n$
with crank $k$ are equinumerous with partitions of $n$ with
crank $-k$.
We also show that self-pseudo-conjugate partitions of $n$
(introduced there) are equinumerous with partitions of $n$ into
distinct odd parts.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%INSERT #2
Finally, in Appendix B we outline an alternative proof of the formula 
\tagref{5.17}. This proof was communicated to us by George Andrews \cite{Y}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%END SECTION 1

\setcounter{equation}{0}
\section{Polynomial analogues of Euler's pentagonal number theorem}
\label{sec:2}
%%%%%%SECTION 2

We say that a partition $\pi$ is in the box $[L,M]$ if its largest part
does not exceed $L$ and the number of parts does not exceed $M$. In other words,
\begin{align*}
\lambda(\pi) &\le L,\\
\nu(\pi) &\le M.
\end{align*}
It is well known \cite{A2} that the generating functions for partitions
in the box $[L,M]$
is
$$
\left[
\matrix L + M \\
L
\endmatrix
\right]_q.
$$
Let us define $Q_m^L(q)$ as
\begin{equation}
Q_m^L(q) = \sum_{n\ge1} \qtwid_m^L(n) q^n,
\tag{2.1}
\end{equation}
where $\qtwid_m^L(n)$ is the number of partitions of $n$ with
rank $\ge m$ and largest part $\le L$. As before, we assume that
$\qtwid_m^L(0)=0$. Clearly, $Q_m^L(q)=0$ whenever $L \le m$. We now prove
that
\begin{equation}
Q_m^L(q) - Q_m^{L-1}(q) = q^L \qbin{2L -m -1}{L}.
\tag{2.2}
\end{equation}
To this end we observe that the left side of \tagref{2.2} counts
partitions with rank $\ge m$ and $\lambda(\pi)=L$.
We note that these partitions are in the box $[L,L-m]$. 
If we remove the largest part $L$ from one of those partitions
we obtain a partition in the
box $[L, L-m-1]$, and this partition is counted by the $q$-binomial coefficient
on the right side of \tagref{2.2}, as desired.

We now move on to derive the bounded analogues of \tagref{1.18}, \tagref{1.19},
namely:
\begin{equation}
Q_m^L(q) + Q_{1-m}^{L-m}(q) + 1 = \qbin{2L-m}{L},\qquad L >m,
\tag{2.3}
\end{equation}
and 
\begin{equation}
Q_m^L(q) = q^{m+1}\left(Q_{-2-m}^{L-1-m}(q) + 1\right),\qquad L>m\ge0.
\tag{2.4}
\end{equation}
To prove \tagref{2.3} we note that any given nonempty partition
$\pi$, counted by $\qbin{2L-m}{m}$, has either rank $\ge m$
or rank $<m$. Moreover, $\pi$ is in the box $[L,L-m]$.
Now if $\rank(\pi)\ge m$, then $\pi$ is counted by $Q_m^L(q)$.
If $\rank(\pi)< m$, then we conjugate $\pi$ to get $\pi^{*}$.
Obviously, $\pi^{*}$ is counted by $Q_{1-m}^{L-m}(q)$. Finally,
the empty partition is counted by $1$ and by the $q$-binomial coefficient
on the left and right sides of \tagref{2.3}, respectively.

Next, to prove \tagref{2.4} we observe that any partition $\pi$ counted
by $Q_m^L(q)$ is in the box $[L,L-m]$.  Performing Dyson's
transformation $\pi \to \pi'$, as explained in the previous section,
we see that $\lambda(\pi')\le L-1-m$
and $\rank(\pi')\ge -2 -m$. Therefore, if $|\pi| \ne m+1$, then
$\pi'$ is counted by $Q_{-2-m}^{L-1-m}(q)$. If $|\pi| = m+1$,
then $\pi'$ is empty. In this case it is counted by $1$ on the right side
of \tagref{2.4}.

%
% L -> L+2, m -> m+3 in (2.3) and use (2.4)
%
Combining \tagref{2.3} and \tagref{2.4} yields
\begin{equation}
Q_m^L(q) + q^{m+1} Q_{m+3}^{L+2}(q) 
= q^{m+1} \qbin{2L-m+1}{L+2}, \quad m\ge0.
\tag{2.5}
\end{equation}
We remark that when $L\le m$ the above formula becomes $0+0=0$,
and when $L$ tends to infinity \tagref{2.5} reduces to \tagref{1.24}.
Actually, it is possible to derive another bounded analogue of \tagref{1.24}.
To this end we employ \tagref{2.2} together with the
well known recurrence
\begin{equation}
\qbin{n+m}{n} = q^n \qbin{n+m-1}{n} + \qbin{n+m-1}{m-1}
\tag{2.6}
\end{equation}
to transform \tagref{2.5} as
\begin{align}
Q_m^L(q) + q^{m+1} Q_{m+3}^{L+1}(q)
&= q^{m+1}\left\{
\qbin{2L-m+1}{L+2} - q^{L+2} \qbin{2L-m}{L+2}\right\}\tag{2.7}\\
&= q^{m+1} \qbin{2L-m}{L+1},\quad m\ge 0.
\nonumber
\end{align}

The power of \tagref{2.5} and \tagref{2.7} lies in the fact that these
transformations can be employed to generate an infinite binary tree
of representations for $Q_m^L(q)$.
First we consider four special cases, namely:
\begin{align}
Q_m^L(q) &= \sum_{j\ge1} (-1)^{j-1} q^{\frac{j(3j-1)}{2} + mj}
\qbin{2L-m+j}{L-m-j}, \quad m\ge0, \tag{2.8}\\
Q_m^L(q) &= \sum_{j\ge1} (-1)^{j-1} q^{\frac{j(3j-1)}{2} + mj}
\qbin{2L-m-j+1}{L+j}, \quad m\ge0, \tag{2.9}\\
Q_m^L(q) &= \sum_{j\ge1} (-1)^{j-1} q^{\frac{j(3j-1)}{2} + mj}
\qbin{2L-m+1}{L-\lfloor-\tfrac{3j}{2} \rfloor}, \quad m\ge0, \tag{2.10}\\
Q_m^L(q) &= \sum_{j\ge1} (-1)^{j-1} q^{\frac{j(3j-1)}{2} + mj}
\qbin{2L-m}{L+\lfloor\tfrac{3j}{2} \rfloor}, \quad m\ge0. 
\tag{2.11}
\end{align}
To derive \tagref{2.8}--\tagref{2.11} we use the
iteration schemes which we denote symbolically as
\begin{align}
&\tagref{2.5} - \tagref{2.5} - \tagref{2.5} - \tagref{2.5} - \tagref{2.5} - \tagref{2.5} - \cdots, \tag{2.12}\\
&\tagref{2.7} - \tagref{2.7} - \tagref{2.7} - \tagref{2.7} - \tagref{2.7} - \tagref{2.7} - \cdots, \tag{2.13}\\
&\tagref{2.5} - \tagref{2.7} - \tagref{2.5} - \tagref{2.7} - \tagref{2.5} - \tagref{2.7} - \cdots, \tag{2.14}\\
&\tagref{2.7} - \tagref{2.5} - \tagref{2.7} - \tagref{2.5} - \tagref{2.7} - \tagref{2.5} - \cdots, \tag{2.15}
\end{align}
respectively.
For example, the scheme \tagref{2.12} means each transformation uses
only equation \tagref{2.5}, and the scheme \tagref{2.14} means
that we use both \tagref{2.5} and \tagref{2.7} in an alternating
fashion with \tagref{2.5} being used first.

Now, \tagref{2.3} with $m=0$ yields
\begin{equation}
Q_0^L(q) + Q_{1}^{L}(q) + 1 = \qbin{2L}{L}.
\tag{2.16}
\end{equation}
Equation \tagref{1.28} then follows by using \tagref{2.8} with $m=0$
and \tagref{2.9} with $m=1$. 

Schur's formula \tagref{1.27} follows in a similar fashion.
We use \tagref{2.16}, \tagref{2.10} with $m=1$, \tagref{2.11}
with $m=0$ and the fact that 
\begin{equation}
\qbin{2L}{L+a}=\qbin{2L}{L-a}.
\tag{2.17}
\end{equation}

%%%%%INSERT II
To prove \tagref{1.30} we need to consider the following periodic
iterations:
\begin{equation}
\underbrace{
\tagref{2.7} - \tagref{2.7} - \cdots - \tagref{2.7} - \tagref{2.5}}_n
-
\underbrace{
\tagref{2.7} - \tagref{2.7} - \cdots - \tagref{2.7} - \tagref{2.5}}_n
- \cdots,
\tag{2.18}
\end{equation}
and
\begin{equation}
\underbrace{
\tagref{2.5} - \tagref{2.5} - \cdots - \tagref{2.5} - \tagref{2.7}}_n
-
\underbrace{
\tagref{2.5} - \tagref{2.5} - \cdots - \tagref{2.5} - \tagref{2.7}}_n
- \cdots.
\tag{2.19}
\end{equation}
The iteration schemes \tagref{2.18} and \tagref{2.19} yield for $m\ge0$
\begin{equation}
Q_m^L(q) = \sum_{j\ge1} (-1)^{j-1} q^{\frac{j(3j-1)}{2} + mj}
\qbin{2L-m-\au(n,j)}{L+\lfloor\tfrac{n+1}{n}j \rfloor},
\tag{2.20}
\end{equation}
and
\begin{equation}
Q_m^L(q) = \sum_{j\ge1} (-1)^{j-1} q^{\frac{j(3j-1)}{2} + mj}
\qbin{2L+1-m+\au(n,j)}{L+1-m+\lfloor-\tfrac{n+1}{n}j \rfloor},
\tag{2.21}
\end{equation}
respectively.
If we employ \tagref{2.3}, \tagref{2.20} with $m=0$
and \tagref{2.21} with $m=1$ we obtain
\tagref{1.30} with $\sigma=0$. On the other hand, \tagref{2.3}, \tagref{2.20} 
with $m=1$ and $L\to L+1$ together with
\tagref{2.21} with $m=0$ give       
\tagref{1.30} with $\sigma=1$.
To prove \tagref{1.30} with $\sigma=-1$ we note that \tagref{2.7} and 
\tagref{2.3} with $m=-1$ can be combined to give
\begin{equation}
\delta_{m,-1} +
Q_m^L(q) + q^{m+1} Q_{m+3}^{L+1}(q)
= 
q^{m+1} \qbin{2L-m}{L+1},\quad m\ge -1.
\tag{2.22}
\end{equation}
Therefore \tagref{2.20} can be slightly generalized as
\begin{equation}
\delta_{m,-1} + Q_m^L(q) = \sum_{j\ge1} (-1)^{j-1} q^{\frac{j(3j-1)}{2} + mj}
\qbin{2L-m-\au(n,j)}{L+\lfloor\tfrac{n+1}{n}j \rfloor},\quad m\ge -1.
\tag{2.23}
\end{equation}

Next, equation \tagref{2.3} with $m=-1$ and $L\to L-1$ becomes
\begin{equation}
1+ Q_{-1}^{L-1}(q) + Q_{2}^{L}(q) = \qbin{2L-1}{L-1}.
\tag{2.24}
\end{equation}
The last equation together with \tagref{2.21} with $m=2$
and \tagref{2.23} with $m=-1$ and $L\to L-1$ gives
\tagref{1.30} with $\sigma=-1$, as desired.
%%% END INSERT II%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

We now move on to generalize \tagref{1.11}. To this end we define
$P_m^L(q)$ as
\begin{equation}
P_m^L(q) = \sum_{n\ge1} p_m^L(n) q^n,
\tag{2.25}
\end{equation}
where $p_m^L(n)$ is the number of partitions of $n$ with
largest part $\le L$ and rank $m$. Obviously,
\begin{equation}
P_m^L(q) = Q_m^L(q) - Q_{m+1}^L(q).
\tag{2.26}
\end{equation}
So using \tagref{2.10}, \tagref{2.11} and \tagref{2.17}
we obtain
\begin{equation}
P_m^L(q) = 
\sum_{j\ge1} (-1)^{j-1} q^{\frac{j(3j-1)}{2} + mj}
\qbin{2L-m}{L+\lfloor\tfrac{3j}{2} \rfloor}
-\sum_{j\ge1} (-1)^{j-1} q^{\frac{j(3j+1)}{2} + mj}
\qbin{2L-m}{L-\lfloor-\tfrac{3j}{2} \rfloor},
\tag{2.27}
\end{equation}
provided $m\ge0$.
Using the obvious conjugation symmetry
\begin{equation}
P_{-|m|}^L(q) = P_{|m|}^{L+|m|}(q)
\tag{2.28}
\end{equation}
it is straightforward to extend \tagref{2.27} to negative $m$.
This way we obtain the following polynomial analogue of
\tagref{1.11}
\begin{align}
P_m^L(q) &= 
\sum_{j\ge1} (-1)^{j-1} q^{\frac{j(3j-1)}{2} + mj}
\qbin{2L-m}{L+\sign(m)\lfloor\tfrac{3j}{2} \rfloor}\tag{2.29}\\
&\quad -\sum_{j\ge1} (-1)^{j-1} q^{\frac{j(3j+1)}{2} + mj}
\qbin{2L-m}{L-\sign(m)\lfloor-\tfrac{3j}{2} \rfloor},
\nonumber  
\end{align}    
where
$$
\sign(m) = 
\begin{cases}
  1, &\mbox{if $m\ge0$}, \\
  -1, &\mbox{otherwise.}       
\end{cases}
$$
%%%%%END SECTION 2

\setcounter{equation}{0}
\section{Partitions with prescribed cranks}
\label{sec:3}
%%%%%%SECTION 3

Dyson \cite{D1} conjectured that the generating function for the
crank should have a form similar to \tagref{1.11}, and it does
as can be seen from the following formula
\begin{equation}
\Chat_k(q) = 
\frac{1}{(q)_\infty} 
\sum_{j\ge1} (-1)^{j-1} q^{T_{j-1} + j|k|}(1-q^j)
+ q (\delta_{k,0} - \delta_{k,1}),
\tag{3.1}
\end{equation}
where
\begin{equation}
T_j = \frac{j(j+1)}{2},
\tag{3.2}
\end{equation}
and
\begin{equation}
\Chat_k(q) = \sum_{n\ge0} \chat_k(n) q^n,
\tag{3.3}
\end{equation}
with $\chat_k(n)$ denoting the number of partitions of $n$ with
crank $k$. 
In \tagref{3.3} we adopt the convention that $\chat_k(0)=\delta_{k,0}$.
Formula \tagref{3.1} is a consequence of
Theorem (7.19) in \cite{Ga1} and Theorem 1 in \cite{AG}.

To explain \tagref{3.1} in a combinatorial fashion Dyson \cite{D4}
introduced the concept of the rank-set $R(\pi)$ of
a partition
$\pi = p_1 + p_2 + p_3 + \cdots$ with
parts $p_1\ge p_2\ge p_3\ge\cdots$.
$R(\pi)$ is defined as
\begin{equation}
R(\pi) = [j - p_{j+1}, j=0,1,2,\dots].
\tag{3.4}
\end{equation}

To prove \tagref{3.1} Dyson first established that
\begin{equation}
C_k(q) = G_k(q) + q \delta_{k,0},
\tag{3.5}
\end{equation}
where
\begin{align}
C_k(q) = \sum_{n\ge0} c_k(n) q^n, \tag{3.6}\\
G_k(q) = \sum_{n\ge0} g_k(n) q^n, 
\tag{3.7}
\end{align}
with $c_k(n)$, $g_k(n)$ denoting the number of partitions of $n$
with crank $\le k$ and $k$ in the rank-set of these partitions,
respectively. 
In \tagref{3.6}--\tagref{3.7} we use the convention that
$c_k(0)=g_k(0)=1$ if $k\ge0$ and $0$, otherwise.
He then showed that
\begin{equation}
G_{-k}(q) + G_{k-1}(q) = \frac{1}{(q)_\infty},
\tag{3.8}
\end{equation}
and
\begin{equation}
G_{k}(q) + q^{k+1} G_{k+1}(q) = \frac{1}{(q)_\infty},\quad k\ge-1.
\tag{3.9}
\end{equation}

Iteration of \tagref{3.9} yields
\begin{equation}
G_k(q) = \frac{1}{(q)_\infty}
\sum_{j\ge0} (-1)^j q^{T_j + kj}, \quad k\ge-1.
\tag{3.10}
\end{equation}
Now \tagref{3.10}, \tagref{3.5} and the obvious relation
\begin{equation}
\Chat_k(q) = C_k(q) - C_{k-1}(q)
\tag{3.11}
\end{equation}
together imply that
\begin{equation}
\Chat_k(q) = \frac{1}{(q)_\infty}
\sum_{j\ge0} (-1)^{j-1} \left(q^{T_{j-1} + kj} - q^{T_{j} + kj}\right)
+ q(\delta_{k,0} - \delta_{k,1}),\quad k\ge0,
\tag{3.12}
\end{equation}
which is \tagref{3.1} with $k\ge0$. To extend \tagref{3.12} to negative $k$,
we observe that \tagref{3.8} implies that
\begin{equation}
\Ghat_{-k}(q) = \Ghat_k(q),
\tag{3.13}
\end{equation}
where
\begin{equation}
\Ghat_k(q) = G_k(q) - G_{k-1}(q).
\tag{3.14}
\end{equation}
From \tagref{3.5} we deduce that
\begin{equation}
\Chat_k(q) = \Ghat_k(q) + q(\delta_{k,0} - \delta_{k,1}).
\tag{3.15}
\end{equation}
%%If we now replace $k$ by $-k$ in \tagref{3.15} with $k>0$ and use
%%\tagref{3.13} we obtain
%%\begin{equation}
%%\Chat_{-k}(q) = \Ghat_{-k}(q) = \Ghat_k(q),\quad k>0.
%%\tag{3.16}
%%\end{equation}
%%The last equation together with \tagref{3.10}, \tagref{3.14} gives
%%\tagref{3.1} for $k < 0$.
If we now replace $k$ by $-k$ in \tagref{3.15} with $k\ge0$ and use
\tagref{3.13} we obtain
\begin{equation}
\Chat_{-k}(q) = \Ghat_{-k}(q) +q \delta_{k,0}= \Ghat_k(q)+q\delta_{k,0},
\quad k\ge0.
\tag{3.16}
\end{equation}
This equation together with \tagref{3.10}, \tagref{3.14} gives
\tagref{3.1} for $k < 0$.
In addition, using \tagref{3.15} we see that \tagref{3.16} implies
that
\begin{equation}
\Chat_{-k}(q) = \Chat_k(q) + q \delta_{k,1}, \quad k\ge0.
\tag{3.17}
\end{equation}
In the appendix we give a direct combinatorial proof of \tagref{3.17}
without using \tagref{3.15}.

To deal with \tagref{3.8}, \tagref{3.9} Dyson introduced a simple graphical
tool to determine whether or not $k\in R(\pi)$. To explain it we follow
Dyson \cite{D4} and define the boundary of the Ferrers graph of $\pi$
as the infinite zig-zag line consisting of vertical and
horizontal segments each of unit length (see Fig.\ 3).

\begin{figure}[h!]
\begin{center}
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\mycap{3}{\enskip Graph of $\pi=5+2+1$, the boundary $B(\pi)$ is indicated
by the thick line.}
\end{figure}

Next, we draw two $45^{\mbox{o}}$ lines, namely
$$
y=k+x, \qquad y = 1 + k + x,
$$
as shown in Fig.\ 4 and Fig.\ 5.


\begin{figure}[h!]
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\mycap{4}{\enskip Graph of $\pi=3+2+1+1$ with $k=2\in R(\pi)$.}
\end{figure}


\begin{figure}[h!]
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\mycap{5}{\enskip Graph of $\pi=2+1$ with $k=1\not\in R(\pi)$.}
\end{figure}

Let $BS_k(\pi)$ denote the segment of $B(\pi)$ lying in the strip
$$
k+x \le y \le k+1+x
$$
determined by these two lines. 
Now if $BS_k(\pi)$ is vertical, then $k\in R(\pi)$, otherwise $k\not\in R(\pi)$.
Using this criterion it is easy to verify that $\nu(\pi)\ne1+k$,
whenever $k\in R(\pi)$.

We are now ready to prove \tagref{3.8}.
First, it is obvious that any given partition $\pi$ counted by 
$\frac{1}{(q)_\infty}$ has either $-k\in R(\pi)$ or $-k\not\in R(\pi)$. 
In the first case, $\pi$ is counted by $G_{-k}(q)$
in \tagref{3.8}.
In the second case, $BS_{-k}(\pi)$ is a horizontal segment, and so if we
conjugate $\pi$ to get $\pi^{*}$, then it is clear that
$BS_{k-1}(\pi^{*})$ is vertical. Therefore, $k-1\in R(\pi^{*})$
and consequently $\pi^{*}$ is counted by $G_{k-1}(q)$ in \tagref{3.8}.

To prove \tagref{3.9}, we remove the row containing the segment $BS_k(\pi)$
from some given partition $\pi$ counted by $G_k(q)$. Next,
we insert a vertical column of height $j+k$ to the right of the
rectangle $[j,j+k]$, where $j$ is the length of the row removed.
This procedure is illustrated in Fig.\ 6.


\begin{figure}[h!]
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\mycap{6}{\enskip The transformation $\pi\to\pi'$ used in the proof of
\tagref{3.9} ($k\ge0$).}
\end{figure}

Let us call the resulting partition $\pi'$. It is easy to see that
$$
|\pi'| = |\pi| + k,
$$
and, because $BS_{-1+k}(\pi')$ is a horizontal segment,
$$
k-1 \not\in R(\pi').
$$
Since the map $\pi\to\pi'$ is reversible, we immediately infer that
\begin{equation}
g_k(n) = p(n+k) - g_{k-1}(n + k), \qquad k\ge0,
\tag{3.18}
\end{equation}
where $n=|\pi|$. The last equation can be easily transformed in \tagref{3.9}.

In \cite{D4}, Dyson proves \tagref{3.5} first by mapping partitions 
$\pi$ with $k\in R(\pi)$ onto
certain vector partitions introduced in \cite{Ga1}, and then 
mapping these vector partitions onto ordinary partitions with crank $\le k$.
This approach involved ten separate cases. Here, we choose to
prove \tagref{3.5} directly, without any reference to vector partitions.
Our analysis requires consideration of only three separate cases, as we
now explain.

\underbar{Case 1.} Here we consider partitions $\pi$ with $k\in R(\pi)$
and $\nu(\pi)\ge k+2$. This case is illustrated in Fig.\ 7.

\begin{figure}[h!]
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\vskip -60pt
\textsc{Fig 7}.\ \parbox[t]{5in}{Map $\pi\to\pi'$ from partitions $\pi$ with
$k\in R(\pi)$, $\nu(\pi)\ge k+2$ to partitions $\pi'$ with $\crank(\pi')\le k$,
$\mu(\pi')>0$, $\nu(\pi')\ge k+2$.}
\end{figure}

We now remove the row bounded by the vertical segment $BS_k(\pi)$
and then add a vertical column representing $j$ ones to the resulting
graph, where $j>0$ is the length of the row removed. We call this
last partition $\pi'$. It is easy to see that
$$
\nu(\pi')\ge k+2,\quad
\mu(\pi')\ge j >0,\quad\mbox{and}\quad \nutwid(\pi') \le j+k,
$$
where $\mu$ and $\nutwid$ were defined in \tagref{1.6}.
Clearly, $\crank(\pi')=\nutwid(\pi') - \mu(\pi')\le k$.
Perhaps, it is not immediately obvious that the map $\pi\to\pi'$
is reversible. To see that it is, we consider partitions $\pi'$
with $\crank(\pi')\le k$, $\mu(\pi')>0$ and $\nu(\pi')\ge k+2$.
Next we define $j$ to be the $x$-coordinate of the intersection
point of the line $y=x+k$ and the boundary $B(\pi')$.
Since $\nu(\pi')\ge k+2$, $j$ is positive. Moreover, $j\le\mu$
because otherwise $\crank(\pi')$ would be $>k$. So we can remove from
$\pi'$ a vertical column of length $j$ representing ones and place it as
a row of length $j$ right underneath the $[j,j+k]$ rectangle.
This way we obtain $\pi$ with $k\in R(\pi)$, $\nu(\pi)\ge k+2$.

\underbar{Case 2.} Here we consider partitions $\pi$ with $\nu(\pi)\le k$
and unique largest part $\lambda(\pi)$. In this case the
segment $BS_k(\pi)$ is necessarily vertical, implying that
$k\in R(\pi)$. We now transform $\pi$ into $\pi'$ as follows.
If $|\pi|>1$, then we add a part of size $1$ to $\pi$
and subtract $1$ from $\lambda(\pi)$, giving
$\lambda(\pi')=\lambda(\pi)-1$, $\nu(\pi')=\nu(\pi)+1$, $\mu(\pi')>0$.
If $|\pi|=1$, then we define $\pi'=\pi$. It is obvious that the
map $\pi\to\pi'$ is reversible and that $\crank(\pi')\le k-1$, $\mu(\pi')>0$,
$\nu(\pi')\le k+1$.

\underbar{Case 3.} Here we consider partitions $\pi$ with $k\ge2$, 
$\nu(\pi)\le k$ and the largest part $\lambda(\pi)$ is repeated.
Once again, it is clear that $k\in R(\pi)$. We now conjugate $\pi$
to get $\pi'=\pi^{*}$. Since the smallest part of $\pi'$ is
at least $2$, we have $\mu(\pi')=0$, and 
$\crank(\pi')=\lambda(\pi') =\nu(\pi)\le k$.

We now recall that $\nu(\pi)\ne k+1$ whenever $k\in R(\pi)$.
Thus the three cases above are exhaustive. Hence, \tagref{3.5} holds
for $k>0$.

If $k<0$ there is no need to consider cases 2 and 3, because there
are no partitions with a negative number of parts. In addition,
case 1 requires no modification. Hence, \tagref{3.5}
is valid in this case $k<0$, as well.

If $k=0$, then there is no need to consider case 3. Once again, case 1
requires no modification. However, in case 2 the map 
$\pi\to\pi'$ is not bijective. To see this, we note that the set
of partitions $\pi$ with $\nu(\pi)\le0$ is empty, but the set of partitions
$\pi'$ with $\crank(\pi')\le-1$, $\mu(\pi')>0$,
$\nu(\pi')\le 1$ consists of the single partition $\pi'$ with $|\pi'|=1$,
$\nu(\pi')=1$ and $\crank(\pi')=-1$.
Thus,
\begin{equation}
c_0(n) = g_0(n) + \delta_{n,1}, \qquad n\ge0.
\tag{3.19}
\end{equation}
The last equation can be easily transformed into \tagref{3.5} with $k=0$.

%%%%%END SECTION 3

\setcounter{equation}{0}
\section{Partitions with bounds on the largest part and the crank}
\label{sec:4}
%%%%%%SECTION 4

Let $C_k^L(q)$, $\Chat_k^L(q)$, $G_k^L(q)$ denote the generating functions
for partitions with crank $\le k$ and  largest part $\le L$, with crank $k$
and largest part $\le L$, with $k$ in the rank-set and  largest part $\le L$,
respectively.  In this section we will establish the following bounded
analogues of \tagref{3.5} and \tagref{3.9}:
\begin{align}    
&C_k^L(q) = G_k^L(q) + \frac{1-q}{(q)_k} + (q-1)\qbin{L+k}{k} +q \delta_{k,0},
\tag{4.1}\\
&G_{k}^L(q) + q^{k+1} G_{k+1}^{L-1}(q) = \frac{1}{(q)_L},
\tag{4.2}
\end{align}    
where for the sake of simplicity here (and throughout this section) we assume 
that $0\le k \le L$,
$L\ne0$, unless otherwise stated.

The proof of \tagref{4.2} is essentially the same as that of \tagref{3.9}.
Iterating \tagref{4.2} we derive
\begin{equation}
G_k^L(q) = \sum_{j=0}^L (-1)^j \frac{q^{T_j + kj}}{(q)_{L-j}}.
\tag{4.3}
\end{equation}
To prove \tagref{4.1} we need to follow the three separate cases of
the map $\pi\to\pi'$ we used to prove \tagref{3.5}.

Case 1 requires no modification. In case 2 the map $\pi\to\pi'$
produces partitions $\pi'$ with $\lambda(\pi')=\lambda(\pi)-1\le L-1$
and, therefore, misses partitions $\pi'$ counted by $C_k^L(q)$ in \tagref{4.1}
such that $\lambda(\pi')=L$, $\mu(\pi')>0$ and $\nu(\pi')\le k+1$,
and when $k=0$ this map also misses 
the partition $\pi'=1$, as discussed earlier. In other words, the
correction term needed in this case is
\begin{equation}
\CT_2 = q^{1+L}\qbin{L+k-1}{k-1} + q \delta_{k,0}.
\tag{4.4}
\end{equation}
In case 3, the map $\pi\to\pi'$ fails to account for partitions $\pi'$
counted by $C_k^L(q)$ such that $\lambda(\pi')\le k$, $\nu(\pi')>L$,
$\mu(\pi')=0$. The correction term needed in this case is
\begin{equation}
\CT_3 = \left\{ \frac{1-q}{(q)_k} - \left( \qbin{L+k}{k} - q\qbin{L-1+k}{k}
\right)\right\} \theta(k>1), 
\tag{4.5}
\end{equation}
where
\begin{equation}
\theta(\mbox{statement}) =
\begin{cases}
  1, &\mbox{if statement is true}, \\
  0, &\mbox{otherwise.}
\end{cases}
\tag{4.6}
\end{equation}
To understand \tagref{4.5} we observe that
$\frac{1-q}{(q)_k}$ is the generating function for partitions without ones
and largest part not exceeding $k$, and
$$
\left( \qbin{L+k}{k} - q\qbin{L-1+k}{k} \right)
$$
is the generating function for partitions $\pitwid$
with 
$\lambda(\pitwid)\le k$, $\nu(\pitwid)\le L$,
$\mu(\pitwid)=0$.
Combining \tagref{4.4} and \tagref{4.5} and using the $q$-binomial recurrence
\tagref{2.6} we get the total correction term
\begin{equation}
T = \CT_2 + \CT_3 = \frac{1-q}{(q)_k} 
+ (q-1)\qbin{L+k}{k} +q \delta_{k,0},
\tag{4.7}
\end{equation}
as desired. Since
\begin{equation}
\Chat_k^L(q) = C_k^L(q) - C_{k-1}^L(q)
\tag{4.8}
\end{equation}
we have for $L\ge k>0$
\begin{align}
\Chat_k^L(q) &= 
 \sum_{j=1}^L (-1)^{j-1} q^{T_{j-1} + kj}\frac{(1-q^j)}{(q)_{L-j}}\tag{4.9}\\
 &\quad
 + \theta(k>1) \frac{1-q}{(q)_k} q^k + (q-1) q^k \qbin{L-1+k}{k},
\nonumber
\end{align}
by using \tagref{4.1} and \tagref{4.3}.

We now derive a very different representation for $\Chat_k^L(q)$ using
\tagref{1.6}. Because the crank is defined in \tagref{1.6}
in a piece-wise fashion we have to treat two separate cases.

\underbar{Case A.} Here we consider partitions $\pio$
with $\crank(\pio)=k>0$, 
$\lambda(\pio)\le L$, and 
$\mu(\pio)>0$. We decompose the graph of some given $\pio$ as shown
in Fig.\ 8 below.

\begin{figure}[h!]
\begin{center}
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\begin{center}
\textsc{Fig 8}.\ \parbox[t]{4in}{Decomposition of partition $\pio$
with $\crank(\pio)=k>0$, $L\ge\lambda(\pio)$, $\mu(\pio)>0$.}
\end{center}
\end{figure}

From this decomposition it is clear that the generating function for these
partitions is
\begin{equation}
A(q) = \sum_{\mu=1}^{L-1} q^{\mu} q^{(\mu+1)(\mu+k)}
\frac{1}{(q^2;q)_{\mu-1}} \qbin{L-1+k}{\mu + k}.
\tag{4.10}
\end{equation}

\underbar{Case B.} 
Here we consider partitions $\pio$ without ones with 
$\crank(\pio)=\lambda(\pio)=k$, $2\le k\le L$.
Clearly, the generating function for these partitions is
\begin{equation}
B(q) = \frac{q^k}{(q^2;q)_{k-1}} \theta(k>1).
\tag{4.11}
\end{equation}
Combining \tagref{4.10}, \tagref{4.11} we find that
\begin{align}
\Chat_k^L(q) &= A(q) + B(q) \tag{4.12}\\
&= \frac{q^k(1-q)}{(q)_k} \theta(k>1)
+ \sum_{\mu=1}^{L-1} \frac{q^{(\mu+1)(\mu+k) + \mu}}
{(q^2;q)_{\mu-1}} \qbin{L-1+k}{\mu + k},
\quad 0< k \le L.
\nonumber 
\end{align}
Comparing \tagref{4.9} and \tagref{4.12} we arrive at the following
identity
\begin{equation}
\sum_{j=1}^L (-1)^{j-1} q^{T_{j-1} + kj}\frac{(1-q^j)}{(q)_{L-j}}
= (1-q)
\sum_{\mu=0}^{L-1} \frac{q^{(\mu+1)(\mu+k) + \mu}}
{(q)_{\mu}} \qbin{L-1+k}{\mu + k}.
\tag{4.13}
\end{equation}
Remarkably, this identity is nothing else but a limiting case of
Heine's second transformation of a ${}_2\phi_1$-series \cite{GR1}:
\begin{equation}
\2phi1{a,b}{c}{q,z} = \frac{ \qinf{\frac{c}{b}} \qinf{bz} }
	                   { \qinf{c} \qinf{z} }
			\2phi1{\frac{abz}{c},b}{bz}{q,\frac{c}{b}},
\tag{4.14}
\end{equation}
where
\begin{equation}
\2phi1{a,b}{c}{q,z} = \sum_{n=1}^\infty \frac{ (a)_n (b)_n }
					     { (c)_n (q)_n }
					z^n.
\tag{4.15}
\end{equation}
To see this we rewrite the left side of \tagref{4.13} in
$q$-hypergeometric form as
\begin{equation}
\sum_{j=1}^L (-1)^{j-1} q^{T_{j-1} + kj}\frac{(1-q^j)}{(q)_{L-j}}
= \frac{q^k(1-q)}{(q)_{L-1}}
  \lim_{c\to0} \2phi1{q^2,q^{1-L}}{c}{q,q^{L+k}},
\tag{4.16}
\end{equation}
Here we have used
\begin{equation}
(q)_{L-j} = \frac{ (q)_{L-1}}{ (q^{1-L} )_{j-1}}
		(-1)^{j-1} q^{T_{j-2} - (L-1)(j-1)},
\tag{4.17}
\end{equation}
and
\begin{equation}
\frac{1-q^{1+j}}{1-q} =  \frac{ (q^2)_j }{ (q)_j },
\tag{4.18}
\end{equation}
along with the trivial relation
\begin{equation}
\lim_{c\to0} (c)_n = 1.
\tag{4.19}
\end{equation}
Next we employ \tagref{4.14} with $a=q^2$, $b=q^{1-L}$, $z=q^{L+k}$
together with
\begin{equation}
\frac{ \qinf{q^{1+k}} }{ \qinf{q^{L+k}} } = (q^{1+k})_{L-1},
\tag{4.20}
\end{equation}
and
\begin{equation}
\lim_{\rho\to\infty} (\rho)_i \rho^{-i} = (-1)^i q^{T_{i-1}}
\tag{4.21}
\end{equation}
to derive
\begin{align}    
&\sum_{j=1}^L (-1)^{j-1} q^{T_{j-1} + kj}\frac{(1-q^j)}{(q)_{L-j}}
= \frac{q^k(1-q)}{(q)_{L-1}} (q^{1+k})_{L-1}
  \lim_{c\to0} \2phi1{\frac{q^{3+k}}{c},q^{1-L}}{q^{1+k}}{q,cq^{L-1}}
\tag{4.22}\\
&=
\frac{q^k(1-q)}{(q)_{L-1}} (q^{1+k})_{L-1}
\sum_{i=0}^{L-1} (-1)^i \frac{ (q^{1-L})_i }{ (q)_i (q^{1+k})_i }
			q^{ T_{i-1} + (L+k+2)i }.
\nonumber
\end{align}
Finally, verifying that
\begin{equation}
(-1)^i \frac{ (q^{1+k})_{L-1} }{ (q)_{L-1} } \frac{ (q^{1-L})_i }{ (q^{1+k})_i}
= q^{T_i - Li} \qbin{L-1+k}{i+k}
\tag{4.23}
\end{equation}
we see that
\begin{equation}
\sum_{j=1}^L (-1)^{j-1} q^{T_{j-1} + kj}\frac{(1-q^j)}{(q)_{L-j}}
= (1-q) \sum_{i=0}^{L-1} \frac{ q^{i^2 + (2+k)i + k} }{ (q)_i }
			\qbin{L-1+k}{i+k}.
\tag{4.24}
\end{equation}
This last equation is essentially \tagref{4.13}, as desired.

The $q$-hypergeometric proof of \tagref{4.13} clearly suggests that our
analysis can be extended further to treat partitions $\pi$ with $\crank(\pi)=k$,
$\lambda(\pi)\le L$ and $\nu(\pi) \le M$. However, we will not pursue this
here.

%%%%%END SECTION 4

\setcounter{equation}{0}
\section{A variant of Dyson's transformation \\
and a new proof of Gauss's formula}
\label{sec:5}

%%%%%%SECTION 5

Let $e(n)$ denote the number of partitions of $n$
into distinct odd parts with all other parts being even.
The generating function $E(q)$ for these partitions can be written
in the form of a product as
\begin{equation}
E(q) = \sum_{n\ge0} e(n) q^n = \frac{ (-q;q^2)_\infty }{ (q^2;q^2)_\infty }.
\tag{5.1}
\end{equation}
We will use MacMahon's graphs with modulus $2$ to depict these partitions.
For example, the mod $2$ graph of the partition
$\pi = 7 + 6 + 6 + 5 + 2$ is given in Fig.\ 9.


\begin{figure}[h!]
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%%\vskip -60pt
\textsc{Fig 9}.\ \parbox[t]{5in}{mod $2$ and regular mod $1$ representations
of $\pi = 7 + 6 + 6 + 5 + 2$}
\end{figure}

A nice thing about mod $2$ representations of the partitions counted
by $E(q)$ is that these representations have certain invariance
properties under conjugation. Namely, if we conjugate
the mod $2$ graph of some given partition counted by $E(q)$
we obtain a partition that is also counted by $E(q)$. For instance
if we conjugate the mod $2$ graph of the partition depicted in Fig.\ 9
we get $\pi^{*}=10+8+7+1$ whose mod $2$ graph is given in Fig.\ 10.


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%%\vskip -60pt
\mycap{10}{mod $2$ representation of $\pi^{*} = 10 + 8 + 7 + 1$}    
\end{figure}

Note that the ordinary Ferrers graph representations do not possess this
invariance property. For example, if we conjugate the mod $1$ graph
in Fig.\ 9 we get the partition $5 + 5 + 4 + 4 + 4 + 3 + 1$,
which has repeated odd part $5$. 

Next, we define the $M_2$-rank of a partition as the
{\it largest row minus the number of rows of its mod $2$ graph}. 
It is easy to check that the $M_2$-rank of the partition,
$7 + 6 + 6 + 5 + 2$, depicted in Fig.\ 9, is equal to $4 - 5 = -1$,
while its rank is $7 - 5 = 2$. Also, it is straightforward to verify that
under conjugation the $M_2$-rank changes its sign, as does the
ordinary rank.

Let us define $\Etwid_r(q)$ as
\begin{equation}
\Etwid_r(q) = \sum_{n\ge1} \etwid_r(q) q^n,
\tag{5.2}
\end{equation}
where $\etwid_r(n)$ denotes the number of partitions of $n$
into distinct odd parts and unrestricted even parts such that
the $M_2$-rank $\ge r$. We assume that $\etwid_r(0)=0$.
We now show that
\begin{equation}
\Etwid_r(q) + \Etwid_{1-r}(q) + 1 = E(q),
\tag{5.3}
\end{equation}
and
\begin{equation}
\Etwid_r(q) = q^{2r+1} \left( \Etwid_{-1-r}(q) + 1\right), \quad r\ge0.
\tag{5.4}
\end{equation}
To prove \tagref{5.3} we will follow a well-trodden path and observe
that any nonempty partition counted by $E(q)$ whose $M_2$-rank
$\ge r$ is also counted by $\Etwid_r(q)$.
Any nonempty partition counted by $E(q)$ whose $M_2$-rank
$< r$ gives rise to a partition with $M_2$-rank
$\ge -1-r$, after conjugation. Thus, this conjugated partition
is counted by $\Etwid_{-1-r}(q)$ in \tagref{5.3}. Finally, the empty
partition is counted by 1 and $E(q)$ on the left and right sides
of \tagref{5.3}, respectively.

The proof of \tagref{5.4} requires modification of Dyson's transformation,
which we now proceed to describe. Let $\pi$ denote the mod $2$ 
graph of some partition counted by $\Etwid_r(q)$ in \tagref{5.4}.
Let $r + \ell(\pi)$ denote the length of the largest row of $\pi$,
and $h(\pi)$ denote the number of rows of $\pi$.
Clearly,
$$
h(\pi) \le \ell(\pi)
$$
for $\pi$ to have $M_2$-rank $\ge r$. Next, we remove the largest row
from $\pi$ to get a mod $2$ graph $\pitwid$. Conjugating $\pitwid$
we obtain $\pitwid^{*}$. Now, if the removed row represented the odd part
$2\ell + 2r - 1$, then we add to $\pitwid^{*}$ a new largest row
of length $\ell - 1$, representing the even part $2\ell - 2$.
On the other hand, if the removed row represented the even part $2\ell + 2r$,
then we add to $\pitwid^{*}$ a new largest row
of length $\ell$, representing the odd  part $2\ell - 1$.
These operations are illustrated in Fig. 11, 12 where the resulting partition
is denoted by $\pi'$.

\begin{figure}[h!]
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%%\vskip -60pt
\textsc{Fig 11}.\ \parbox[t]{4in}{Modification of Dyson's adjoint for
$\pi=7 + 5 + 2$ with $M_2$-rank $= 1 > r = 0$.}
\end{figure}


\begin{figure}[h!]
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\textsc{Fig 12}.\ \parbox[t]{4in}{Modification of Dyson's adjoint for
$\pi=8 + 5 + 2$ with $M_2$-rank $= 1 > r = 0$.}
\end{figure}

Remarkably, regardless of whether the largest part of $\pi$ is even of odd
we have
\begin{equation}
|\pi'| = |\pi| - 2r - 1,
\tag{5.5}
\end{equation}
and
\begin{equation}
\mbox{$M_2$-rank}(\pi') \ge -1 - r.
\tag{5.6}
\end{equation}
It is easy to check that the map 
$\pi\to\pi'$ is reversible, except when $|\pi|=2r+1$. In the last case $\pi'$ is
empty. This concludes the proof of \tagref{5.4}.

Combining \tagref{5.3}, \tagref{5.4} we obtain
\begin{equation}
\Etwid_r(q) + q^{2r+1} \Etwid_{2+r}(q) = q^{2r+1} 
				\frac{ (-q;q^2)_\infty }{ (q^2;q^2)_\infty },
\quad r\ge 0.
\tag{5.7}
\end{equation}
Iteration of \tagref{5.7} yields
\begin{equation}
\Etwid_r(q) = \frac{ (-q;q^2)_\infty }{ (q^2;q^2)_\infty }
	 	\sum_{j\ge1} (-1)^{j-1} q^{2rj + j(2j-1)}, \quad r\ge 0.
\tag{5.8}
\end{equation}
Now \tagref{5.3} with $r=0$ states that
\begin{equation}
\Etwid_0(q) + \Etwid_1(q) + 1 = \frac{ (-q;q^2)_\infty }{ (q^2;q^2)_\infty }.
\tag{5.9}
\end{equation}
Thanks to \tagref{5.8} we may cast \tagref{5.9} in the form
\begin{equation}
1 = \frac{ (-q;q^2)_\infty }{ (q^2;q^2)_\infty }
	\sum_{j=-\infty}^{\infty} (-1)^j q^{j(2j+1)},
\tag{5.10}
\end{equation}
and so
\begin{equation}
\frac{ (q^2;q^2)_\infty }{ (-q;q^2)_\infty }
=        \sum_{j=-\infty}^{\infty} (-1)^j q^{j(2j+1)}.
\tag{5.11}
\end{equation}
Finally, replacing $q$ by $-q$ in \tagref{5.11} we obtain
the Gauss identity
\begin{equation}
\frac{ (q^2;q^2)_\infty }{ (q;q^2)_\infty }
=        \sum_{j=-\infty}^{\infty} q^{j(2j+1)} = \sum_{j\ge0} q^{T_j}.
\tag{5.12}
\end{equation}

Formula \tagref{5.8} implies that
\begin{align}
\Ehat_r(q) &= E(q) - \Etwid_{r+1}(q)\tag{5.13}\\
&= \frac{ (-q;q^2)_\infty}{ (q^2;q^2)_\infty }
   \sum_{j\ge0} (-1)^j q^{2rj + j(2j+1)},\quad r\ge0,
\nonumber
\end{align}
where $\Ehat_r(q)$ denotes the generating function for partitions into
distinct odd, and unrestricted even parts with $\mbox{$M_2$-rank}\le r$.
We now develop very different representations for $\Ehat_r(q)$.
To this end we decompose partitions counted by $\Ehat_r(q)$ into
even and odd parts. Let's assume that this decomposition gives
$\pi_1$ with $j$ distinct odd parts and $\pi_2$ with $i$ even parts.
Clearly, $\lambda(\pi_1) \le 2(i+j+r)-1$
and $\lambda(\pi_2) \le 2(i+j+r)$, and so, for $r\ge0$ we have
\begin{equation}
\Ehat_r(q) = \sum_{i,j\ge0}
q^{j + 2T_{j-1}} 
\qbinb{i+j+r}{j} q^{2i}
\qbinb{i+j+r-1+i}{i}.
\tag{5.14}
\end{equation}
Comparing \tagref{5.13} and \tagref{5.14}, we see that
\begin{equation}
\sum_{i,j\ge0} q^{j^2 + 2i}
\qbinb{i+j+r}{j} \qbinb{2i+j+r-1}{i}
=
\frac{ (-q;q^2)_\infty }{ (q^2;q^2)_\infty }
\sum_{j\ge0} (-1)^j q^{j(2j+1)}\left(q^{2r}\right)^j,\quad r\ge0.
\tag{5.15}
\end{equation}
Next, since 
\begin{equation}
\qbin{n+m}{n} = \frac{ (q^{1+m};q)_n }{ (q;q)_n },
\tag{5.16}
\end{equation}
we can rewrite \tagref{5.13} as
\begin{equation}
\sum_{i,j\ge0} q^{j^2 + 2i}
\frac{ (aq^{2i+2};q^2)_j (aq^{2i+2j};q^2)_i }
{ (q^2;q^2)_j (q^2;q^2)_i}
=
\frac{ (-q;q^2)_\infty }{ (q^2;q^2)_\infty }
\sum_{j\ge0} (-1)^j q^{2j^2+j} a^j,
\tag{5.17}
\end{equation}
where $a=q^{2r}$, $r\ge0$. Since the limit of the sequence
$\{q^{2r}\}$ is equal to zero, we may treat $a$ in \tagref{5.17}
as a free parameter.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%INSERT #3
In Appendix B we discuss an alternative proof of 
\tagref{5.17}. This proof was communicated to us by George Andrews \cite{Y}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

In the past, fundamental as they are, modular representations have not
received the attention they deserve. Recently, Alladi \cite{AL}
used $2$-modular representations to provide an elegant
combinatorial bijection for a variant of G\"ollnitz's partition
theorem. However, in \cite{AL} partitions into only distinct odd parts
are considered, whereas here we allow even parts to appear with
possible repetition. 

In this regard, Alladi pointed out to us that Andrews \cite[ex.6, p.13]{A2}
used mod $2$ representations on the set of partitions treated here, subject
to the extra condition that no part $=1$, in order to establish a partition
theorem, which is equivalent to Cauchy's identity in the form:
\begin{equation}
\sum_{n\ge0} \frac{(-aq;q^2)_n}{(q^2;q^2)_n} t^n q^{2n}
= \frac{ (-atq^3;q^2)_\infty }{ (tq^2;q^2)_\infty }.
\tag{5.18}
\end{equation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%INSERT #4
Andrews's proof of the original Cauchy's identity with base $q$
(instead of base $q^2$ as above) may be found in \cite{Z}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%END SECTION 5


\setcounter{equation}{0}
\section{Open questions}
\label{sec:6}

%%%%%%SECTION 6

In \cite{A3} Andrews proposed a dissection of a partition $\pi$
into successive Durfee squares with sizes
$n_1(\pi) \ge n_2(\pi) \ge n_3(\pi) \ge \cdots$. For example, the partition
$\pi$, depicted in Fig.\ 13, has two Durfee squares with sizes $n_1(\pi)=3$,
$n_2(\pi)=2$.

\begin{figure}[h!]
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%%\vskip -60pt
\textsc{Fig 13}.\ \parbox[t]{5in}{Ferrers graph of $\pi=6 +5 + 4 + 2 + 2 + 1$.
This graph can be dissected into two Durfee squares of sizes $3$ and $2$.
$\mbox{$3$-rank}(\pi) = 2 - 1 = 1$.}
\end{figure}

Garvan \cite{Ga2} introduced a generalization of Dyson's rank
for partitions with at least $k-1$ successive Durfee squares.
He called this generalization the $k$-rank of a partition $\pi$.
The $k$-rank is defined as
\begin{equation}
\mbox{$k$-rank}(\pi) =
\matrix
&\mbox{the number of columns in the Ferrers graph of } \hfill\\
&\mbox{$\pi$ which lie to the right of the first Durfee square} \hfill\\
&\mbox{and whose length $\le n_{k-1}(\pi)$} \hfill\\
&\mbox{minus} \hfill\\
&\mbox{the number of parts of $\pi$ that lie below the} \hfill\\
&\mbox{$(k-1)$-th Durfee square.}\hfill
\endmatrix
\tag{6.1}
\end{equation}
For instance, the partition $\pi$ depicted in Fig.\ 13 has
$\mbox{$3$-rank}(\pi)=2-1=1$. Since any nonempty partition $\pi$
has at least one Durfee square we can easily infer that the $2$-rank
is the same as Dyson's rank.

Formula (1.10) in \cite{Ga2} implies that for $m\ge0$
\begin{equation}
FG_{k,m} (q) = \frac{1}{(q)_\infty} 
\sum_{j=1}^\infty (-1)^{j-1} q^{ \frac{j((2k-1)j-1)}{2} + mj},
\tag{6.2}
\end{equation}
where $FG_{k,m} (q)$ denotes the generating function for partitions
$\pi$ with at least $k-1$ successive Durfee squares and with
$\mbox{$k$-rank}(\pi) \ge m \ge 0$. Using \tagref{6.2} it is easy to
verify that
\begin{equation}
FG_{k,m}(q) + q^{k+m-1} FG_{k,2k-1+m}(q) = \frac{q^{k+m-1}}{ (q)_\infty }.
\tag{6.3}
\end{equation}
We note that \tagref{6.3} with $k=2$ becomes \tagref{1.24}.
Despite its speciously simple appearance the functional equation
\tagref{6.3} with $k > 2$ turned out to be very difficult to prove
in a combinatorial fashion. Perhaps the appropriate generalization
of Dyson's notion of rank-set may provide a key to a combinatorial
proof
of \tagref{6.3}.

We feel that it would be worthwhile to determine the precise
$q$-hypergeometric status of the new polynomial analogues of
Euler's pentagonal number theorem \tagref{1.30} and to explore
more general iteration schemes.
Finally, we would like to pose the problem of finding a natural bounded
extension of formulas \tagref{1.4}, \tagref{1.5}, and
\tagref{1.7}--\tagref{1.9}.
%%%%%END SECTION 6

\setcounter{equation}{0}
\section*{Appendix A}
%%\label{sec:6}
%%%%%%SECTION APPENDIX A
Here we give a direct proof of \tagref{3.17} which we restate as
\begin{equation}
\chat_{-k}(n) = \chat_k(n) + \delta_{n,1} \delta_{k,1}, \quad k\ge0,
\tag{A.1}
\end{equation}
with $\chat_{k}(n)$ denoting the number of partitions of $n$ with
crank $k$. It is easy to check that \tagref{A.1} holds for $n=0$, $1$.

To proceed further let us recall that $\lambda(\pi)$, $\mu(\pi)$
and $\nutwid(\pi)$ denote the largest part of a partition $\pi$,
the number of ones in $\pi$ and the number of parts of $\pi$ which are 
larger than $\mu(\pi)$, respectively. In addition, let $\gamma(\pi)$
be defined by
$$
\gamma(\pi) =
\begin{cases}
  p_1-p_2, &\mbox{if $\nutwid(\pi)\ne1$}, \\
  \lambda(\pi)-\mu(\pi)-1, &\mbox{if $\nutwid(\pi)=1$,}
\end{cases}
$$ 
where 
$\pi = p_1 + p_2 + p_3 + \cdots$ has
parts $p_1\ge p_2\ge p_3\ge\cdots$.
It is easy to check that \tagref{A.1} with $n>1$ is an immediate consequence 
of the following two propositions.

\proclaim{Proposition 1}
The number of partitions $\pi$ with $|\pi|=n>1$,
$\lambda(\pi)=\ell$,
and $\mu(\pi)=0$, equals the number of partitions $\pi'$ with
$|\pi'|=|\pi|$, $\nutwid(\pi')=0$, and $\mu(\pi')=\ell$.
\endproclaim

\proclaim{Proposition 2}
The number of partitions $\pi$ with $|\pi|=n>1$,
$\mu(\pi)=M>0$,
and $\nutwid(\pi)=N>0$,
equals the number of partitions $\pi'$ with
$|\pi'|=|\pi|$, 
$\mu(\pi)=N$,
and $\nutwid(\pi)=M$.
\endproclaim

To prove Proposition 1 we remove the largest row from the graph of $\pi$
and then add a vertical column representing $\lambda(\pi)$ ones to the
resulting graph. Let's call the resulting partition $\pi'$.
Obviously, $\mu(\pi')=\lambda(\pi)$, and
$\nutwid(\pi')=0$. 
Since $n>1$ the map $\pi\to\pi'$ is a bijection and the
result follows. \qed

The proof of Proposition 2 is more involved. Here we need to decompose
$\pi$ as indicated in Fig.\ 14 below.

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\mycap{14}{\enskip Graph of $\pi$ in Proposition 2.}
\end{figure}
%%%
%%%

\noindent
Let us now remove from the graph of $\pi$ in Fig.\ 14 three pieces, namely, 
the vertical columns of height $M$, $N$ and the horizontal row of length 
$\gamma$. Next, we conjugate the resulting graph to get $\pitwid$. 
We now add three pieces to $\pitwid$ as indicated in Fig.\ 15 to get
$\pi'$. 

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\end{center}
\mycap{15}{\enskip Graph of $\pi'$ in Proposition 2.}
\end{figure}
%%%
%%%

\noindent
Clearly, $\mu(\pi')=N$, $\nutwid(\pi')=M$ and $|\pi'|=|\pi|$.
To finish the proof we observe that the map $\pi\to\pi'$ is
a bijection. \qed

Let us call the map employed in the proofs of Propositions 1 and 2
a pseudo-conjugation transformation. We say that a partition $\pi$
with $|\pi|>1$ is self-pseudo-conjugate if it remains invariant
under pseudo-conjugation. In addition, we say the
partitions $\pi=0$, $\pi=1$ are self-pseudo-conjugate.

It is well known that the number of self-conjugate partitions of $n$
equals the number of partitions into distinct odd parts. The generating
function for the last set of partitions is $(-q;q^2)_\infty$.
Remarkably, the same is true for self-pseudo-conjugate partitions,
as we now demonstrate. First, it is obvious that the partitions described in 
Proposition 1 are not self-pseudo-conjugate. Second, the partitions
$\pi$ in Proposition 2 are self-pseudo-conjugate only if $M=N$ and
the conjugate of sub-graph A in Fig.\ 14 is identical to
sub-graph B. Therefore, the generating function $\SPC(q)$ for
self-pseudo-conjugate partitions is
\begin{align}
\SPC(q) &= 1 + q + \sum_{\substack{M\ge1, \\ \gamma\ge0}}     
\frac{q^{M(M+1)+M+\gamma}}{(q^4;q^2)_{M-1}}\tag{A.2}\\
&=1 + q + \sum_{M\ge1}\frac{q^{M(M+1)+M}}{(1-q)(q^4;q^2)_{M-1}}
=(1+q) \sum_{M\ge0}\frac{q^{M(M+1)}}{(q^2;q^2)_{M}}q^M.
\nonumber
\end{align}
Making use of the Euler identity
\begin{equation}
\sum_{j\ge0} \frac{q^{j(j+1)}}{(q^2;q^2)_j} z^j = (-zq^2;q^2)_\infty,
\tag{A.3}
\end{equation}
we find that
\begin{equation}
\SPC(q) = (1+q) (-q^3;q^2)_\infty = (-q;q^2)_\infty,
\tag{A.4}
\end{equation}
as desired.
%%%%%%END SECTION APPENDIX A

\setcounter{equation}{0}
\section*{Appendix B}
%%\label{sec:6}
%%%%%%SECTION APPENDIX B
Here we describe an alternative proof of
\tagref{5.17} communicated to us by George Andrews \cite{Y}.
We begin by expanding the products
$(aq^{2i+2};q^2)_j$ and $(aq^{2i+2j};q^2)_i$
in \tagref{5.17} using the $q$-binomial theorem \cite[(3.3.6), p.36]{A2}
\begin{equation}
\sum_{n\ge0} q^{n^2-n} z^n \qbinb{L}{n}
= (-z;q^2)_L.
\tag{B.1}
\end{equation}
This way we obtain after preforming changes of summation variables the
following expression for the left side of \tagref{5.17}
\begin{equation}
\LHS\tagref{5.17}
=
\sum_{i,j,s,t\ge0}
(-a)^{s+t}
\frac{
q^{ (j+t)^2 + 2(i+s) + 2s(i+j+s+t) + s(s-1) + t^2+t + 2t(i+s) }
}
{
(q^2;q^2)_i
(q^2;q^2)_j
(q^2;q^2)_s
(q^2;q^2)_t
}.
\tag{B.2}
\end{equation}
Next, we use the Euler identity \tagref{A.3} along with another result
of Euler \cite[(2.2.5), p.19]{A2}
\begin{equation}
\sum_{n\ge0} \frac{z^n}{(q)_n} = \frac{1}{(z;q)_\infty}
\tag{B.3}
\end{equation}
to sum out the $j$ and $i$ variables in \tagref{B.2} to get

\begin{equation}
\LHS\tagref{5.17}
=
\sum_{s,t\ge0}
(-a)^{s+t}
\frac{q^{2t^2+4st+3s^2+s+t}}{(q^2;q^2)_s (q^2;q^2)_t}
\frac{(-q^{1+2s+2t};q^2)_\infty }{(q^{2+2s+2t};q^2)_\infty}.
\tag{B.4}
\end{equation}
Since

\begin{equation}
\frac{(-q^{1+2s+2t};q^2)_\infty }{(q^{2+2s+2t};q^2)_\infty}
= \frac{(-q;q^2)_\infty}{(q^2;q^2)_\infty}
\frac{(q^2;q^2)_{s+t}}{(-q;q^2)_{s+t}},
\tag{B.5}
\end{equation}
we can derive

\begin{equation}
\LHS\tagref{5.17}
=
 \frac{(-q;q^2)_\infty}{(q^2;q^2)_\infty}
\sum_{n\ge0} (-a)^n \frac{q^{2n^2+n}}{(-q;q^2)_n}
\sum_{s=0}^n q^{s^2} \qbinb{n}{s}.
\tag{B.6}
\end{equation}
Making use of \tagref{B.1} we can evaluate the inner sum in \tagref{B.6}
to get

\begin{equation}
\LHS\tagref{5.17}
=
\frac{(-q;q^2)_\infty}{(q^2;q^2)_\infty}
\sum_{n\ge0} (-a)^n q^{2n^2+n},
\tag{B.7}
\end{equation}
which is essentially \tagref{5.17}, as desired. 
%%%%%%END SECTION APPENDIX B



{\it Acknowledgements}.
We are grateful to Krishnaswami Alladi 
for interesting discussion and for his comments and suggestions,
and to George E. Andrews
for his help and encouragement.

{\it Note added}. In a recent paper, Warnaar \cite{W1} observed that
\tagref{1.30} with $n=1$ is a limiting case of
a rather non-trivial cubic summation formula
\begin{multline}
\sum_{k=0}^{\lfloor N/2 \rfloor} \frac{1-Aq^{4k}}{1-A}
\frac{(A,Aq^{N+1};q^3)_k}{(q,q^{-N};q)_k}
\frac{(q^{-N};q)_{2k}}{(Aq^{N+1};q)_{2k}}
\frac{(C,D;q)_k}{(Aq^3/C,Aq^3/D;q^3)_k}\, q^k \nonumber\\
=\begin{cases}\displaystyle
\frac{(Aq^3,q^{2-N}/C,q^{2-N}/D;q^3)_{\lfloor N/3 \rfloor}}
{(Aq^3/C,Aq^3/D,q^{2-N}/CD;q^3)_{\lfloor N/3 \rfloor}},
& N\not\equiv 2\pmod{3},\\[3mm]
0, & N\equiv 2\pmod{3},
\end{cases}
\tag{N.1}
\end{multline}
with $CD=Aq^{N+1}$.
More precisely, Warnaar replaced $A\to A^2$, $C\to CA$,
$D\to DA$ and let $A\to0$ in \tagref{N.1} to obtain
\begin{equation}
\sum_{k=0}^{\lfloor N/2 \rfloor} \frac{ (q^{-N};q)_{2k} }{ (q,q^{-N};q)_k } q^k =
\begin{cases}
(-1)^{\lfloor N/3 \rfloor} q^{-\frac{N(N-1)}{6}},& N\not\equiv 2\pmod{3}, \\[3mm]
0, &\mbox{otherwise},
\end{cases}
\tag{N.2}
\end{equation}
which is essentially \tagref{1.30} with $n=1$ and
$L=\lfloor (N+1)/3 \rfloor$. In \cite{W1} Warnaar established \tagref{N.1} by 
setting $p=0$ in his elliptic generalization of \tagref{N.1}
(Corollary 4.13 in \cite{W2}).
Here, we would like to point out that the cubic summation formula \tagref{N.1}
is a special case of the Gasper-Rahman transformation formula (3.19)
in \cite{GR2}. Indeed, setting $ac=d=A$ and $b=cq^{1+N}$ in this formula
we get
\begin{align}
&\sum_{k=0}^{\lfloor N/2 \rfloor} \frac{1-Aq^{4k}}{1-A}
\frac{(A,Aq^{N+1};q^3)_k}{(q,q^{-N};q)_k}
\frac{(q^{-N};q)_{2k}}{(Aq^{N+1};q)_{2k}}
\frac{(cq^{1+N},A/c;q)_k}{(Aq^{2-N}/c,cq^3;q^3)_k}\, q^k \nonumber\\
&\, = \frac{ (Aq;q)_N }{ (q^{-N};q)_N }
 \frac{(q^{1-2N};q^3)_N}{(Aq^{2-N};q^3)_N} \cdot \tag{N.3}\\
&\, \cdot {}_8W_7(Aq^{-1-N};Aq^{1+N},c,Ac^{-1}q^{-1-N},q^{1-N},q^{-N};q^3,q^3)
\nonumber
\end{align}
with ${}_{r+1}W_r(a_1;a_4,\dots,a_{r+1};q,z)$ defined as in \cite[(2.11.11)]{GR1}.
Note that for $N\equiv2\pmod{3}$, $N>0$
\begin{equation}
(q^{1-2N};q^3)_N = 0,
\tag{N.4}
\end{equation}
and, consequently, the right side of \tagref{N.3} becomes zero.
When $N\not\equiv2\pmod{3}$, the series ${}_8W_7$ in \tagref{N.3}
can be summed thanks to Jackson's $q$-Dougall's summation \cite[(II.22)]{GR1}.
As a result, we obtain \tagref{N.1} with $C=cq^{1+N}$ and $D=A/c$.

%%% HERE %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{thebibliography}{999}
\bibitem{AL}
	K.~Alladi,    
	\emph{A variation on a theme of Sylvester --- a smoother road to
	G\"ollnitz's (Big) theorem},
	Discrete Math. \textbf{196} (1999), 1--11.    

\bibitem{Z}
	G.~E.~Andrews,
	\emph{On a calculus of partition functions},
	Pac. J. Math. \textbf{31} (1969), 565--562.

\bibitem{X}
	G.~E.~Andrews,
	\emph{Two theorems of Gauss and allied indentities proved
	arithmetically},
	Pac. J. Math. \textbf{41} (1972), 563--578.

\bibitem{A1}
	G.~E.~Andrews,
	\emph{On a partition theorem of N.~J.~Fine},
	J. Nat. Acad. Math. India \textbf{1} (1983), 105--107.

\bibitem{A2}
	G.~E.~Andrews,
      \emph{ The Theory of Partitions},
           Encyclopedia of Mathematics and Its Applications, Vol.~2
(G.~-~C.~Rota, ed.), Addison-Wesley, Reading, Mass., 1976. (Reissued:
Cambridge Univ. Press, London and New York, 1985)

\bibitem{A3}
	G.~E.~Andrews,
	\emph{Partitions and Durfee dissection},
	Amer. J. Math. \textbf{101} (1979), 735--742.

\bibitem{Y}
	G.~E.~Andrews,
	private communication.

\bibitem{AG}
	G.~E.~Andrews and F.~G.~Garvan,
	\emph{Dyson's crank of a partition},
	Bull. Amer. Math. Soc. \textbf{18} (1988), 167--171.

\bibitem{ASD}
	A.~O.~L.~Atkin and P.~Swinnerton-Dyer,
	\emph{Some properties of partitions},
	Proc. London Math. Soc.
	\textbf{4} (1954), 84--106.

\bibitem{D1}
	F.~J.~Dyson,
	\emph{Some guesses in the theory of partitions},
	Eureka (Cambridge) \textbf{8} (1944), 10--15.

\bibitem{D2}
	F.~J.~Dyson,
	\emph{A walk through Ramanujan's garden},
	in ``Ramanujan revisited'' (Urbana-Champaign, Ill., 1987), 
	Academic Press, Boston, MA, 1988.

\bibitem{D3}
	F.~J.~Dyson,
	\emph{A new symmetry of partitions},
	J. Combinatorial Theory
	\textbf{7} (1969), 56--61.

\bibitem{D4}
	F.~J.~Dyson,
	\emph{Mappings and symmetries of partitions},
	J. Combin. Theory Ser. A \textbf{51} (1989), 169--180.

\bibitem{Ga1}
	F.~G.~Garvan, 
	\emph{New combinatorial interpretations of Ramanujan's partition
	congruences mod $5$, $7$ and $11$},
	Trans. Amer. Math. Soc. \textbf{305} (1988), 47--77.

\bibitem{Ga2}
	F.~G.~Garvan, 
	\emph{Generalizations of Dyson's rank and non-Rogers-Ramanujan 
	partitions},
	Manuscripta Math. {\bf 84} (1994), 343--359. 

\bibitem{GR1}
	G.~Gasper\ and\ M.~Rahman, 
	\emph{Basic hypergeometric series}, 
	Cambridge Univ.\ Press, Cambridge, 1990.

\bibitem{GR2}
	G.~Gasper\ and\ M.~Rahman, 
	\emph{An indefinite bibasic summation formula and some quadratic, cubic
        and quartic summation and transformation formulas}, 
        Canad. J. Math. {\bf 42} (1990), 1--27.

\bibitem{Ra1}
	S.~Ramanujan,
	\emph{Some properties of $p(n)$, the number of partitions of $n$},
	 Proc. Cambridge Philos. Soc. \textbf{19} (1919), 207--210.

\bibitem{Ra2}
	S.~Ramanujan,
	\emph{Congruence properties of partitions},
	Math. Zeitschr. \textbf{9} (1921), 147--153.

\bibitem{W1}
	S.~O.~Warnaar,
        \emph{$q$-Hypergeometric proofs of polynomial analogues of the triple 
	product identity, Lebesgue's identity and Euler's pentagonal number 
	theorem}, 
        arXiv:math.CO/0203229, preprint. 

\bibitem{W2}
	S.~O.~Warnaar,
        \emph{Summation and transformation formulas for elliptic 
	hypergeometric series}, 
        arXiv:math.QA/0001006, to appear in Constructive Approximation.

\bibitem{Win}
	L.~Winquist
        \emph{An elementary proof of $p(11m+6)\equiv 0$ (mod $11$)},
	J. Combin. Theory \textbf{6} (1969), 56--59.

%%
%%   \bibitem{sF90}
%%      Soo-Key Foo, \emph{Lattice Constructions}, Ph.D. thesis, University 
%%      of Winnebago, Winnebago, MN, December 1990.
%%
%%   \bibitem{gM68}
%%      George~A. Menuhin, \emph{Universal Algebra}, D.~van Nostrand,
%%      Princeton-Toronto-London-Mel\-bourne, 1968.
%%
%%   \bibitem{eM57}
%%      Ernest~T. Moynahan, \emph{On a problem of M.~H. Stone}, Acta Math.
%%       Acad.Sci. Hungar. \textbf{8} (1957), 455--460.
%%
%%   \bibitem{eM57a}
%%      \bysame, \emph{Ideals and congruence relations in lattices.~II},
%%     Magyar Tud. Akad. Mat. Fiz. Oszt. K\"{o}zl. \textbf{9} (1957), 
%%     417--434  (Hungarian).
%%
%%   \bibitem{fR82}
%%      Ferenc~R. Richardson, \emph{General Lattice Theory}, Mir, Moscow, 
%%      expanded and revised ed., 1982 (Russian).
%%
\end{thebibliography}


\end{document}

math0203111.extracted_bib
\begin{thebibliography}{999}
\bibitem{AL}
	K.~Alladi,    
	\emph{A variation on a theme of Sylvester --- a smoother road to
	G\"ollnitz's (Big) theorem},
	Discrete Math. \textbf{196} (1999), 1--11.    

\bibitem{Z}
	G.~E.~Andrews,
	\emph{On a calculus of partition functions},
	Pac. J. Math. \textbf{31} (1969), 565--562.

\bibitem{X}
	G.~E.~Andrews,
	\emph{Two theorems of Gauss and allied indentities proved
	arithmetically},
	Pac. J. Math. \textbf{41} (1972), 563--578.

\bibitem{A1}
	G.~E.~Andrews,
	\emph{On a partition theorem of N.~J.~Fine},
	J. Nat. Acad. Math. India \textbf{1} (1983), 105--107.

\bibitem{A2}
	G.~E.~Andrews,
      \emph{ The Theory of Partitions},
           Encyclopedia of Mathematics and Its Applications, Vol.~2
(G.~-~C.~Rota, ed.), Addison-Wesley, Reading, Mass., 1976. (Reissued:
Cambridge Univ. Press, London and New York, 1985)

\bibitem{A3}
	G.~E.~Andrews,
	\emph{Partitions and Durfee dissection},
	Amer. J. Math. \textbf{101} (1979), 735--742.

\bibitem{Y}
	G.~E.~Andrews,
	private communication.

\bibitem{AG}
	G.~E.~Andrews and F.~G.~Garvan,
	\emph{Dyson's crank of a partition},
	Bull. Amer. Math. Soc. \textbf{18} (1988), 167--171.

\bibitem{ASD}
	A.~O.~L.~Atkin and P.~Swinnerton-Dyer,
	\emph{Some properties of partitions},
	Proc. London Math. Soc.
	\textbf{4} (1954), 84--106.

\bibitem{D1}
	F.~J.~Dyson,
	\emph{Some guesses in the theory of partitions},
	Eureka (Cambridge) \textbf{8} (1944), 10--15.

\bibitem{D2}
	F.~J.~Dyson,
	\emph{A walk through Ramanujan's garden},
	in ``Ramanujan revisited'' (Urbana-Champaign, Ill., 1987), 
	Academic Press, Boston, MA, 1988.

\bibitem{D3}
	F.~J.~Dyson,
	\emph{A new symmetry of partitions},
	J. Combinatorial Theory
	\textbf{7} (1969), 56--61.

\bibitem{D4}
	F.~J.~Dyson,
	\emph{Mappings and symmetries of partitions},
	J. Combin. Theory Ser. A \textbf{51} (1989), 169--180.

\bibitem{Ga1}
	F.~G.~Garvan, 
	\emph{New combinatorial interpretations of Ramanujan's partition
	congruences mod $5$, $7$ and $11$},
	Trans. Amer. Math. Soc. \textbf{305} (1988), 47--77.

\bibitem{Ga2}
	F.~G.~Garvan, 
	\emph{Generalizations of Dyson's rank and non-Rogers-Ramanujan 
	partitions},
	Manuscripta Math. {\bf 84} (1994), 343--359. 

\bibitem{GR1}
	G.~Gasper\ and\ M.~Rahman, 
	\emph{Basic hypergeometric series}, 
	Cambridge Univ.\ Press, Cambridge, 1990.

\bibitem{GR2}
	G.~Gasper\ and\ M.~Rahman, 
	\emph{An indefinite bibasic summation formula and some quadratic, cubic
        and quartic summation and transformation formulas}, 
        Canad. J. Math. {\bf 42} (1990), 1--27.

\bibitem{Ra1}
	S.~Ramanujan,
	\emph{Some properties of $p(n)$, the number of partitions of $n$},
	 Proc. Cambridge Philos. Soc. \textbf{19} (1919), 207--210.

\bibitem{Ra2}
	S.~Ramanujan,
	\emph{Congruence properties of partitions},
	Math. Zeitschr. \textbf{9} (1921), 147--153.

\bibitem{W1}
	S.~O.~Warnaar,
        \emph{$q$-Hypergeometric proofs of polynomial analogues of the triple 
	product identity, Lebesgue's identity and Euler's pentagonal number 
	theorem}, 
        arXiv:math.CO/0203229, preprint. 

\bibitem{W2}
	S.~O.~Warnaar,
        \emph{Summation and transformation formulas for elliptic 
	hypergeometric series}, 
        arXiv:math.QA/0001006, to appear in Constructive Approximation.

\bibitem{Win}
	L.~Winquist
        \emph{An elementary proof of $p(11m+6)\equiv 0$ (mod $11$)},
	J. Combin. Theory \textbf{6} (1969), 56--59.

%%
%%   \bibitem{sF90}
%%      Soo-Key Foo, \emph{Lattice Constructions}, Ph.D. thesis, University 
%%      of Winnebago, Winnebago, MN, December 1990.
%%
%%   \bibitem{gM68}
%%      George~A. Menuhin, \emph{Universal Algebra}, D.~van Nostrand,
%%      Princeton-Toronto-London-Mel\-bourne, 1968.
%%
%%   \bibitem{eM57}
%%      Ernest~T. Moynahan, \emph{On a problem of M.~H. Stone}, Acta Math.
%%       Acad.Sci. Hungar. \textbf{8} (1957), 455--460.
%%
%%   \bibitem{eM57a}
%%      \bysame, \emph{Ideals and congruence relations in lattices.~II},
%%     Magyar Tud. Akad. Mat. Fiz. Oszt. K\"{o}zl. \textbf{9} (1957), 
%%     417--434  (Hungarian).
%%
%%   \bibitem{fR82}
%%      Ferenc~R. Richardson, \emph{General Lattice Theory}, Mir, Moscow, 
%%      expanded and revised ed., 1982 (Russian).
%%
\end{thebibliography}