%%  This article has been submitted and accepted for publication in
%%  THE FOATA FESTSCHRIFT, 
%%  a special issue of THE ELECTRONIC JOURNAL OF COMBINATORICS
%%  dedicated to DOMINIQUE FOATA at the occasion of his 60th birthday.
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%%  Author(s):		DAVID A. BRESSOUD
%%  Title:		THE BORWEIN CONJECTURE AND PARTITIONS WITH 
%%			PRESCRIBED HOOK DIFFERENCES
%%  Date of submission:	March 10, 1995
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%%  Number of pages:	14
%%  Author's email:	bressoud@MACALSTR.EDU
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\documentstyle{article}
\newtheorem{conjecture}{Conjecture}
\newtheorem{proposition}{Proposition}
\newtheorem{lemma}{Lemma}
\newcommand{\qed}{{\bigskip\begin{flushright}{\bf
Q.E.D.}\end{flushright}}\vspace{3mm}}
\newcommand{\GP}[2]{\left[  #1 \atop #2  \right]}
\newcommand{\lfrob}{\left(\begin{array}}
\newcommand{\rfrob}{\end{array}\right)}
\newcommand{\alfbar}{\overline{\alpha}}
\newcommand{\betbar}{\overline{\beta}}
\newcommand{\abar}{\overline{a}}
\newcommand{\bbar}{\overline{b}}
\newcommand{\cala}{{\cal{A}}}
\title{The Borwein conjecture and partitions with prescribed hook differences}
\author{David M.\ Bressoud}
\date{\small Submitted: March 10, 1995; Accepted: May 11, 1995}
\begin{document}
\pagestyle{myheadings} 
\markright{\sc the electronic journal of combinatorics 3 (2) (1996), \#R4\hfill} 
\thispagestyle{empty}
 
\maketitle

\begin{center}{\bf Dedicated to Dominique Foata: teacher, mentor, and
friend}\end{center}

\begin{abstract}\noindent Peter Borwein has conjectured that certain polynomials
have non-negative coefficients. In this paper we look at some generalizations
of this conjecture and observe how they relate to the study of generating
functions for partitions with prescribed hook differences. A
combinatorial proof of the generating function for partitions with
prescribed hook differences is given.\end{abstract}

\section{Introduction}

In a personal communication to George Andrews in 1990, Peter Borwein made the
following three conjectures. We use the notation 
	\begin{eqnarray*} (a;q)_n &=& \prod_{j=0}^{n-1} (1-a\,q^j), \\
\GP{N}{M} & = & \frac{(q^{N-M+1};q)_M}{(q;q)_M}.\end{eqnarray*}

\begin{conjecture}The polynomials $A_n(q)$, $B_n(q)$, and $C_n(q)$ defined by
	\begin{equation} (q;q^3)_n(q^2;q^3)_n = A_n(q^3) - qB_n(q^3) - q^2C_n(q^3)
\end{equation} have non-negative coefficients.\end{conjecture}

\begin{conjecture}The polynomials $A^*_n(q)$, $B^*_n(q)$, and $C^*_n(q)$
defined by
\begin{equation}	 (q;q^3)_n^2(q^2;q^3)_n^2 = A^*_n(q^3) - qB^*_n(q^3) -
q^2C^*_n(q^3)\label{eqn1}\end{equation}

 have non-negative coefficients.\end{conjecture}

\begin{conjecture}The polynomials $A^{\star}_n(q)$, $B^{\star}_n(q)$,
$C^{\star}_n(q), D^{\star}_n(q)$ and $E^{\star}_n(q)$ defined by
\begin{eqnarray} \qquad &&
\hspace{-25mm}	 (q;q^5)_n(q^2;q^5)_n(q^3;q^5)_n(q^4;q^5)_n \nonumber \\
&=&
A^{\star}_n(q^5) - qB^{\star}_n(q^5) - q^2C^{\star}_n(q^5) -
q^3D^{\star}_n(q^5) - q^4E^{\star}_n(q^5) \end{eqnarray}
 have non-negative coefficients.\end{conjecture}

George Andrews \cite{And} has generalized the first two conjectures:

\begin{conjecture}For $m \geq 1$, the polynomials $A^{\dagger}(m,n,t,q)$,
$B^{\dagger}(m,n,t,q)$, and $C^{\dagger}(m,n,t,q)$ defined by
\begin{eqnarray} \qquad &&
\hspace{-25mm}(q;q^3)_m(q^2;q^3)_m(zq;q^3)_n(zq^2;q^3)_n \nonumber
\\
	&=&
\sum_{t=0}^{2n}z^t\left[A^{\dagger}(m,n,t,q^3)-q\,B^{\dagger}(m,n,t,q^3)-q^2\,
C^{\dagger}(m,n,t,q^3)\right]
\end{eqnarray} have non-negative coefficients.
\end{conjecture}

\noindent Dennis Stanton has discovered a generalization of the first
conjecture. We can use the $q$-binomial theorem to expand ($k$ odd, $1 \leq a <
k/2$)
\begin{equation}
(q^a;q^k)_m(q^{k-a};q^k)_n =
\sum_{\nu=(1-k)/2}^{(k-1)/2}(-1)^{\nu}q^{k(\nu^2+\nu)/2 - a\nu}F_{\nu}(q^k),
\end{equation}
where
\begin{equation}
F_{\nu}(q) =
\sum_{j=-\infty}^{\infty}(-1)^jq^{j(k^2j+2k\nu+k-2a)/2}\GP{m+n}{m+\nu+kj}.
\label{eqn:6}
\end{equation}
Each monomial in $q^{k(\nu^2+\nu)/2 - a\nu}F_{\nu}(q^k)$ involves a power of
$q$ for which the exponent is congruent to $-a\nu$ modulo $k$. 

\begin{conjecture}If $a$ is relatively prime to $k$ and $m=n$, then the
coefficients of $F_{\nu}(q)$ are non-negative.\end{conjecture}

The polynomial $F_{\nu}(q)$ appears to be a special case of the generating
function for partitions ``with prescribed hook differences,'' \cite{And2}. In
particular, it is shown in that paper that if $\alpha+\beta < 2K$ and $-K+\beta
\leq n-m \leq K-\alpha$, then
	\begin{equation} G(\alpha,\beta,K;q)=\sum_j (-1)^j
q^{j[K(\alpha+\beta)j+K(\alpha-\beta)]/2}\GP{m+n}{m+Kj} \label{eqn7}
\end{equation} is the generating function for partitions inside an $m \times n$
rectangle with ``hook difference conditions'' specificed by $\alpha$, $\beta$,
and $K$. The polynomial $F_{\nu}(q)$ is simply the special case \[ K=k,\quad
\alpha=\nu+\frac{k+1}{2}-\frac{a}{k},\quad
\beta=-\nu+\frac{k-1}{2}+\frac{a}{k}.\]
Since we know that this is a generating function, it follows that the
coefficients are non-negative.

The only problem with this analysis is that the hook difference conditions
defined in \cite{And2} only make sense for integer values of $\alpha$, $\beta$,
and $K$. In section 2, we will examine these hook difference
conditions, and in section 3 we will consider what is involved in extending the
definition to non-integer values. We are not able to show that $F_{\nu}(q)$ is a
generating function. However, it is possible to construct a family of partition
generating functions, $\cala_{m,n}(q)$, that are remarkably close to $A_n(q)$
when $m=n$. Furthermore, it appears that conjecture 5 can be strengthened to the
following.

\begin{conjecture} Let $\alpha$ and $\beta$ be positive rational numbers and $K$
an integer greater than 1 such that $\alpha K$ and $\beta K$ are integers. If $1
\leq \alpha+\beta \leq 2K-1$ (with strict inequalities when $K=2$) and $-K+\beta
\leq n-m \leq K-\alpha$, then 
$G(\alpha,\beta,K;q)$ has non-negative coefficients.
\end{conjecture}

This conjecture is justified heuristically by the arguments of section 3. Several
special cases have been verified. For $K=2$ and $m=n$, this author has proven
identities that imply the conjecture for $\alpha = 1$, $\beta = 1/2$ and $\alpha
= 3/2$, $\beta = 1$
\cite{me}:
\begin{eqnarray}
G(1,1/2,2;q) & = & \sum_{j=0}^m q^{jm} \GP{m}{j}, \\
G(3/2,1,2;q) & = & \sum_{j=0}^m q^{j^2} \GP{m}{j}.
\end{eqnarray}
Mourad Ismail and Dennis Stanton \cite{im} have proven that the conjecture holds if
\[ \alpha + \beta = K \qquad {\rm and} \qquad \alpha -\beta = m-n+1. \]
Experimentally, it appears that the bounds on $n-m$ are sharp. For example,
$A_n(q) = G(5/3,4/3,3;q)$ with $m=n$. The conjecture states that $G(5/3,4/3,3;q)$
has non-negative coefficients when $|n-m| \leq 1$:

\[ \begin{array}{lll} m & n & G(5/3,4/3,3;q) \\[5pt] 2 & 1 & 1+q+q^2 \\
& 2 & 1+q+2q^2+q^3+q^4 \\
& 3 & 1+q+2q^2+2q^3+2q^4+q^6 \\
& 4 & 1+q+2q^2+2q^3+3q^4+q^5+q^6-q^9-q^{10} \\[5pt]
3 & 1 & 1+q+q^2+q^3-q^4 \\
& 2 & 1+q+2q^2+2q^3+q^4+q^5+q^6 \\
& 3 & 1+q+2q^2+3q^3+2q^4+2q^5+3q^6+2q^7+q^8+q^9 \\
& 4 & 1+q+2q^2+3q^3+3q^4+3q^5+4q^6+3q^7+3q^8+2q^9+q^{10}+q^{12} \\
& 5 &
1+q+2q^2+3q^3+3q^4+4q^5+5q^6+4q^7+4q^8+3q^9+2q^{10}-q^{13} \\
&& \quad -q^{14}-q^{15}-q^{16}-q^{17} \\[5pt]
4 & 2 & 1+q+2q^2+2q^3+2q^4+q^5+q^6-q^9 \\
& 3 & 1+q+2q^2+3q^3+3q^4+2q^5+4q^6+3q^7+3q^8+2q^9+q^{10}+q^{11} \\ && \quad
+q^{12}
\\ & 4 &
1+q+2q^2+3q^3+4q^4+3q^5+5q^6+5q^7+6q^8+5q^9+5q^{10}+3q^{11}\\ &&
\quad +4q^{12}+3q^{13} +2q^{14}+q^{15}+q^{16} \\ & 5 &
1+q+2q^2+3q^3+4q^4+4q^5+6q^6+6q^7+8q^8+7q^9+8q^{10}+6q^{11}\\ &&
\quad +6q^{12}+5q^{13} +5q^{14}+3q^{15}+3q^{16}+q^{17}+q^{18}+q^{20} \\ & 6 &
1+q+2q^2+3q^3+4q^4+4q^5+7q^6+7q^7+9q^8+9q^9+10q^{10}+8q^{11}\\ &&
\quad +9q^{12}+6q^{13} +6q^{14}+4q^{15}+3q^{16}-q^{19}-2q^{20}-2q^{21}
-2q^{22}-2q^{23}\\ && \quad -q^{24}-q^{25}-q^{26}
\end{array}\]

I wish to acknowledge Dennis Stanton's contribution to this paper in the form of
many fruitful discussions.

\section{Partitions with prescribed hook differences}

Given a partition $\lambda$ whose $i$th largest part is $\lambda_i$, we define
$\lambda'_i$ to be the $i$th largest part in the conjugate partition
($\lambda'_i$ is the number of parts that are greater than or equal to $i$). We
say that $\lambda$ fits inside an $m \times n$ rectangle if $m \geq \lambda_1'$
and $n \geq \lambda_1$. If $(i,j) \in \lambda$ (equivalently, if $\lambda_i
\geq j$), then we define the {\bf hook difference} at position $(i,j)$ to be
$\lambda_i-\lambda'_j$. The {\bf diagonal} $\delta$ is the set of all positions
$(i,j) \in\ \lambda$ for which $i-j=\delta$. The following proposition is a
special case of theorem 1 in \cite{And2}.

\begin{proposition}
If $-K+\beta \leq n-m \leq K-\alpha$ where $\alpha$, $\beta$, and $K$ are
positive integers, $\alpha + \beta < 2K$, then
$G(\alpha,\beta,K;q)$ as defined in equation~(\ref{eqn7}) is the generating
function for partitions inside an
$m
\times n$ rectangle for which the hook differences on diagonal $\alpha-1$ are
less than or equal to $K-\alpha-1$ and the hook differences on diagonal
$1-\beta$ are greater than or equal to $\beta+1-K$.
\end{proposition}

The proof of this proposition given in \cite{And2} relies on
recurrences and does not lend itself to non-integer values of $\alpha$ or
$\beta$. However, as we shall demonstrate, there is a combinatorial proof of this
proposition that uses the approach of \cite{Bre}. It is this proof that
appears to be amenable to generalization.
\bigskip

\noindent{\bf Proof:} We shall use the Frobenius representation of a partition,
\[ \lambda = \lfrob{cccc}a_1,&a_2,&\ldots,&a_t\\ b_1,&b_2,&\ldots,&b_t\rfrob, \]
where $a_i = \lambda_i - i$, $b_i = \lambda'_i - i$, and $t$ is the
largest integer for which $\lambda_t \geq t$. We note that $a_1 > a_2 > \cdots
> a_t \geq 0$, $b_1 > b_2 > \cdots > b_t \geq 0$, and the number being
partitioned is $t+\sum(a_i+b_i)$. We want to show that $G(\alpha,\beta,K;q)$ is
the generating function for partitions whose Frobenius representation satisfies
\begin{eqnarray}
a_1 < n, &\quad & b_1 < m \nonumber \\
a_i - b_{i-\alpha+1} \leq K-2\alpha, && b_i - a_{i-\beta+1} \leq K-2\beta, \quad
{\rm for\ all}\ i.
\end{eqnarray}

We shall say that a partition has an $(\alpha,\beta,K)$ {\bf positive oscillation
of length
$j$},
$j\geq 1$, if we can find a sequence $i_1 < i_2 < \cdots < i_j$ for which
\begin{eqnarray}
&& a_{i_1} - b_{i_1-\alpha+1} \ >\ K-2\alpha, \nonumber \\
&& b_{i_2} - a_{i_2-\beta+1} \ >\  K-2\beta, \nonumber \\
&& \qquad \vdots \nonumber \\
&& \left\{ \begin{array}{ll} a_{i_j} - b_{i_j-\alpha+1} > K-2\alpha, & j\ {\rm
odd,}\\ b_{i_j} - a_{i_j-\beta+1} > K-2\beta, & j\ {\rm
even.}\end{array}\right. \label{pososc}
\end{eqnarray}
A partition has  an $(\alpha,\beta,K)$ {\bf negative oscillation of length $j$},
$j\geq 1$, if we can find a sequence $i_1 < i_2 < \cdots < i_j$ for which
\begin{eqnarray}
&& b_{i_1} - a_{i_1-\beta+1} \ >\  K-2\beta, \nonumber \\
&& a_{i_2} - b_{i_2-\alpha+1} \ >\ K-2\alpha, \nonumber \\
&& \qquad \vdots \nonumber \\
&& \left\{ \begin{array}{ll} a_{i_j} - b_{i_j-\alpha+1} > K-2\alpha, & j\ {\rm
even,}\\ b_{i_j} - a_{i_j-\beta+1} > K-2\beta, & j\ {\rm
odd.}\end{array}\right.
\end{eqnarray} 

\begin{lemma}
If $\alpha$, $\beta$, and $K$ are positive integers with $\alpha + \beta < 2K$ and
if $-K+\beta \leq n-m \leq K-\alpha$, then the generating function for partitions
inside an $m\times n$ rectangle with an
$(\alpha,\beta, K)$ positive oscillation of length $j$ is
\begin{equation}
f^+_{\alpha,\beta,
K}(j;q)=q^{j[K(\alpha+\beta)j+K(\alpha-\beta)]/2}\GP{m+n}{m+Kj}.
\end{equation}
\end{lemma}

By conjugating the partition (interchanging the $a$s and $b$s in the Frobenius
representation), this lemma implies that the generating function for partitions
inside an $m\times n$ rectangle with an $(\alpha,\beta, K)$ negative oscillation of
length $j$ is 
\begin{equation}
f^-_{\alpha,\beta,
K}(j;q)=q^{j[K(\alpha+\beta)j-K(\alpha-\beta)]/2}\GP{m+n}{m-Kj}.
\end{equation}
Lemma 1 thus implies that
\begin{equation}
G(\alpha,\beta,K;q) = \GP{m+n}{m} + \sum_{j=1}^{\infty}(-1)^j\left[ f^+_{\alpha,\beta,
K}(j;q)+f^-_{\alpha,\beta,K}(j;q)\right]. \label{eqn:13}
\end{equation}
Proposition 1 is an immediate consequence of equation~(\ref{eqn:13}):
$\GP{m+n}{m}$ is the generating function for all partitions that sit inside an
$m\times n$ box, and if such a partition has a positive or negative
oscillation and if $j$ is the length of the longest such oscillation, then the
alternating sum will count it with a total weight of
	\[ -2\left\lfloor\frac{j}{2}\right\rfloor +
2\left\lfloor\frac{j-1}{2}\right\rfloor + (-1)^j = -1. \]

\bigskip

\noindent{\bf Proof of lemma 1:} Let $\lambda$ be a partition into at most $m+Kj$
parts, each part less than or equal to $n-Kj$. If $i$ is greater than the
number of parts in $\lambda$, then we define $\lambda_i = 0$. We let $t$ be the
largest integer ($\geq 0$) such that
\begin{equation}
\lambda_{2\lceil j/2 \rceil\alpha + 2 \lfloor j/2 \rfloor \beta + t} \geq t,  
\end{equation}
and then define sequences $a_1, \ldots, a_{\mu}$ and $b_1, \ldots, b_{\mu}$,
$\mu =\lceil j/2 \rceil \alpha + \lfloor j/2 \rfloor \beta + t$,  as follows. If
$j$ is even, then
\begin{eqnarray}
a_i & = & \left\{ \begin{array}{lr} \lambda_i +jK -i, & 1 \leq i \leq \alpha, \\
\lambda_{i+\alpha+\beta} + (j-2)K + \alpha+\beta -i, & \alpha +1 \leq i \leq
2\alpha + \beta, \nonumber \\
\lambda_{i+2(\alpha+\beta)} + (j-4)K + 2(\alpha+\beta) -i, & 2\alpha+\beta +1
\leq i \leq 3\alpha + 2\beta, \nonumber \\
\quad \vdots \nonumber \\
\lambda_{i+j(\alpha+\beta)/2} + j(\alpha+\beta)/2 - i,\\ &\hspace{-1in}
j(\alpha+\beta)/2-\beta + 1 \leq i \leq j(\alpha+\beta)/2+t,\end{array}\right.
\nonumber \\ \label{eqn15} \\  
b_i &=& \left\{ \begin{array}{lr} \lambda_{\alpha+i} + (j-1)K + \alpha-i, & 1
\leq i \leq \alpha+\beta, \nonumber \\
\lambda_{2\alpha+\beta+i} + (j-3)K + 2\alpha+\beta-i, &\hspace{-.15in} \alpha +
\beta+ 1 \leq i
\leq 2(\alpha+\beta), \nonumber \\
\lambda_{3\alpha+2\beta+i} + (j-5)K + 3\alpha+2\beta-i, &\hspace{-1.5in}
2(\alpha + \beta)+ 1
\leq i \leq 3(\alpha+\beta), \nonumber \\ 
\quad \vdots \\
\lambda_{j(\alpha+\beta)/2-\beta+i} + K + j(\alpha+\beta)/2-\beta-i, \nonumber \\
& \hspace{-1.5in} (\frac{j}{2}-1)(\alpha +
\beta)+ 1
\leq i \leq j(\alpha+\beta)/2, \nonumber \\ 
\lambda'_{i-j(\alpha+\beta)/2}-j(\alpha+\beta)/2-i, \nonumber \\ &
\hspace{-1in} j(\alpha+\beta)/2+1
\leq i
\leq j(\alpha+\beta)/2+t. 
\end{array}\right.\nonumber  \\ \label{eqn16}
\end{eqnarray}
We note that 
\begin{eqnarray}
	\frac{j}{2}(\alpha+\beta) + t + \sum_i (a_i + b_i) & = & 
\sum_{i=1}^{j(\alpha+\beta)+t} \lambda_i\ +\ \sum_{i=1}^t\lambda'_i\ -\ 
t[j(\alpha+\beta)+t]   \nonumber \\
	&& \qquad + \frac{j}{2}[K(\alpha+\beta)j+K(\alpha-\beta)] \nonumber \\
	& = & \sum_{i \geq 1}\lambda_i\ +\
\frac{j}{2}[K(\alpha+\beta)j+K(\alpha-\beta)]. \nonumber \\
\end{eqnarray}
If $j$ is odd then,
\begin{eqnarray}
a_i & = & \left\{ \begin{array}{lr} \lambda_i +jK -i, & 1 \leq i \leq \alpha, \\
\lambda_{i+\alpha+\beta} + (j-2)K + \alpha+\beta -i, & \alpha +1 \leq i \leq
2\alpha + \beta, \nonumber \\
\quad \vdots \nonumber \\
\lambda_{i+(j-1)(\alpha+\beta)/2} + K + (j-1)(\alpha+\beta)/2 - i,\\
&\hspace{-2.2in} (j-1)(\alpha+\beta)/2-\beta + 1 \leq i \leq
\alpha + (j-1)(\alpha+\beta)/2, \\
\lambda'_{i-\alpha-(j-1)(\alpha+\beta)/2}-\alpha - (j-1)(\alpha+\beta)/2 - i, \\
&\hspace{-2.5in} \alpha + (j-1)(\alpha+\beta)/2 + 1 \leq i \leq
\alpha + (j-1)(\alpha+\beta)/2 + t,
\end{array}\right.
\nonumber \\ \label{eqn18} \\  
b_i &=& \left\{ \begin{array}{lr} \lambda_{\alpha+i} + (j-1)K + \alpha-i, & 1
\leq i \leq \alpha+\beta, \nonumber \\
\lambda_{2\alpha+\beta+i} + (j-3)K + 2\alpha+\beta-i, &\hspace{-.8in} \alpha +
\beta+ 1 \leq i
\leq 2(\alpha+\beta), \nonumber \\
\quad \vdots \\
\lambda_{\alpha + (j-1)(\alpha+\beta)/2+i} + \alpha +
(j-1)(\alpha+\beta)/2-i,
\nonumber
\\ & \hspace{-2.5in} (j-1)(\alpha +
\beta)/2+ 1
\leq i \leq (j-1)(\alpha+\beta)/2+ \alpha + t. \nonumber \\
\end{array}\right.\nonumber  \\ \label{eqn19}
\end{eqnarray}
Here we have that 
\begin{eqnarray}
\alpha +	\frac{j-1}{2}(\alpha+\beta) + t + \sum_i (a_i + b_i) & = & 
\sum_{i=1}^{(j+1)\alpha + (j-1)\beta+t} \lambda_i\ +\ \sum_{i=1}^t\lambda'_i\
\nonumber \\
&&\qquad -\  t[(j+1)\alpha+(j-1)\beta+t]   \nonumber \\
	&& \qquad +\ \frac{j}{2}[K(\alpha+\beta)j+K(\alpha-\beta)] \nonumber \\
	& = & \sum_{i \geq 1}\lambda_i\ +\
\frac{j}{2}[K(\alpha+\beta)j+K(\alpha-\beta)]. \nonumber \\ 
\end{eqnarray}

The $\alpha_i$ and $\beta_i$ are non-negative integers because of the choice of
$t$. Since $\lambda_1 \leq n - jK$ and $n-m \leq K-\alpha$, we have that
\begin{eqnarray}
a_1 & = & \lambda_1 + jK -1 \ <\ n, \\
b_1 & = & \lambda_{\alpha+1} +(j-1)K +\alpha - 1 < m.
\end{eqnarray}
Furthermore, if we take $i_1 = \alpha$, $i_2 = \alpha+\beta$, $i_3 =
2\alpha+\beta$, $i_4 = 2\alpha+2\beta$, \ldots, $i_j = \lceil j/2 \rceil \alpha +
\lfloor j/2 \rfloor \beta$, then these sequences satisfy the inequalities in
(\ref{pososc}) that characterize a partition with an $(\alpha,
\beta, K)$ positive oscillation of length $j$. The only reason why these
sequences might not represent a partition with  an $(\alpha,
\beta, K)$ positive oscillation of length $j$ is that we might have
	\[ b_{j(\alpha+\beta)/2}\ \leq\ b_{j(\alpha+\beta)/2+1} \]
when $j$ is even or
	\[ a_{\alpha+(j-1)(\alpha+\beta)/2}\ \leq\ a_{\alpha+(j-1)(\alpha+\beta)/2+1} \]
when $j$ is odd.

\subsection{Combinatorial proof of equivalence: first direction}

We now perform a shifting operation that is done, successively, for each
integer value of $r$ from $j$ down through 1. Initially, we take $i_{j+1}$ to be
$\infty$, $i_r = \lceil r/2 \rceil \alpha + \lfloor r/2 \rfloor \beta$ for $j \geq
r \geq 1$. Our objective is to define a bijection
between the pairs of sequences given above and the pairs of sequences that give
the Frobenius representation for partitions inside an
$m\times n$ rectangle with an $(\alpha,\beta,K)$ positive oscillation of length
$j$. 
\bigskip

\noindent {\bf If $r$ is even:}
\begin{eqnarray}
\tau & = & b_{i_r} - a_{i_r-\beta+1}, \\
\kappa & = & \max\{ \nu \:|\: i_r < \nu \leq i_{r+1}-\alpha,\
b_{\nu}-a_{\nu-\beta+1} > \tau\}, \\
\gamma(\nu) & = & \max\{b_i-a_{i-\beta+1}-\tau \:|\: \nu \leq i \leq \kappa\}.
\end{eqnarray}
If the set that defines $\kappa$ is empty, then we do no shifting for this
value of $r$. Otherwise, for $i_r < \nu \leq \kappa$, we set
\begin{eqnarray}
b_{\nu} & \longleftarrow & b_{\nu} - \gamma(\nu), \\
a_{\nu-\beta} & \longleftarrow & a_{\nu-\beta} + \gamma(\nu), 
\end{eqnarray}
and then reset the value of $i_r$ to $\kappa$. 

\bigskip

\noindent {\bf If $r$ is odd:}
\begin{eqnarray}
\tau & = & a_{i_r} - b_{i_r-\alpha+1}, \\
\kappa & = & \max\{ \nu \:|\: i_r < \nu \leq i_{r+1}-\beta,\
a_{\nu}-b_{\nu-\alpha+1} > \tau\}, \\
\gamma(\nu) & = & \max\{a_i-b_{i-\alpha+1}-\tau \:|\: \nu \leq i \leq \kappa\}.
\end{eqnarray}
If the set that defines $\kappa$ is empty, then we do no shifting for this
value of $r$. Otherwise, for $i_r < \nu \leq \kappa$, we set
\begin{eqnarray}
a_{\nu} & \longleftarrow & a_{\nu} - \gamma(\nu), \\
b_{\nu-\alpha} & \longleftarrow & b_{\nu-\alpha} + \gamma(\nu), 
\end{eqnarray}
and then reset the value of $i_r$ to $\kappa$.

We claim that this shifting procedure yields a pair of
sequences that give the Frobenius representation for a partition inside an
$m\times n$ rectangle with an $(\alpha,\beta,K)$ positive oscillation of length
$j$.

If $j$ is even, then after doing the shift for $r=j$, the new value of
$b_{j(\alpha+\beta)/2}$ is strictly larger than the new value of 
$b_{j(\alpha+\beta)/2+1}$. To see this, we observe that the value of
$b_{j(\alpha+\beta)/2}$ does not change, and if the old value of
$b_{j(\alpha+\beta)/2+1}$ is greater than or equal to the old value of
$b_{j(\alpha+\beta)/2}$, then 
\begin{equation}
\gamma(j(\alpha+\beta)/2+1) \geq
b_{j(\alpha+\beta)/2+1} - a_{j(\alpha+\beta)/2-\beta+2} - \left(
b_{j(\alpha+\beta)/2} - a_{j(\alpha+\beta)/2-\beta+1} \right), \end{equation}
so that the new value of $b_{j(\alpha+\beta)/2+1}$ equals
\begin{eqnarray}
b_{j(\alpha+\beta)/2+1} - \gamma(j(\alpha+\beta)/2+1) & \leq &
b_{j(\alpha+\beta)/2} + a_{j(\alpha+\beta)/2-\beta+2} -
a_{j(\alpha+\beta)/2-\beta+1} \nonumber \\ &<& b_{j(\alpha+\beta)/2}.
\end{eqnarray}

The function $\gamma$ is weakly decreasing, and so the new values
of $a_{\nu-\beta}$ are still strictly decreasing. We do need to verify that
	\[ b_{\nu} - \gamma(\nu) > b_{\nu+1} - \gamma(\nu+1). \]
This will be true if $\gamma(\nu) = \gamma(\nu+1)$. If these values of $\gamma$ are
not equal, then by definition:
\begin{equation}
		\gamma(\nu)\ =\ b_{\nu} - a_{\nu-\beta+1} - \tau, \qquad \gamma(\nu+1)\ \geq\
b_{\nu+1} - a_{\nu-\beta+2} - \tau.
\end{equation} 
It follows that
\begin{equation}
b_{\nu} - \gamma(\nu)\ =\ a_{\nu-\beta+1} + \tau \ >\ a_{\nu-\beta+2} + \tau\
\geq\ b_{\nu+1} - \gamma(\nu+1).
\end{equation}

We note that after the shift for $r=j$, the value of
$a_{j(\alpha+\beta)/2-\beta}$ might be less than or equal to the new value of
$a_{j(\alpha+\beta)/2-\beta+1}$ which will be bounded by
	\begin{equation} a_{j(\alpha+\beta)/2-\beta+1} < \lambda'_1
- j\alpha - (j-2)\beta - K. 
\end{equation}
After the next shift, for $r=j-1$, the new value of $a_{j(\alpha+\beta)/2-\beta}$
will be strictly greater than the new value of $a_{j(\alpha+\beta)/2-\beta+1}$. 

The same argument holds {\it mutatis mutandis\/} if $j$ is odd and for each
successive value of $r$. If $r$ is even, then the new value of $a_{\lceil
r/2\rceil\alpha + \lfloor r/2\rfloor\beta-\beta+1}$ is bounded by
\begin{equation}
a_{\lceil r/2\rceil\alpha + \lfloor r/2\rfloor\beta-\beta+1} < \lambda'_1
- r\alpha - (r-2)\beta - (j-r+1)K.
\end{equation}
If $r$ is odd, then the new value of $b_{\lceil r/2\rceil\alpha
+ \lfloor r/2\rfloor\beta-\alpha+1}$ is bounded by
\begin{equation}
b_{\lceil r/2\rceil\alpha + \lfloor r/2\rfloor\beta-\alpha+1} < \lambda'_1
- (r-1)\alpha - (r-1)\beta - (j-r+1)K.
\end{equation}

We observe that the value of $a_1$ is left unchanged and so is strictly less than
$n$, and that the final value of $b_1$ (after the shift that corresponds to $r=1$)
is strictly less than $\lambda'_1 - jK \leq m$.

\subsection{Combinatorial proof of equivalence: other direction}

To see that we do, in fact, have a bijection we note that we can uniquely
reconstruct the values of $\tau$, $\kappa$, and $\gamma(\nu)$ for each shift as
$r$ runs from 1 back up to $j$. We first choose the sequence
$i_1 < i_2 < \cdots < i_j$ maximally. That is to say, we find the largest integer
$i_j$ for which
$b_{i_j}-a_{i_j-\beta+1} > K-2\beta$ (if
$j$ is even) or $a_{i_j} - b_{i_j-\alpha+1} > K-2\alpha$ (if $j$ is odd), and
then after each $i_r$ is chosen, we choose the largest possible value for
$i_{r-1}$. To reverse the shifting process, we perform the following operation
for each $r$ from 1 through $j$.

\bigskip

\noindent {\bf If $r$ is even:}
\begin{eqnarray}
\tau^* & = & \max\{b_{\nu} - a_{\nu-\beta+1} \:|\: r(\alpha+\beta)/2
\leq \nu \leq i_r \}, \\
\kappa^* & = & \min\{ \nu \:|\: r(\alpha+\beta)/2
\leq \nu \leq i_r,\ b_{\nu} - a_{\nu-\beta+1} = \tau^*\}, \\
\gamma^*(\nu) & = & \min\{\tau^* - (b_{i} - a_{i-\beta+1}) \:|\:
r(\alpha+\beta)/2 \leq i < \nu \}.
\end{eqnarray}
For $r(\alpha+\beta)/2  < \nu \leq \kappa^*$, we set
\begin{eqnarray}
b_{\nu} & \longleftarrow & b_{\nu} + \gamma^*(\nu), \\
a_{\nu-\beta} & \longleftarrow & a_{\nu-\beta} - \gamma^*(\nu).
\end{eqnarray}

\bigskip

\noindent {\bf If $r$ is odd:}
\begin{eqnarray}
\tau^* & = & \max\{a_{\nu} - b_{\nu-\alpha+1} \:|\: \lceil r/2 \rceil \alpha + 
\lfloor r/2 \rfloor \beta
\leq \nu \leq i_r \}, \\
\kappa^* & = & \min\{ \nu \:|\: \lceil r/2 \rceil \alpha + \lfloor r/2 \rfloor 
\beta
\leq \nu \leq i_r,\ a_{\nu} - b_{\nu-\alpha+1} = \tau^*\}, \\
\gamma^*(\nu) & = & \min\{\tau^* - (a_{i} - b_{i-\alpha+1}) \:|\:
\lceil r/2 \rceil \alpha + \lfloor r/2 \rfloor \beta \leq i < \nu \}.
\end{eqnarray}
For $\lceil r/2 \rceil \alpha + \lfloor r/2 \rfloor 
\beta < \nu \leq \kappa^*$, we set
\begin{eqnarray}
a_{\nu} & \longleftarrow & a_{\nu} + \gamma^*(\nu), \\
b_{\nu-\alpha} & \longleftarrow & b_{\nu-\alpha} - \gamma^*(\nu).
\end{eqnarray}

It is left to the reader to verify that this does uniquely reverse the shifting
done in section 2.1. To prove that we have a bijection between pairs of sequences
generated by $f^+_{\alpha,\beta,K}(j;q)$ and partitions inside an $m\times n$
rectangle with an $(\alpha,\beta,K)$ positive oscillation of lenth $j$, we need to
verify that if we start with an arbitrary partition, we get a pair of sequences
generated by $f^+_{\alpha,\beta,K}(j;q)$. The only condition on these sequences
that is not straightforward to verify is that they are strictly decreasing with
the possible exception that if $j$ is even then we might have
$b_{j(\alpha+\beta)/2+1} \geq b_{j(\alpha+\beta)/2}$ and if $j$ is odd then we
might have $a_{\lceil j/2\rceil\alpha+\lfloor j/2\rfloor\beta+1} \geq a_{\lceil
j/2\rceil\alpha+\lfloor j/2\rfloor\beta}$.

We observe that in applying the shift given above for $r=1$, the new value for
$a_{\alpha}$ might be less than or equal to the new value for $a_{\alpha+1}$. If
$j=1$, there is no problem. If $j$ is larger than 1, then on the $r=2$ shift we
replace $a_{\alpha+1}$ with 
	\[ a_{\alpha+1} \longleftarrow a_{\alpha+1} - (\tau^* -
b_{\alpha+\beta}+a_{\alpha+1})\ =\ b_{\alpha+\beta} - \tau^*. \]
We note that if the new value of $a_{\alpha+1}$ is greater than or
equal to the new value of $a_{\alpha}$, then the value of
$b_1$ after the $r=1$ shift is strictly less than its original value. This implies
that after the $r=1$ shift we have
$a_{\alpha} - b_1 > K-2\alpha$. We combine this observation with the following
inequalities:
\begin{eqnarray}
b_1 - b_{\alpha+\beta} &\geq& \alpha+\beta-1, \\
\tau^* & \geq& K - 2\beta, \\
2K & > & \alpha + \beta,
\end{eqnarray}
to see that
\begin{equation}
 b_{\alpha+\beta} - \tau^*\ < \ a_{\alpha}.
\end{equation} 
The new value of $a_{\alpha+1}$ after the $r=2$ shift is strictly less than
$a_{\alpha}$. This argument continues to hold for each $r < j$ so that after all
of the shifting we have at most one pair of consecutive elements in the sequences
for which we do not have strict decrease.

\qed

\section{Prescribed Hook Differences with non-integer parameters}

We want to define a prescribed hook difference condition when $\alpha$ and
$\beta$ are not integers. While there does not seem to be hope for doing this in
general, the particular case
\begin{equation}
\alpha = \nu + \frac{K+1}{2} - \frac{a}{K}, \qquad \beta = -\nu + \frac{K-1}{2} +
\frac{a}{K}
\end{equation}
does hold promise. In particular, let
\begin{equation}
	\alpha = \alfbar - a/K, \qquad \beta = \betbar + a/K,
\end{equation}
where $\alfbar$ and $\betbar$ are positive integers and $a$ is a positive
integer less than or equal to $\min\{\alfbar, \betbar\}$. Let $\{\abar_i\}$ and
$\{\bbar_i\}$ be the pair of sequences generated by 
\[f^+_{\alfbar,\betbar,K}(j;q)
= q^{j[K(\alfbar+\betbar)j+K(\alfbar-\betbar)]/2}\GP{m+n}{m+Kj}
\] as given in equations (\ref{eqn15}--\ref{eqn16}) and (\ref{eqn18}--\ref{eqn19}).
We now define
\begin{eqnarray}
	a_i & = & \abar_i\ -\ \left\{ \begin{array}{ll} 1,& {\rm if}\ \alfbar - a +1 \leq 
(i\ \bmod\ \alfbar+\betbar)
\leq \alfbar \\[3pt]  & \quad {\rm and}\ i\leq \lceil j/2
\rceil(\alfbar+\betbar)-\betbar,
\\[5pt] 0, & {\rm otherwise}, \end{array} \right. \\ 
 b_i & = & \bbar_i\ -\ \left\{ \begin{array}{ll} 1,& {\rm if}\ \alfbar+\betbar - a
+1
\leq (i\
\bmod\ \alfbar+\betbar)
\leq \alfbar+\betbar, \\[3pt] & \quad {\rm and}\ i \leq \lfloor j/2
\rfloor(\alfbar+\betbar),
\\[5pt] 0, & {\rm otherwise,}
\end{array}\right. 
\end{eqnarray}
where $(i\ \bmod\ \alfbar+\betbar)$ is the least positive residue of $i$ modulo
$(\alfbar+\betbar = \alpha+\beta)$. We have subtracted a total of $aj$ from the
pair of sequences. We are left with a pair of sequences generated by
\[ f^+_{\alpha,\beta,K}(j;q) =
q^{j[K(\alpha+\beta)j+K(\alpha-\beta)]/2}\GP{m+n}{m+Kj}. \]

To get a pair of sequences generated by 
$ f^-_{\alpha,\beta,K}(j;q) $, we find the sequences generated by
$f^+_{\beta,\alpha,K}(j;q)$ and then interchange the $a$s and $b$s. This means
that we start with the pair of sequences generated by
$f^+_{\betbar,\alfbar,K}(j;q)$ and then add $aj$ to them by defining
\begin{eqnarray}
	a_i & = & \abar_i\ +\ \left\{ \begin{array}{ll} 1,& {\rm if}\ \betbar +1 \leq
(i\ \bmod\ \betbar+\alfbar) \leq \betbar + a \\[3pt]  & \quad {\rm and}\ i\leq
\lfloor j/2
\rfloor(\betbar+\alfbar), \\[5pt] 0, & {\rm otherwise}, \end{array} \right. \\ 
 b_i & = & \bbar_i\ +\ \left\{ \begin{array}{ll} 1,& {\rm if}\ 1 \leq (i\ \bmod\
\betbar+\alfbar) \leq a, \\[3pt] & \quad {\rm and}\ i \leq \lceil j/2
\rceil \betbar + \lfloor j/2 \rfloor \alfbar, \\[5pt] 0, & {\rm otherwise.}
\end{array}\right. 
\end{eqnarray}

It is not clear to what partitions these pairs of sequences correspond. The
sequences generated by $f^+_{\alpha,\beta,K}(j;q)$ satisfy a weakened form of the
oscillating condition. If we set $i_r = \lceil r/2\rceil \alfbar + \lfloor r/2
\rfloor \betbar$, then
\begin{eqnarray}
a_{i_r} - b_{i_r-\alfbar+1} & \geq & K-2\alfbar, \qquad j\ {\rm odd}, \\
b_{i_r} - a_{i_r - \betbar +1} & \geq & K - 2\betbar, \qquad j\ {\rm even}.
\end{eqnarray} 
We also introduce additional inequalities:
\begin{eqnarray}
a_{i_r - a} & > & a_{i_r-a+1}, \qquad j\ {\rm odd}, \\
b_{i_r -a} & > & b_{i_r-a+1}, \qquad j\ {\rm even}.
\end{eqnarray}

\subsection{$\cala_{m,n}(q)$: a related partition generating function}

If we restrict our attention to the polynomial $A_n(q)=G(5/3,4/3,3;q)$ given in
conjecture 1---see equations~(\ref{eqn1}) and (\ref{eqn7})---then we have
\[ K=3,\quad a=1, \quad, \alfbar=2, \quad \betbar=1. \]
A family of partitions whose generating function appears to be very closely
related to $A_n(q)$ consists of those that fit inside an $m\times n$ rectangle
and satisfy the following conditions for all $i$ such that $\lambda_i \geq
i$:
\begin{eqnarray}
{\rm either}\quad \lambda_i-\lambda_i'\ \geq\ 0 \hspace{6.3mm} & {\rm or} &
\lambda_i\ =\
\lambda_{i+1}, \\
{\rm either}\quad \lambda_{i+1}-\lambda_i'\ \leq\ -1 & {\rm or} & \lambda_i\ =\
\lambda_{i+1}.
\end{eqnarray}
These two conditions can be combined into
\begin{equation}
{\rm either}\quad \lambda_i\ =\ \lambda_{i+1} \quad {\rm or} \quad \lambda_i\ \geq
\ \lambda_i'\ >\ \lambda_{i+1}.
\end{equation}
If we designate the generating function for these partitions by $\cala_{m,n}(q)$,
then we have a simple recursion from which they can be computed:
\begin{eqnarray}
\cala_{m,n} & = & 0, \quad {\rm if}\ m<0\quad {\rm or}\quad n<0, \\
\cala_{m,n} & = & 1, \quad {\rm if}\ mn=0,\quad m\geq 0,\ n \geq 0 \\ 
\cala_{1,n} & = & \GP{n+1}{1},\quad {\rm if}\  n \geq 0, \\
\cala_{m,n} & = & \cala_{m-1,n} + \cala_{m,n-1} - \cala_{m-1,n-1} \nonumber
\\ && +\ q^{m+n-1}\left[ \cala_{m-1,n-1} - \cala_{m-1,n-2} +
\chi(m\leq n)\cala_{m-1,m-2}\right], \nonumber \\
&& \hspace{1in}{\rm if}\ m\geq 2, \quad n \geq 1.
\end{eqnarray}

We compare $\cala_{n,n}$ with $A_n$:
\begin{eqnarray}
A_n(q) - \cala_{n,n}(q) & = & 0, \quad n \leq 4, \\
A_5(q) - \cala_{5,5}(q) & = & q^{11} - q^{12} - q^{13} + q^{14}, \\
A_6(q) - \cala_{6,6}(q) & = & q^{11} - q^{13} - q^{15} + q^{17}  + q^{19} -
q^{21} - q^{23} + q^{25}, \\
A_7(q) - \cala_{7,7}(q) & = & q^{11} - q^{15} - q^{16} + q^{19}  + q^{20} +
2q^{22} - q^{23} - 2q^{24} \nonumber \\ && -\ 2q^{25} - q^{26} + 2q^{27} +
q^{29} + q^{30} - q^{33} - q^{34} + q^{38}. \nonumber \\
\end{eqnarray}
It is easily verified by induction that
\begin{equation}
\cala_{m,n}(1) \ =\ \left\{ \begin{array}{cl} 2\times 3^{n-1}, &
m=n, \\[5pt] 3^{{\rm min}(m,n)}, & |m-n| = 1.
\end{array}\right.
\end{equation}
so that in fact
\begin{equation}
A_n(1)\ = \ \cala_{n,n}(1).
\end{equation}
As $n$ increases, the coefficients of $A_n(q)-\cala_{n,n}(q)$ do increase, but
remain substantially less than the coefficients of either $A_n(q)$ or
$\cala_{n,n}(q)$. Plots of these coefficients for
$n = 8, 10, 12, 15,$ and 18 are included in the file {\bf borwein.ps}. While we do
get coefficients on the order of $4000$ when
$n=18$, this is less than 1/500 of the corresponding coefficient in
either $A_n(q)$ or $\cala_{n,n}(q)$. 

One interesting pattern does emerge. For $5 \leq n \leq 17$, 
	\[ P_n(q)\ =\ \frac{A_n(q) - \cala_{n,n}(q)}{q^{11}(1-q)(1-q^2)} \]
is a symmetric, unimodal, monic polynomial of degree $n^2-25$ with strictly
positive coefficients. The fact that it is symmetric and monic of degree $n^2-25$
follows from the fact that both $\cala_{n,n}(q)$ and $A_n(q)$ are symmetric
polynomials. There is no apparent reason why it should be unimodal with strictly
positive coefficients.





\begin{thebibliography}{99}
\bibitem{And} George E.\ Andrews, On a conjecture of Peter Borwein, preprint.
\bibitem{And2} George E.\ Andrews {\it et al}, Partitions with prescribed hook
differences, {\it Europ.\ J.\ Combinatorics\/} (1987) {\bf 8}, 341--350.
\bibitem{Bre} David M.\ Bressoud, Extension of the partition sieve, {\it J.\
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\bibitem{me} David M.\ Bressoud, Some identities for terminating $q$ series, {\it
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\bibitem{im} Mourad Ismail and Dennis Stanton, private communication.
\end{thebibliography}

\noindent{\it Department of Mathematics and Computer Science, Macalester College,
Saint Paul, MN 55105, USA}
\end{document}





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