% EJC papers *must* begin with the following two lines.
\documentclass[12pt]{article}
\usepackage{e-jc}

% Please remove all other commands that change parameters such as
% margins or pagesizes.

% only use standard LaTeX packages
% only include packages that you actually need

% we recommend these ams packages
\usepackage{amsthm,amsmath,amssymb}

% we recommend the graphicx package for importing figures
\usepackage{graphicx}

% use this command to create hyperlinks (optional and recommended)
\usepackage[colorlinks=true,citecolor=black,linkcolor=black,urlcolor=blue]{hyperref}

% use these commands for typesetting doi and arXiv references in the bibliography
\newcommand{\doi}[1]{\href{http://dx.doi.org/#1}{\texttt{doi:#1}}}
\newcommand{\arxiv}[1]{\href{http://arxiv.org/abs/#1}{\texttt{arXiv:#1}}}

% all overfull boxes must be fixed;
% i.e. there must be no text protruding into the margins


% declare theorem-like environments
\theoremstyle{plain}
\newtheorem{theorem}{Theorem}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{fact}[theorem]{Fact}
\newtheorem{observation}[theorem]{Observation}
\newtheorem{claim}[theorem]{Claim}

\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}
\newtheorem{conjecture}[theorem]{Conjecture}
\newtheorem{open}[theorem]{Open Problem}
\newtheorem{problem}[theorem]{Problem}
\newtheorem{question}[theorem]{Question}

\theoremstyle{remark}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{note}[theorem]{Note}

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

% if needed include a line break (\\) at an appropriate place in the title

\title{\bf On a certain vector crank modulo 7}

% input author, affilliation, address and support information as follows;
% the address should include the country, and does not have to include
% the street address

\author{Michael D. Hirschhorn\\
\small School of Mathematics and Statistics\\[-0.8ex]
\small University of New South Wales\\[-0.8ex]
\small Sydney, NSW, 2052, Australia\\
\small\tt m.hirschhorn@unsw.edu.au\\
\and
Pee Choon Toh\thanks{Supported by the NIE Academic Research Fund RI 3/12 TPC.}\\
\small Mathematics \& Mathematics Education,\\[-0.8ex]
\small National Institute of Education,
Nanyang Technological University\\[-0.8ex]
\small 1 Nanyang Walk, Singapore 637616\\
\small\tt peechoon.toh@nie.edu.sg
}

% \date{\dateline{submission date}{acceptance date}\\
% \small Mathematics Subject Classifications: comma separated list of
% MSC codes available from http://www.ams.org/mathscinet/freeTools.html}

\date{\dateline{Apr 25, 2014}{?}\\
\small Mathematics Subject Classifications: 11P83, 05A17, }

\begin{document}

\maketitle

% E-JC papers must include an abstract. The abstract should consist of a
% succinct statement of background followed by a listing of the
% principal new results that are to be found in the paper. The abstract
% should be informative, clear, and as complete as possible. Phrases
% like "we investigate..." or "we study..." should be kept to a minimum
% in favor of "we prove that..."  or "we show that...".  Do not
% include equation numbers, unexpanded citations (such as "[23]"), or
% any other references to things in the paper that are not defined in
% the abstract. The abstract will be distributed without the rest of the
% paper so it must be entirely self-contained.

\begin{abstract}
  We define a vector crank to provide a combinatorial interpretation for a certain Ramanujan type congruence modulo 7.

  % keywords are optional
  \bigskip\noindent \textbf{Keywords:} partitions; congruences; crank
  \end{abstract}

\section{Introduction}
In \cite{toh}, one of the authors established several new Ramanujan type identities and congruences modulo 3, 5 and 7 for certain types of partition functions. For example,
define $ Q_{po,\overline{p}} (n)$ as the number of partitions of $n$ into two colors, where the red colored parts form a partition into odd parts and the blue colored parts form an overpartition.
Using the standard notation
\begin{align*}
&(a;q)_n = \prod_{j=0}^{n-1} (1-aq^j),\\
&(a;q)_\infty = \lim_{n \rightarrow \infty} (a;q)_n,\\
&(a_1,\ldots,a_m;q)_\infty =  (a_1;q)_\infty \cdots  (a_m;q)_\infty,
\end{align*}
for $|q| < 1$ and $a, a_1, \ldots, a_m \ne 0$, we can write the generating function of $ Q_{po,\overline{p}} (n)$ as
$$ \sum_{n=0}^\infty Q_{po,\overline{p}} (n) q^n = \frac{1}{(q;q^2)_\infty} \times \frac{(-q;q)_\infty}{(q;q)_\infty} =\frac{(-q,-q;q)_\infty}{(q;q)_\infty}.$$
Toh \cite{toh} proved that
\begin{equation}\label{toh7}
 \sum_{n=0}^\infty Q_{po,\overline{p}} (7n+2) q^n \equiv 0 \pmod{7}.
\end{equation}
Zhou \cite{zhou} subsequently provided alternative proofs of all of the congruences in \cite{toh} with the exception of \eqref{toh7}.
She re-interpreted these partition functions as partitions into multi-colors, introduced what she termed as \emph{multiranks} -- which are essentially vector cranks as defined by Garvan \cite{Garvan} -- and proved that these vector cranks divided the partitions into equinumerous parts. The aim of this article
is to define a vector crank that will explain \eqref{toh7} combinatorially.

\section{A vector crank}
If $\lambda$ is a partition, we define $\sigma(\lambda)$ and $n(\lambda)$  as the sum of the parts and the number of parts of $\lambda$ respectively.
Let $\mathcal{D}, \mathcal{O}$, $\mathcal{P}$ denote the sets of partitions into distinct parts, partitions into odd parts, and unrestricted partitions respectively.
Define the cartesian product
$$ \mathcal{V} =  \mathcal{D}\times\mathcal{D} \times\mathcal{O}\times\mathcal{O} \times\mathcal{P}\times\mathcal{P}.$$
For a vector partition $\vec{\lambda} = (\lambda_1, \lambda_2, \lambda_3, \lambda_4, \lambda_5, \lambda_6) \in \mathcal{V}$ define a sum of parts $s$,
a weight $w$ and a crank $r$ by
\begin{subequations}
\begin{align}
s(\vec{\lambda}) &= 2\sigma(\lambda_1) + \sigma(\lambda_2) +\sigma(\lambda_3) +\sigma(\lambda_4) +2\sigma(\lambda_5) +2\sigma(\lambda_6), \\
w(\vec{\lambda}) &= (-1)^{n(\lambda_1)}, \\
r(\vec{\lambda}) &=  2n(\lambda_3) -2n(\lambda_4) + n(\lambda_5) - n(\lambda_6).
\end{align}
\end{subequations}


The weighted count of vector partitions of $n$ with crank $m$, denoted by $N_\mathcal{V}(m,n)$, is given by
\begin{equation}
N_\mathcal{V}( m,n) = \sum_{\substack{\vec\lambda \in \mathcal{V} \\ s(\vec\lambda)=n \\ r(\vec\lambda) = m}} w(\vec\lambda).
\end{equation}
We also define the weighted count of vector partitions of $n$ with crank congruent to $k$ modulo $t$ by
\begin{equation}
N_\mathcal{V}( k, t, n) = \sum_{m=-\infty}^\infty N_\mathcal{V}(mt+k,n) = \sum_{\substack{\vec\lambda \in \mathcal{V} \\ s(\vec\lambda)=n \\ r(\vec\lambda)\equiv k \!\!\!\!\pmod{t}}} w(\vec\lambda).
\end{equation}
Finally, we have the following generating function for $N_\mathcal{V}(m,n)$,
\begin{equation} \label{crankgf}
\sum_{m= -\infty}^\infty \sum_{n=0}^{\infty} N_\mathcal{V}(m,n) z^m q^n  = \frac{(q^2;q^2)_\infty (-q;q)_\infty}{(z^2q;q^2)_\infty(z^{-2}q;q^2)_\infty(zq^2;q^2)_\infty(z^{-1}q^2;q^2)_\infty}.
\end{equation}

\begin{theorem}\label{main}
The following equation holds for all nonnegative integers $n$.
$$ N_\mathcal{V}(0,7,7n+2) = N_\mathcal{V}(1,7,7n+2) = \cdots = N_\mathcal{V}(6,7,7n+2) = \frac{Q_{po,\overline{p}} (7n+2) }{7}.$$
\end{theorem}

The main ingredient in the proof of the theorem is Winquist's identity \cite{winquist}, which is a variant of the $B_2$ case of the Macdonald identities \cite{macdonald}. We state the identity in the following symmetric
form \cite[Eq. (3.1)]{tohmac}. If we define
\begin{subequations}
\begin{align}
F_1(x) &= \sum_{j =-\infty}^\infty (-1)^j q^{3j^2}(x^{3j} +x^{-3j}), \\
F_2(x) &= \sum_{k =-\infty}^\infty (-1)^k q^{3k^2+2k}(x^{3k+1} +x^{-3k-1}),
\end{align}
we have
\begin{align} \label{winq}
&F_1(x)F_2(y)-F_1(y)F_2(x)
= -\frac{2}{x} \left(xq, \frac{q}{x} , yq, \frac{q}{y}, xy, \frac{q^2}{xy} , \frac{x}{y},  \frac{yq^2}{x}, q^2, q^2;q^2\right)_\infty.
\end{align}
\end{subequations}


\begin{proof}[Proof of Theorem \ref{main}]
If we set $\zeta = \exp(2\pi i/7)$ in \eqref{crankgf}, we obtain
\begin{align*}
&  \sum_{t=0}^{6} \zeta^t \sum_{n=0}^\infty N_\mathcal{V}(t,7,n)q^n  \\
&=\sum_{m= -\infty}^\infty \sum_{n=0}^{\infty} N_\mathcal{V}(m,n) \zeta^m q^n\\
&= \frac{(q^2;q^2)_\infty }{(q,\zeta^2q, q/\zeta^{2}, \zeta q^2, q^2/\zeta;q^2)_\infty} \\
&= \frac{(\zeta q, q/\zeta, \zeta^3 q, q/\zeta^{3};q^2)_\infty}{(q^7;q^{14})_\infty } \times
 \frac{(q^2, q^2, \zeta^2 q^2, q^2/\zeta^2, \zeta^3 q^2, q^2/\zeta^{3};q^2)_\infty}{(q^{14};q^{14})_\infty }\\
&=\frac{ F_1 (\zeta^3)F_2(\zeta)-F_1(\zeta)F_2(\zeta^3)}{2\zeta(1-\zeta^2)(1-\zeta^3) (q^{7};q^{7})_\infty},
\end{align*}
where we used \eqref{winq} with $x=\zeta^3$ and $y=\zeta$.

Since $3j^2 \equiv 0, 3, 5, 6 \pmod{7}$ and $3k^2+2k \equiv 0, 1, 2, 5 \pmod{7}$, the power of $q$ in $q^{3j^2+3k^2+2k}$ is congruent to $2$ modulo $7$ exactly when $j \equiv 0 \pmod{7}$ and $k \equiv 2 \pmod{7}.$ This means that the coefficient of $q^{7n+2}$ in
$$F_1 (\zeta^3)F_2(\zeta)-F_1(\zeta)F_2(\zeta^3)$$
is zero since
$$
(-1)^{j+k} (\zeta^{9j}+\zeta^{-9j})(\zeta^{3k+1}+\zeta^{-3k-1}) -  (-1)^{j+k}(\zeta^{3j}+\zeta^{-3j})(\zeta^{9k+3}+\zeta^{-9k-3}) = 0
$$
when $j \equiv 0 \pmod{7}$ and $k \equiv 2 \pmod{7}$. Thus
\begin{equation}
\sum_{t=0}^{6} N_\mathcal{V}(t,7,7n+2) \zeta^t = 0.
\end{equation}
Since the minimal polynomial for $\zeta$ over the rational numbers is
$$p(x) = 1+x+x^2+\cdots+x^6,$$
we conclude that
$$ N_\mathcal{V}(0,7,7n+2) = N_\mathcal{V}(1,7,7n+2) = \cdots = N_\mathcal{V}(6,7,7n+2). \qedhere $$
\end{proof}

We end by indicating how one may prove \eqref{toh7} directly as the details were omitted in \cite{toh}. This can be done by observing that
\begin{align}
\sum_{n=0}^\infty Q_{po,\overline{p}} (n) q^n  &= \frac{(q^2;q^2)_\infty^2}{ (q;q)_\infty^3 } \nonumber \\
&\equiv    \frac{(q^2;q^2)_\infty^9}{ (q;q)_\infty^3 } \times \frac{1}{(q^{14};q^{14})_\infty} \pmod{7}.
\end{align}
Thus \eqref{toh7} is equivalent to proving the coefficients of $q^{7n+2}$ in
$$ \frac{(q^2;q^2)_\infty^9}{ (q;q)_\infty^3 }$$
are all divisible by 7. We offer three alternative ways of doing this. The easiest way is to appeal directly to \cite[Th. 2]{CooperHL}. Alternatively,
we can use one of the Macdonald identities associated with the $C^\vee_2$ root system \cite[p. 137]{macdonald} or \cite[Eq. 3.12]{tohmac}, to express
\begin{align*}
\frac{(q^2;q^2)_\infty^9}{ (q;q)_\infty^3 }
&= \sum_{\substack{\alpha \equiv 1 \pmod{8}\\\beta \equiv 3 \pmod{8} }} \frac{1}{8}(\beta^2-\alpha^2) q^{ \frac{\alpha^2+\beta^2-10}{16}}.
\end{align*}
If the exponent of $q$ is congruent to 2 modulo 7, we have
\begin{align*}
\alpha^2+\beta^2 \equiv 16(2)+10 \equiv 0 \pmod{7}.
\end{align*}
Since $-1$ is a quadratic nonresidue modulo 7, 7 must divide both $\alpha$ and $\beta$. The third way is to  apply the Hecke operator $T_7$
to $\frac{\eta(16\tau)^9}{\eta(8\tau)^3}$, a weight 3 cusp form of level 128. One can refer to \cite{Ahlgren}
for examples of how this may be done.

%-------------------------------------------------------------------------

\subsection*{Acknowledgements}
We thank the anonymous referee for helpful comments that improved the article. In particular, the referee pointed out
that the coefficients $N_\mathcal{V}(m,n)$ are nonnegative which can be proved using the following corollary \cite[Eq. (4.1)]{Berkovich} of
the $q$-binomial theorem.
\begin{proposition} If $|q|, |t| < 1$ then
$$ \frac{(at;q)_\infty}{(a;q)_\infty(t;q)_\infty} = \frac{1}{(a;q)_\infty} + \sum_{n=1}^\infty \frac{t^n}{ (aq^n;q)_\infty (q;q)_n}.$$
\end{proposition}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \bibliographystyle{plain}
% \bibliography{myBibFile}
% If you use BibTeX to create a bibliography
% then copy and past the contents of your .bbl file into your .tex file

\begin{thebibliography}{10}


\bibitem{Ahlgren}
S. Ahlgren.
\newblock
Multiplicative relations in powers of Euler's product.
\newblock
{\em J. Number Theory}, 89:222--233, 2001.

\bibitem{Berkovich}
A. Berkovich, and F.~G. Garvan.
\newblock
K. Saito's Conjecture for nonnegative eta products and analogous results for other infinite products.
\newblock
{\em J. Number Theory}, 128:1731--1748, 2008.


\bibitem{CooperHL}
S. Cooper, M.~D. Hirschhorn, and R. Lewis.
\newblock
Powers of Euler's product and related identities.
\newblock
{\em Ramanujan J.}, 4:137--155, 2000.

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


\bibitem{macdonald}
I.~G. Macdonald.
\newblock
Affine root systems and Dedekind's $\eta$-function.
\newblock
{\em Invent. Math.}, 15:91--143, 1972.

\bibitem{tohmac}
P.~C. Toh.
\newblock
Generalized $m$th order Jacobi theta functions and the Macdonald identities.
\newblock
{\em Int. J. Number Theory}, 4:461--474, 2008.
%
\bibitem{toh}
P.~C. Toh.
\newblock
Ramanujan type identities and congruences for partition pairs.
\newblock
{\em Discrete Math.}, 312:1244--1250, 2012.


\bibitem{winquist}
L. Winquist.
\newblock
An elementary proof of $p(11m+6) \equiv 0 \pmod{11}$.
\newblock
{\em J. Combin. Theory}, 6:56--59, 1969.

\bibitem{zhou}
R.~R. Zhou.
\newblock
Multiranks for partitions into multi-colors.
\newblock
{\em Electron. J. Combin.}, 19:\#P8, 2012.


\end{thebibliography}

\end{document}
