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\title{\bf Rank and crank analogs for some colored partitions}

% 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{Roberta R. Zhou
        \thanks{The first author's research was partially supported by
        %$^\ast$
%the scientific research foundation (XNB201503)
%of Northeastern University at Qinhuangdao,
the Fundamental Research Funds for the Central Universities (N142303009),
%and
the Natural Science Foundation of Hebei Province (A2015501066)
and NSFC(11501090).}\\
\small School of Mathematics and Statistics\\[-0.8ex]
\small Northeastern University at Qinhuangdao\\[-0.8ex]
\small Hebei 066004, P.~R.~China\\
\small\tt zhourui@neuq.edu.cn\\
\and  Wenlong Zhang
\thanks{The second author's research was partially supported by
the Fundamental Research Funds for the Central Universities.}\\
\small School of Mathematical Sciences\\[-0.8ex]
\small Dalian University of Technology\\[-0.8ex]
\small Dalian 116024, P. R. China\\
\small\tt wenlongzhang@dlut.edu.cn
}

% \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{Jan 1, 2012}{Jan 2, 2012}\\
\small  Mathematics Subject Classifications: Primary 11P83 and\\
Secondary 05A17, 05A19, 11F03, 11P81}

\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
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% 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 establish some rank and crank analogs
for partitions into distinct colors and give combinatorial interpretations
for colored partitions such as partitions defined by Toh, Zhang and Wang
congruences modulo 5, 7.
  % keywords are optional

\bigskip\noindent \textbf{Keywords:} Partition congruences;
          rank analogs;
          Jacobi's triple product identity;
          Winquist's product identity
\end{abstract}

\section{Introduction and Motivation}

Let $p(n)$ be the number of unrestricted partitions of $n$, where
$n$ is nonnegative integer.
In 1921, Ramanujan~\cite{R1921mz} discovered the following
congruences \bnm
p(5n+4)&\equiv&0\pmod5\\
p(7n+5)&\equiv&0\pmod7.\enm
There exist many proofs in mathematical
literature,
%for which one can be found,
for example
%, in
~\cite{berndt-x,berndt-ono,%chu05a,ewell82,
ramanujan}.
%watson1938}.

In 1944, F.~J.~Dyson~\cite{dyson44} defined the rank of a partition
as the largest part minus the number of parts. Let $N(m,n)$ denote
the number of partitions of $n$ with rank $m$ and let $N(m,t,n)$
denote the number of partitions of $n$ with rank congruent to $m$
modulo $t$. In 1953, A.~O.~L. Atkin and H.~P.~F.
Swinnerton-Dyer~\cite{atkin.Swinnerton} proved
\[N(0,5,5n+4)=N(1,5,5n+4)=\cdots=N(4,5,5n+4)=\frac{p(5n+4)}5\]
and
\[N(0,7,7n+5)=N(1,7,7n+5)=\cdots=N(6,7,7n+5)=\frac{p(7n+5)}7.\]
Following from the fact that the operation of conjugation reverses
the sign of the rank, the trivial consequences are
\[N(m,n)=N(-m,n)\quad\text{and}\quad N(m,t,n)=N(t-m,t,n).\]

Hammond and Lewis~\cite{hammond-lewis} %investigated some elementary
%results for 2-colored partitions mod 5. They
defined birank
and
explained that the residue of the birank mod $5$ divided %the
2-colored partitions of $n$ into $5$ equal classes provided
$n\equiv2,3\:\:\text{or}\:\:4\pmod{5}$. F. G. Garvan~\cite{garvan10}
found two other analogs the Dyson-birank and the 5-core-birank.

In 2010, Chan~\cite{chan18rj} introduced the cubic partition
$a(n)$ as the number of 2-color partitions
of n with colors red and blue subjecting to the restriction that the color blue appears
only in even parts, and obtained the
following congruence
\[a(3n+2)\equiv0\pmod3.\]
Another proof has been given by B. Kim~\cite{kim}. He
defined a crank analog $M'(m,N,n)$ for $a(n)$ and proved that
\[M'(0,3,3n+2)\equiv M'(1,3,3n+2)\equiv M'(2,3,3n+2)\pmod{3},\]
for all nonnegative integers $n$, where $M'(m,N,n)$ is the number of
partition of $n$ with crank congruent to $m$ modulo $N$.
Later, B. Kim~\cite{kim11} gave two partition statistics which explained the
partition congruences about cubic partition pairs $b(n)$.
Here, $b(n)$ is the number of 4-color partitions of $n$
with colors red, yellow, orange, and blue subjecting
to the restriction that the colors orange and blue appear only in even parts.

About further research of arithmetic properties of cubic partitions,
overcubic partitions and other colored partitions,
some interesting results can be found in
~\cite{lin14,toh,zhang-wang,zhao-zhong}.
The first author~\cite{zhou12} of the present paper generalized Hammond-Lewis birank
and gave combinatorial interpretations for some colored partitions.

The paper is organized as follows. In Section 2, we introduce necessary notation and
some preliminary results. In Section 3, we aim to provide two partition statistics
for two colored partition congruences modulo 5. We establish six rank or crank analogs
for six colored partition with modulus 7 and give combinatorial interpretations
in Section 4.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Preliminary results}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
For the two indeterminates $q$ and $z$ with $|q|<1$, the $q$-shifted
factorial of infinite order %and the modified Jacobi theta function are
is defined %respectively
by
\[(z;q)_\infty=\prod_{n=0}^\infty(1-zq^n)
%\quad\text{and}\quad \ang{z;q}_\infty=(z;q)_\infty(q/z;q)_\infty
\]
where the multi-parameter expression for the former will be
abbreviated as
\[\ffat{\alp,\bet,\cdots,\gam}{q}{\infty}
=(\alp;q)_\infty(\bet;q)_\infty \cdots(\gam;q)_\infty.\]
%For brevity we denote $E_m%=\prod_{k=1}^\infty(1-q^{km})
%=(q^m;q^m)_\infty$.

The main purpose of this paper is to define rank and crank analogs for partition
into colors and prove colored partition congruences applying the
method of~\cite{garvan88}, which uses roots of unity.
Jacobi triple product identity,
the modified Jacobi triple product identity and Winquist
product identity
are given as follows:
\begin{itemize}
\item Jacobi triple product identity%~\cite{jacobi}
~\cite{and-86x,bailey,b3r,%ewell81,
gas=90x,har=79x%,lewis84
}:
\eqn{\sum_{n=-\infty}^{+\infty} (-1)^{n}\:q^{\zbnm{n}{2}}\:x^{n}
\:=\:\label{jacobi} \ffat{q,x,q/x}{q}{\infty}.}

\item Modified Jacobi triple product identity~\cite{hammond-lewis}:
\eqn{\ffat{q,zq,q/z}{q}{\infty}
=\label{modi-Jacobi}\sum_{n\geq0}(-1)^n(z^n+z^{n-1}
    +\cdots+z^{-n})q^{\zbnm{n+1}{2}}.}
\item Winquist product identity~\cite{chu04b,winquist}:
\bmn\xxqdn
\+\+\nnm(q;q)_{\infty}^4\ffat{x,q/x}{q}{\infty}^2\ffat{x^2,q/{x^2}}{q}{\infty}\\
\+=\+\label{winq}\sum_{i,j={-\infty}}^{+\infty}
(-1)^{i+j}q^{3\zbnm{i}{2}+3\zbnm{j}{2}+j}(1-3i+3j)
\{x^{3i+3j}-x^{4-3i-3j}\}. \emn
\end{itemize}

By replacing $q$ by $q^2$ in \eqref{winq},
splitting into two bilateral sums on right hand side of the resulting equation,
and replacing $j\to j-1$ in the first double sum,
and $i\to i+1$ in the second double sum,
the resulting
formula can be transformed as
\begin{itemize}
\item Modified Winquist product identity
\bmn\xxqdn
\+\+\nnm(q^2;q^2)_{\infty}^4\ffat{x,q^2/x}{q^2}{\infty}^2
\ffat{x^2,q^2/{x^2}}{q^2}{\infty}\\
\+=\+\label{modify-winq}\sum_{i,j={-\infty}}^{+\infty}
(-1)^{i+j}q^{6\zbnm{i}{2}+6\zbnm{j}{2}+3i-j+1}(2+3i-3j)
\{(x/q)^{3i+3j-3}-(x/q)^{1-3i-3j}\}. \emn
\end{itemize}

Dividing both sides by $1+x$ in~\eqref{winq}
and applying L'H\^opital's rule for the limit $x\to -1$,
we have
\bmn\xxqdn
(q;q)_{\infty}^2(q^2;q^2)_{\infty}^4
=\label{winq-1}\sum_{i,j={-\infty}}^{+\infty}
q^{3\zbnm{i}{2}+3\zbnm{j}{2}+j}
\frac{(1-3i+3j)(2-3i-3j)}{4}. \emn

Divide both sides by $1-q^2/{x^2}$ in~\eqref{modify-winq}
and utilize L'H\^opital's rule for the limit $x\to q$
to obtain
\bmn\xxqdn
(q;q)_{\infty}^4(q^2;q^2)_{\infty}^2
=\label{modify-winqq}\sum_{i,j={-\infty}}^{+\infty}
(-1)^{i+j}q^{6\zbnm{i}{2}+6\zbnm{j}{2}+3i-j+1}
(2+3i-3j)(3i+3j-2). \emn

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

After Andrews and Garvan~\cite{andrews-garvan-88}, for a partition
$\lam$, we define $\#(\lam)$ is the number of parts in $\lam$ and
$\sigma(\lam)$ is the sum of the parts of $\lam$ with the convention
$\#(\lam)=\sigma(\lam)=0$ for the empty partition $\lam$. Let
$\mc{P}$ be the set of all ordinary partitions,
%$\mc{D}$ be the set of all partitions into distinct parts,
%$\mc{O}$ be the set of all partitions into odd parts,
$\mc{D}\mc{O}$ be the set of all
partitions into distinct odd parts.

For a given partition
$\lam$, the crank $c(\lam)$ of a partition is defined as
\begin{equation*}
c(\lam):=\begin{cases}
\ell(\lam),&\text{if}\:r=0;\\
\omega(\lam)-r,&\text{if}\:r\geq1,
\end{cases}
\end{equation*}
where $r$ is the number of 1's in $\lam$,
$\omega(\lam)$ is the number of parts in $\lam$ that are strictly larger
than $r$ and $\ell(\lam)$ is the largest part in $\lam$.
By extending the set of partitions $\mc{P}$
to a new set $\mc{P^*}$ by adding two additional copies of
the partition $1$, say $1^*$ and $1^{**}$, B. Kim~\cite{kim,kim11} obtains
\begin{equation*}
\frac{(q;q)_\infty}{\ffat{zq,z^{-1}q}{q}{\infty}}
=\sum_{\lam\in \mc{P^*}}wt(\lam)z^{c^*(\lam)}q^{\sigma^*(\lam)},
\end{equation*}
where $wt(\lam)$, $c^*(\lam)$, and $\sigma^*(\lam)$
are defined as follows.
Denote the weight $wt(\lam)$ for $\lam\in \mc{P^*}$ by
\begin{equation*}
wt(\lam):=\begin{cases}
1,&\text{if}\:\lam\in \mc{P}, \lam=1^*, \text{or}\: \lam=1^{**};\\
-1,&\text{if}\:\lam=1,
\end{cases}
\end{equation*}
and denote the extended crank $c^*(\lam)$ by
\begin{equation*}
c^*(\lam):=\begin{cases}
c(\lam),&\text{if}\:\lam\in \mc{P};\\
0,&\text{if}\:\lam=1;\\
1,&\text{if}\:\lam=1^*;\\
-1,&\text{if}\:\lam=1^{**}.
\end{cases}
\end{equation*}
Finally, denote the extended sum parts function
$\sigma^*(\lam)$ in the following way:
\begin{equation*}
\sigma^*(\lam):=\begin{cases}
\sigma(\lam),&\text{if}\:\lam\in \mc{P};\\
1,&\text{otherwise}.
\end{cases}
\end{equation*}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Rank analogs for colored partitions congruences modulo 5}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

In this section, we establish two statistics that divide the relevant
partitions into equinumerous classes and present the combinatorial
interpretation for colored partition congruences modulo 5.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

We denote
\[C_{1^23^2}=\{(\lam_1,\lam_2,3\lam_3,3\lam_4)|
\lam_1,\lam_2,\lam_3,\lam_4\in\mc{P}\}.\] For $\lam\in
C_{\!1^2\!3^2\!}$, we define the sum of parts $s_{\!1^2\!3^2\!}(\lam)$,
and rank analog
$r_{\!1^2\!3^2\!}(\!\lam)$ by
\bnm
s_{\!1^2\!3^2\!}(\lam)&=&\sigma(\lam_1)+\sigma(\lam_2)+3\sigma(\lam_3)+3\sigma(\lam_4)\\
r_{1^23^2}(\lam)&=&\#(\lam_1)-\#(\lam_2)+\#(\lam_3)-\#(\lam_4).
\enm
The number of 4-colored partitions of $n$ if $s_{\!1^2\!3^2\!}(\lam)\!=\!n$
having $r_{\!1^2\!3^2\!}(\!\lam)\!=\!m$ will be written as
$N_{\!C_{\!1^2\!3^2}\!}(m,n)$, and $N_{\!C_{1^2\!3^2}\!}(m,t,n)$
is the number of such 4-colored partitions of $n$ having rank analog
$r_{\!1^2\!3^2\!}(\lam)\equiv m\pmod t$. Now, summing over all
4-colored partitions $\!\lam\in C_{1^2\!3^2}$ gives
\[N_{\!C_{\!1^2\!3^2}\!}(m,n)
\!=\!\sum_{\double{\lam\in
C_{1^23^2},s_{\!1^2\!3^2\!}(\lam)=n,}{r_{1^23^2}(\lam)=m}}1.\]

Since
\[r_{1^23^2}(\lam_1,\lam_2,\lam_3,\lam_4)
=-r_{1^23^2}(\lam_2,\lam_1,\lam_4,\lam_3),\] hence
\[N_{\!C_{1^23^2}\!}(m,n)
\!=\!N_{\!C_{1^23^2}\!}(\!-\!m,n) \quad\text{and}\quad
N_{\!C_{1^23^2}\!}(m,t,n)
\!=\!N_{\!C_{1^23^2}\!}(t\!-\!m,t,n).\]
Then we have
\eqn{\sum_{m\in
\mb{Z}}\sum_{n=0}^{\infty}N_{C_{1^23^2}}(m,n)z^mq^n
=\label{NC1^23^2gf}
\frac{1}
{(zq;q)_\infty(z^{-1}q;q)_\infty(zq^3;q^3)_\infty(z^{-1}q^3;q^3)_\infty}.}
By putting $z=1$ in the identity~\eqref{NC1^23^2gf}, we find
\[\sum_{m=-\infty}^{\infty}N_{C_{1^23^2}}(m,n)=c(n),\]
where $c(n)$ is defined by
$\sum_{n=0}^\infty c(n)q^n=\frac1{(q;q)^2_\infty(q^3;q^3)^2_\infty}$.

\begin{theorem}\label{c}
For $n\geq0$,
\[N_{\!C_{1^23^2}\!}(0,5,5n+2)\!=\!N_{\!C_{1^23^2}\!}(1,5,5n+2)
\!=\!N_{\!C_{1^23^2}\!}(2,5,5n+2)
\!=\!\frac{c(5n\!+\!2)}5.\]
\end{theorem}
It can also prove the identity in Zhang and Wang~\cite{zhang-wang}:
$c(5n+2)\equiv0\pmod5$.
\begin{proof}
Suppose $\zeta$ is primitive $5th$ root of unity. By setting $z=\zeta$
in \eqref{NC1^23^2gf},
we write
\bnm
\+\+\sum_{m\in
\mb{Z}}\sum_{n=0}^{\infty}N_{C_{1^23^2}}(m,n)\zeta^mq^n
=\frac{1}{\ffat{\zeta q,\zeta^{-1}q}{q}{\infty}
      \ffat{\zeta q^3,\zeta^{-1}q^3}{q^3}{\infty}}\\
\+=\+\frac{1}
        {(q^5;q^{5})_\infty(q^{15};q^{15})_\infty}
        \times\ffat{q,\zeta^2 q, \zeta^{-2}q}{q}{\infty}
            \ffat{q^3,\zeta^2q^3,\zeta^{-2}q^3}{q^3}{\infty}. \enm

Using modified Jacobi triple product identity~\eqref{modi-Jacobi},
the last two infinite products have the
following series representation
\[\sum_{i,j=0}^\infty (-1)^{i+j}q^{\zbnm{i+1}{2}+3\zbnm{j+1}{2}}
      \{\zeta^{2i}+\zeta^{2i-2}+\cdots+\zeta^{-2i}\}
      \{\zeta^{2j}+\zeta^{2j-2}+\cdots+\zeta^{-2j}\}.\]
Observe the congruence relation
\[\binm{i+1}{2}+3\binm{j+1}{2}+3
\equiv8\Big\{\binm{i+1}{2}+3\binm{j+1}{2}+3\Big\}
\equiv (2i+1)^2+3(2j+1)^2\equiv0\pmod5,\]
which can be reached only if
$i\equiv2\pmod5$ and $j\equiv2\pmod5$
since the corresponding residues modulo $5$ read respectively as
\[(2i+1)^2\equiv0,1,4\pmod5,\quad\text{and}\quad3(2j+1)^2\equiv0,2,3\pmod5.\]
When $i\equiv2\pmod5$ and $j\equiv2\pmod5$, we have
$\{\zeta^{2i}+\zeta^{2i-2}+\cdots+\zeta^{-2i}\}
      \{\zeta^{2j}+\zeta^{2j-2}+\cdots+\zeta^{-2j}\}=0$.
We see that in the $q$-expansion on the right side of the last
equation the coefficient of $q^n$ is zero when $n\equiv2\pmod5$.
The proof of Theorem~\ref{c} has been finished.
\end{proof}

Let
\[C_{2^23^2}=\{(2\lam_1,2\lam_2,3\lam_3,3\lam_4)|
\lam_1,\lam_2,\lam_3,\lam_4\in\mc{P}\}.\] For $\lam\in
C_{\!2^2\!3^2\!}$, we define the sum of parts $s_{\!2^2\!3^2\!}(\!\lam\!)$,
and rank analog
$r_{\!2^2\!3^2\!}(\!\lam)$ by
\bnm
s_{\!2^2\!3^2\!}(\lam)&=&2\sigma(\lam_1)+2\sigma(\lam_2)+3\sigma(\lam_3)+3\sigma(\lam_4)\\
r_{2^23^2}(\lam)&=&\#(\lam_1)-\#(\lam_2)+\#(\lam_3)-\#(\lam_4).
\enm
Define $N_{\!C_{\!2^2\!3^2}\!}(m,n)$ as
the number of 4-colored partitions of $n$ if $s_{\!2^2\!3^2\!}(\lam)\!=\!n$
having $r_{\!2^2\!3^2\!}(\!\lam)\!=\!m$, and $N_{\!C_{2^2\!3^2}\!}(m,t,n)$
as the number of such 4-colored partitions of $n$ having rank analog
$r_{\!2^2\!3^2\!}(\lam)\equiv m\pmod t$. Now, summing over all
4-colored partitions $\!\lam\in C_{2^2\!3^2}$ gives
\[N_{\!C_{\!2^2\!3^2}\!}(m,n)
\!=\!\sum_{\double{\lam\in
C_{2^23^2},s_{\!2^2\!3^2\!}(\lam)=n,}{r_{2^23^2}(\lam)=m}}1.\]

Since
\[r_{2^23^2}(\lam_1,\lam_2,\lam_3,\lam_4)
=-r_{2^23^2}(\lam_2,\lam_1,\lam_4,\lam_3),\] hence
\[N_{\!C_{2^23^2}\!}(m,n)
\!=\!N_{\!C_{2^23^2}\!}(\!-\!m,n) \quad\text{and}\quad
N_{\!C_{2^23^2}\!}(m,t,n)
\!=\!N_{\!C_{2^23^2}\!}(t\!-\!m,t,n).\]
Then we have
\eqn{\sum_{m\in
\mb{Z}}\sum_{n=0}^{\infty}N_{C_{2^23^2}}(m,n)z^mq^n
=\label{NC2^23^2gf}
\frac{1}
{(zq^2;q^2)_\infty(z^{-1}q^2;q^2)_\infty(zq^3;q^3)_\infty(z^{-1}q^3;q^3)_\infty}.}
By putting $z=1$ in the identity~\eqref{NC2^23^2gf}, we find
\[\sum_{m=-\infty}^{\infty}N_{C_{2^23^2}}(m,n)=\rho(n),\]
where $\rho(n)$ is defined by
$\sum_{n=0}^\infty \rho(n)q^n=\frac1{(q^2;q^2)^2_\infty(q^3;q^3)^2_\infty}$.

\begin{theorem}\label{rho}
For $n\geq0$,
\[N_{\!C_{2^23^2}\!}(0,5,5n+k)\!=\!N_{\!C_{2^23^2}\!}(1,5,5n+k)
\!=\!N_{\!C_{2^23^2}\!}(2,5,5n+k)
\!=\!\frac{\rho(5n\!+\!k)}5; \:k=1,4.\]
\end{theorem}
It can also prove the identity in Zhang and Wang~\cite{zhang-wang}:
$\rho(5n+1)\equiv0\pmod5$ and $\rho(5n+4)\equiv0\pmod5$.
\begin{proof}
The proof of Theorem~\ref{rho} is similar to Theorem~\ref{c}.
Replacing $z$ by $\zeta$
in \eqref{NC2^23^2gf},
we get
\bnm
\+\+\sum_{m\in
\mb{Z}}\sum_{n=0}^{\infty}N_{C_{2^23^2}}(m,n)\zeta^mq^n
=\frac{1}{\ffat{\zeta q^2,\zeta^{-1}q^2}{q^2}{\infty}
      \ffat{\zeta q^3,\zeta^{-1}q^3}{q^3}{\infty}}\\
\+=\+\frac{1}
        {(q^{10};q^{10})_\infty(q^{15};q^{15})_\infty}
        \times\ffat{q^2,\zeta^2 q^2, \zeta^{-2}q^2}{q^2}{\infty}
            \ffat{q^3,\zeta^2q^3,\zeta^{-2}q^3}{q^3}{\infty}. \enm

Applying modified Jacobi triple product identity \eqref{modi-Jacobi},
we transform the last two infinite products as follows
\[\sum_{i,j=0}^\infty (-1)^{i+j}q^{2\zbnm{i+1}{2}+3\zbnm{j+1}{2}}
      \{\zeta^{2i}+\zeta^{2i-2}+\cdots+\zeta^{-2i}\}
      \{\zeta^{2j}+\zeta^{2j-2}+\cdots+\zeta^{-2j}\}.\]
It is not hard to check that
the residues of q-exponent in the formal power series just displayed
$2\binm{i+1}{2}+3\binm{j+1}{2}$ modulo $5$ are given by the following table:
\centro{\begin{tabular}{||c|c|c|c|c|c||} \hline
\hline
$j\backslash i$&$0$&$1$&$2$&$3$&$4$\\
\hline
$0$&$0$&$2$&$1$&$2$&$0$\\
\hline
$1$&$3$&$0$&$4$&$0$&$3$\\
\hline
$2$&$4$&$1$&$0$&$1$&$4$\\
\hline
$3$&$3$&$0$&$4$&$0$&$3$\\
\hline
$4$&$0$&$2$&$1$&$2$&$0$\\
\hline \hline
\end{tabular}}

When $i\equiv2\pmod5$ or $j\equiv2\pmod5$, we have
$\{\zeta^{2i}+\zeta^{2i-2}+\cdots+\zeta^{-2i}\}
      \{\zeta^{2j}+\zeta^{2j-2}+\cdots+\zeta^{-2j}\}=0$.
We observe that in the $q$-expansion on the right side of the last
equation the coefficient of $q^n$ is zero when $n\equiv1\pmod5$
and $n\equiv4\pmod5$. The
proof of Theorem~\ref{rho} has been completed.
\end{proof}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Rank and crank analogs for colored partitions congruences modulo 7}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
In this section, we define statistics that divide the relevant
partitions into equinumerous classes and provide the combinatorial
interpretation according to~\cite{kim11}
for colored partitions congruences modulo 7 given in
Toh~\cite{toh}, Zhang and Wang~\cite{zhang-wang}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

If we denote
\[C_{1^32^{-2}}
=\{(2\lam_1,2\lam_2,\lam_3,\lam_4,\lam_5)|\lam_1,\lam_2\in\mc{P},
\lam_3,\lam_4,\lam_5\in\mc{P^*}\},\] then we can call
them as partitions into 5-colors.

For the set of the colored partitions, we define
the sum of parts $s_{1^32^{-2}}(\lam)$, a weight $wt_{1^32^{-2}}(\lam)$ and a
crank analog $c_{1^32^{-2}}(\lam)$ by
\bnm
s_{1^32^{-2}}(\lam)
&=&2\sigma(\lam_1)+2\sigma(\lam_2)+\sigma^*(\lam_3)
+\sigma^*(\lam_4)+\sigma^*(\lam_5)\\
wt_{1^32^{-2}}(\lam)&=&(-1)^{\#(\lam_1)+\#(\lam_2)}
wt(\lam_3)wt(\lam_4)wt(\lam_5)\\
c_{1^32^{-2}}(\lam)&=&c^*(\lam_3)+2c^*(\lam_4)+3c^*(\lam_5),
\enm
where the definitions of $\mc{P^*}$, $\sigma^*(\lam)$, $wt(\lam)$
and $c^*(\lam)$ are presented in section 2.
Let
$M_{\!C_{\!1^3\!2^{\!-\!2}}\!}(m,n)$ denote the number of
5-colored partitions of $n$ if
$s_{\!1^3\!2^{\!-\!2}}(\!\lam)\!=\!n$ (counted according
to the weight $wt_{\!1^3\!2^{\!-\!2}}(\lam)$) with analog of crank
$c_{1^32^{-2}}(\lam)=m$, and
$M_{\!C_{\!1^3\!2^{\!-\!2}}\!}(m,t,n)$ denote the number of
5-colored partitions of $n$ with analog of crank $c_{1^32^{-2}}(\lam)$
congruent to $m \pmod t$,
so that
\[M_{C_{1^32^{-2}}}(m,n)
=\sum_{\double{\lam\in
C_{\!1^3\!2^{\!-\!2}},s_{\!1^3\!2^{\!-\!2}}(\lam)=n,}{c_{1^32^{-2}}(\lam)=m}}
 wt_{1^32^{-2}}(\lam).\]
Then we have
\eqn{\sum_{m\in
\mb{Z}}\sum_{n=0}^{\infty}M_{C_{1^32^{-2}}}(m,n)z^mq^n
=\label{MC1^32^-2gf}
\frac{(q^2;q^2)^2_\infty(q;q)^3_\infty}
{\ffat{zq,z^{-1}q,z^2q,z^{-2}q,z^{3}q,z^{-3}q}{q}{\infty}}.}
By putting $z=1$ in the identity~\eqref{MC1^32^-2gf}, we find
\[\sum_{m=-\infty}^{\infty}M_{C_{1^32^{-2}}}(m,n)
=Q_{(po,\overline{p})}(n),\]
where $Q_{(po,\overline{p})}(n)$ is defined by
$\sum_{n=0}^\infty Q_{(po,\overline{p})}(n)q^n
:=\frac{(-q;q)_\infty}{(q;q^2)_\infty(q;q)_\infty}
=\frac{(q^2;q^2)^2_\infty}{(q;q)^3_\infty}$.

Suppose $\varpi$ is primitive seventh root of unity. By letting $z=\varpi$
in \eqref{MC1^32^-2gf}, we have
\bnm
\sum_{m\in
\mb{Z}}\sum_{n=0}^{\infty}M_{C_{\!1^3\!2^{\!-\!2}}}(m,n)\varpi^mq^n
\!=\!\frac{(q^2;q^2)^2_\infty(q;q)^3_\infty}
{\ffat{\varpi q,\varpi^{\!-\!1}q,\varpi^2q,\varpi^{\!-\!2}q,
\varpi^{3}q,\varpi^{\!-\!3}q}{q}{\infty}}
\!=\!\frac{(q^2;q^2)^2_\infty(q;q)^4_\infty}{(q^7;q^7)_\infty}.
\enm
Utilizing product identity
\eqref{modify-winqq}, we compute the numerator of
the right hand side of last identity as follows
\eqn{\xxqdn
\sum_{i,j={-\infty}}^{+\infty}
(-1)^{i+j}q^{6\zbnm{i}{2}+6\zbnm{j}{2}+3i-j+1}
(2+3i-3j)(3i+3j-2)\label{1422}.}
We illustrate that
the residues of q-exponent in the formal power series just displayed
$6\binm{i}{2}+6\binm{j}{2}+3i-j+1$ modulo $7$ are given by the following table:
\centro{\begin{tabular}{||c|c|c|c|c|c|c|c||} \hline
\hline
$j\backslash i$&$0$&$1$&$2$&$3$&$4$&$5$&$6$\\
\hline
$0$&$1$&$4$&$6$&$0$&$0$&$6$&$4$\\
\hline
$1$&$0$&$3$&$5$&$6$&$6$&$5$&$3$\\
\hline
$2$&$5$&$1$&$3$&$4$&$4$&$3$&$1$\\
\hline
$3$&$2$&$5$&$0$&$1$&$1$&$0$&$5$\\
\hline
$4$&$5$&$1$&$3$&$4$&$4$&$3$&$1$\\
\hline
$5$&$0$&$3$&$5$&$6$&$6$&$5$&$3$\\
\hline
$6$&$1$&$4$&$6$&$0$&$0$&$6$&$4$\\
\hline \hline
\end{tabular}}
The power of $q$ is congruent to $2$ modulo $7$ only when
$i\equiv_70$ and $j\equiv_73$.
Since the coefficient of $q^n$ on the right side of the last
identity is a multiple of $7$ when $n\equiv2\pmod7$,
and $1+\varpi+\varpi^2+\varpi^3+\varpi^4+\varpi^5+\varpi^6$ is a
minimal polynomial in $\mf{Z}[\varpi]$, we must have the result
following as
\begin{theorem}
For $n\geq0$ and $0\leq i< j\leq 6$, we have
%\bnm
%&&M_{\!C_{1^32^{\!-2}}\!}(0,7,7n+2)\!\equiv\!M_{\!C_{1^32^{\!-2}}\!}(1,7,7n+2)
%\!\equiv\!M_{\!C_{1^32^{\!-2}}\!}(2,7,7n+2)\\
%&\equiv&M_{\!C_{1^32^{\!-2}}\!}(3,7,7n+2)\equiv\cdots\equiv
%M_{\!C_{1^32^{\!-2}}\!}(6,7,7n+2)
%\!\equiv\!0\pmod7.
%\enm
\[M_{\!C_{1^32^{\!-2}}\!}(i,7,7n+2)
\!\equiv\!M_{\!C_{1^32^{\!-2}}\!}(j,7,7n+2)\pmod7.\]
\end{theorem}
It can also prove the identity in Toh~\cite{toh}:
$Q_{(po,\overline{p})}(7n+2)\equiv0\pmod7$.

Next we define
\[C_{1^44^42^{\!-\!7}}\!=\!\{\!(\!\lam_1,\lam_2,\lam_3,\lam_4,2\!\lam_5
,2\!\lam_6,2\!\lam_7,2\!\lam_8,2\!\lam_9\!)|\lam_1\!,\lam_2\!,\lam_3\!,\lam_4\!\in\mc{D}\mc{O},
\lam_5\!,\lam_6\!\in\mc{P},\lam_7\!,\lam_8\!,\lam_9\!\in\mc{P^*}\!\}.\]
For $\lam\!=\!(\!\lam_1\!,\lam_2\!,\lam_3\!,\lam_4\!,2\!\lam_5\!
,2\!\lam_6\!,2\!\lam_7\!,2\!\lam_8\!,2\!\lam_9\!)$, we
denote the sum of parts $s_{\!1^4\!4^4\!2^{\!-\!7}\!}(\!\lam\!)$,
a weight $wt_{\!1^4\!4^4\!2^{\!-\!7}\!}(\!\lam\!)$ and a crank analog
$c_{\!1^4\!4^4\!2^{\!-\!7}\!}(\!\lam\!)$ by
\bnm
s_{\!1^4\!4^4\!2^{\!-\!7}\!}(\!\lam\!)
&=&\sigma(\!\lam_1\!)\!+\!\sigma(\!\lam_2\!)
\!+\!\sigma(\!\lam_3\!)\!+\!\sigma(\!\lam_4\!)
\!+\!2\sigma(\!\lam_5\!)\!+\!2\sigma(\!\lam_6\!)
\!+\!2\sigma^*(\!\lam_7\!)\!+\!2\sigma^*(\!\lam_8\!)\!+\!2\sigma^*(\!\lam_9\!)\\
wt_{\!1^4\!4^4\!2^{\!-\!7}\!}(\!\lam\!)
&=&(-1)^{\#(\lam_5)+\#(\lam_6)}
wt(\lam_7)wt(\lam_8)wt(\lam_9)\\
c_{\!1^4\!4^4\!2^{\!-\!7}\!}(\!\lam\!)
&=&c^*(\lam_7)+2c^*(\lam_8)+3c^*(\lam_9). \enm
Finally define
$M_{\!C_{\!1^4\!4^4\!2^{\!-\!7}\!}\!}(m,n)$ as the number of
9-colored partitions of $n$ if
$s_{\!1^4\!4^4\!2^{\!-\!7}\!}(\!\lam\!)\!=\!n$
with crank analog
$c_{\!1^4\!4^4\!2^{\!-\!7}\!}(\!\lam\!)=m$
counted according to the weight $wt_{\!1^4\!4^4\!2^{\!-\!7}\!}(\lam)$ as follows:,
\[M_{C_{\!1^4\!4^4\!2^{\!-\!7}\!}}(m,n)
=\sum_{\double{\lam\in
C_{\!1^4\!4^4\!2^{\!-\!7}\!},s_{\!1^4\!4^4\!2^{\!-\!7}\!}(\lam)=n,}
{c_{\!1^4\!4^4\!2^{\!-\!7}\!}(\lam)=m}}
 wt_{\!1^4\!4^4\!2^{\!-\!7}\!}(\lam).\]
Let $M_{\!C_{\!1^4\!4^4\!2^{\!-\!7}\!}\!}(m,t,n)$ denote the number of
9-colored partitions of $n$ with
crank analog $c_{\!1^4\!4^4\!2^{\!-\!7}\!}(\!\lam\!)$
congruent to $m \pmod t$.

Then we have
\eqn{
\sum_{m\in
\mb{Z}}\sum_{n=0}^{\infty}M_{\!C_{\!1^4\!4^4\!2^{\!-\!7}}\!}(m,n)z^mq^n
=\label{MC1^44^42^-7gf}
\frac{(-q;q^2)_\infty^4(q^2;q^2)_\infty^5}
{\ffat{zq^2,z^{\!-\!1}q^2,z^{2}q^2,z^{\!-\!2}q^2,
z^{3}q^2,z^{\!-\!3}q^2}{q^2}{\infty}}.}
By replacing $z$ by $1$ in the identity~\eqref{MC1^44^42^-7gf}, we
discover
\[\sum_{m=-\infty}^{\infty}M_{C_{1^44^42^{-7}}}(m,n)=\gamma(n),\]
where $\gamma(n)$ is defined by
$\sum_{n=0}^\infty\gamma(n)q^n=\frac{(-q;q^2)^4_\infty}{(q^2;q^2)_\infty}$.
(see \cite{zhang-wang}).
\begin{theorem}
For $n\geq0$,
\[M_{C_{\!1^44^42^{\!-\!7}}\!}(0,7,7n\!+\!2)
\!\equiv\!M_{\!C_{1^44^42^{\!-\!7}}\!}(1,7,7n\!+\!2)
\equiv\cdots
\!\equiv\!M_{C_{\!1^44^42^{\!-\!7}}\!}(6,7,7n\!+\!2)\pmod7.\]
\end{theorem}
It can also prove the identity
$\gamma(7n+2)\equiv0\pmod7$.
\begin{proof}
Put $z\!=\!\varpi$ in \eqref{MC1^44^42^-7gf} and apply
product identity \eqref{modify-winqq} substituting $q\to-q$ to obtain
\bnm
&&\sum_{m\in
\mb{Z}}\sum_{n=0}^{\infty}M_{C_{1^44^42^{-7}}}(m,n)\varpi^mq^n
=\frac{(-q;q^2)_\infty^4(q^2;q^2)_\infty^5}
{\ffat{\varpi q^2,\varpi^{\!-\!1}q^2,\varpi^{2}q^2,\varpi^{\!-\!2}q^2,
\varpi^{3}q^2,\varpi^{\!-\!3}q^2}{q^2}{\infty}}\\
&=&\frac{(-q;q^2)_\infty^4(q^2;q^2)_\infty^6}{(q^{14};q^{14})_\infty}
=\frac{-1}{(q^{14};q^{14})_\infty}
\sum_{i,j={-\infty}}^{+\infty}
q^{6\zbnm{i}{2}+6\zbnm{j}{2}+3i-j+1}
(2+3i-3j)(3i+3j-2).
\enm
We discover that the double sum of the last identity
is similar as~\eqref{1422}. Then we can
use the same congruence relations.
Since the coefficient of $q^n$ on the right side of the last
identity is a multiple of $7$ when $n\equiv2\pmod7$,
and $1+\varpi+\varpi^2+\varpi^3+\varpi^4+\varpi^5+\varpi^6$ is a
minimal polynomial in $\mf{Z}[\varpi]$,
we deduce the theorem.
\end{proof}

If we denote
\[C_{1^22}=\{(\lam_1,\lam_2,2\lam_3,2\lam_4,2\lam_5)|
\lam_1,\lam_2,\lam_3,\lam_4,\lam_5\in\mc{P}\}.\]
It can be said as partitions into
5-colors.
For $\lam=(\lam_1,\lam_2,2\lam_3,2\lam_4,2\lam_5)\in C_{1^22}$, we
define the sum of parts $s_{1^22}(\lam)$, a weight $w_{1^22}(\lam)$ and a
rank analog $r_{1^22}(\lam)$ by
\bnm
s_{1^22}(\lam)&=&\sigma(\lam_1)+\sigma(\lam_2)+2\sigma(\lam_3)
+2\sigma(\lam_4)+2\sigma(\lam_5)\\
w_{1^22}(\lam)&=&(-1)^{\#(\lam_5)}\\
r_{1^22}(\lam)&=&\#(\lam_1)-\#(\lam_2)+3\#(\lam_3)-3\#(\lam_4). \enm
Let
$N_{\!C_{1^2\!2}\!}(m,n)$ denote the number of 5-colored
partitions of $n$ if $s_{1^2\!2}(\lam)=n$ (counted according to the weight
$w_{1^2\!2}(\lam)$) with rank analog $r_{1^22}(\lam)=m$, and
$N_{\!C_{1^2\!2}}(\!m,t,n\!)$ denote the number of
5-colored partitions of $n$ with rank analog $r_{1^22}(\lam)$ congruent to
$m \pmod{\!t}$, hence
\[N_{C_{1^2\!2}\!}(m,n)
=\sum_{\double{\lam\in
C_{1^22},s(\lam)=n,}{r_{1^22}(\lam)=m}}
w_{1^22}(\lam).\]
By considering the transformation that
interchanges $\lam_1$ and $\lam_2$, $\lam_3$ and $\lam_4$, we get
\[N_{C_{1^22}}(m,n)\!=\!N_{C_{1^22}}(-m,n),
\quad N_{C_{1^22}}(m,t,n)\!=\!N_{C_{1^22}}(t-m,t,n).\]
Then we have
\eqn{\sum_{m\in
\mb{Z}}\sum_{n=0}^{\infty}N_{C_{1^22}}(m,n)z^mq^n
=\label{NC1^22gf}
\frac{(q^2;q^2)_\infty}
{(zq;q)_\infty(z^{-1}q;q)_\infty
(z^3q^2;q^2)_\infty(z^{-3}q^2;q^2)_\infty}.} By
putting $z=1$ in the identity~\eqref{NC1^22gf}, we check
\[\sum_{m=-\infty}^{\infty}N_{C_{1^22}}(m,n)=\alpha(n),\]
where $\alpha(n)$ is defined by
$\sum_{n=0}^\infty\alpha(n)q^n=\frac{1}{(q;q)_\infty^2(q^2;q^2)_\infty}$.
(see~\cite{zhang-wang}).

Suppose $\varpi$ is primitive $7th$ root of unity. Substituting
$z=\varpi$ into \eqref{NC1^22gf},
we have
\bnm \+\+\sum_{m\in
\mb{Z}}\sum_{n=0}^{\infty}N_{\!C_{\!1^2\!2}\!}(m,n)\varpi^mq^n
=\frac{(q^2;q^2)_\infty}{\ffat{\varpi q,q/\!\varpi}{q}{\infty}
\ffat{\varpi^3 q^2,q^2/\!{\varpi^3}}{q^2}{\infty}} \\
\+=\+\frac{\ffat{q,\varpi^2 q,q/\!{\varpi^2}}{q}{\infty}
          \ffat{q^2,\varpi^3 q,q/\!{\varpi^3}}{q^2}{\infty}
          }
          {(q^7;q^7)_\infty}\\
\+=\+\frac {\sum_{i\geq0}(-1)^i(\varpi^{2i}+\varpi^{2i-2}
                 +\cdots+\varpi^{-2i})q^{\zbnm{i+1}{2}}
\sum_{j=-\infty}^\infty(-1)^j\varpi^{3j}q^{2\zbnm{j}{2}+j}}
      {(q^7;q^7)_\infty}.
\enm
The last line depends only on modified Jacobi identity
\eqref{modi-Jacobi} and classical Jacobi identity~\eqref{jacobi}.

If and only if $i\equiv_73$, we have
$\varpi^{2i}+\varpi^{2i-2}+\cdots+\varpi^{-2i}=0$. Obviously
\begin{equation}\label{1221}
\binm{i+1}{2} \equiv_7\begin{cases}
0,&i\equiv_70,6;\\
1,&i\equiv_71,5;\\
3,&i\equiv_72,4;\\
6,&i\equiv_73;
\end{cases}
\quad\text{and}\quad
2\binm{j}{2}+j \equiv_7\begin{cases}
0,&j\equiv_70;\\
1,&j\equiv_71,6;\\
4,&j\equiv_72,5;\\
2,&m\equiv_73,4.
\end{cases}
\end{equation}
The power of $q$ is congruent to 6 modulo 7 only when
$\binm{i+1}{2}\equiv_76$
and $2\binm{j}{2}+j\equiv_70$ in which
case $i\equiv_73$ and $j\equiv_70$ and the coefficient of $q^{7n+6}$
in the last identity is zero. Since
$1+\varpi+\varpi^2+\varpi^3+\varpi^4+\varpi^5+\varpi^6$ is a minimal polynomial in
$\mf{Z}[\varpi]$, our main result is as follows.
\begin{theorem}~\label{alpha}
For $n\geq0$,
\[N_{C_{1^2\!2}}(0,7,7n\!+\!6)\!
=\!N_{C_{1^2\!2}}(1,7,7n\!+\!6)
\!=\!N_{C_{1^2\!6}}(2,7,7n\!+\!6)\!
=\!\cdots
=\!N_{C_{1^2\!6}}(6,7,7n\!+\!6)\!
=\!\frac{\alpha(7n\!+\!6)}7.\]
\end{theorem}
It can also prove the identity
$\alpha(7n+6)\equiv0\pmod7$.

Denote
\[C_{1^54^52^{\!-\!7}}=\{(\lam_1,\lam_2,\lam_3,\lam_4,\lam_5,2\lam_6,2\lam_7,2\lam_8)|
                 \lam_1,\lam_2,\lam_3,\lam_4,\lam_5\in\mc{D}\mc{O},
\lam_6,\lam_7,\lam_8\in\mc{P^*}\}.\]
We call the elements of
$C_{1^54^52^{\!-\!7}}$ 8-colored partitions.
For $\lam\in
C_{1^54^52^{\!-\!7}}$, we define the sum of parts
$s_{1^54^52^{\!-\!7}}(\!\lam\!)$, a weight
$wt_{1^54^52^{\!-\!7}}(\!\lam\!)$ and a crank analog
$c_{1^54^52^{\!-\!7}}(\!\lam\!)$ by
\bnm
s_{1^54^52^{\!-\!7}}(\lam)&=&\sigma(\lam_1)+\sigma(\lam_2)+\sigma(\lam_3)+\sigma(\lam_4)
+\sigma(\lam_5)+2\sigma^*(\lam_6)+2\sigma^*(\lam_7)+2\sigma^*(\lam_8)\\
wt_{1^54^52^{\!-\!7}}(\lam)&=&wt(\lam_6)wt(\lam_7)wt(\lam_8)\\
c_{1^54^52^{\!-\!7}}(\lam)&=&c^*(\lam_6)+2c^*(\lam_7)+3c^*(\lam_8).
\enm
Let $M_{\!C_{\!1^5\!4^5\!2^{\!-\!7}}\!}(m,n)$ denote the number
of 8-colored partitions of $n$ if $s_{1^54^52^{-7}}(\lam)\!=\!n$ (counted according
to the weight $wt_{\!1^5\!4^5\!2^{\!-\!7}\!}(\!\lam\!)$) with
crank analog $c_{1^54^52^{\!-\!7}}(\lam)=m$,
and $M_{\!C_{\!1^5\!4^5\!2^{\!-\!7}}\!}(m,t,n)$ denote the number of
8-colored partitions of $n$ with crank analog
$c_{1^54^52^{\!-\!7}}(\lam)\equiv t\pmod m$, so
that
\[M_{C_{1^54^52^{\!-\!7}}}(m,n)
=\sum_{\double{\lam\in
C_{1^54^52^{\!-\!7}},s_{1^54^52^{\!-\!7}}(\lam)=n,}{c_{1^54^52^{\!-\!7}}(\lam)=m}}
    wt_{1^54^52^{\!-\!7}}(\lam).\]
Then the generating function is
\eqn{\sum_{m\in
\mb{Z}}\sum_{n=0}^{\infty}M_{C_{1^54^52^{\!-\!7}}}(m,n)z^mq^n
=\label{MC1^54^52-7gf}
\frac{(-q;q^2)_\infty^5(q^2;q^2)^3_\infty}
{\ffat{zq^2,z^{-1}q^2,z^2q^2,z^{-2}q^2,z^3q^2,z^{-3}q^2}{q^2}{\infty}}.}
By putting $z=1$ in the identity~\eqref{MC1^54^52-7gf},
we discover
\[\sum_{m=-\infty}^{\infty}M_{C_{1^54^52^{\!-\!7}}}(m,n)=\nu(n),\]
where $\nu(n)$ is defined by
$\sum_{n=0}^\infty\nu(n)q^n=\frac{(-q,q^2)^5_\infty}{(q^2;q^2)^3_\infty}$.
\begin{theorem}~\label{nu}
For $n\geq0$,
\[M_{\!C_{\!1^5\!4^5\!2^{\!-\!7}\!}}(0,7,7n\!+\!6)
\!\equiv\!M_{\!C_{\!1^5\!4^5\!2^{\!-\!7}\!}}(1,7,7n\!+\!6)
\equiv\cdots
\!\equiv\!M_{\!C_{\!1^5\!4^5\!2^{\!-\!7}\!}}(6,7,7n\!+\!6)
\pmod7.\]
\end{theorem}
It can also prove the identity
$\nu(7n+6)\equiv0\pmod7$.
\begin{proof}
By replacing $z$ by $\varpi$ in \eqref{MC1^54^52-7gf},
we write
\bnm
\+\+\sum_{m\in
\mb{Z}}\sum_{n=0}^{\infty}N_{C_{1^34^12^{-3}}}(m,n)\varpi^mq^n
=\frac{(-q;q^2)_\infty^5(q^2;q^2)^3_\infty}
{\ffat{\varpi q^2,\varpi^{-1}q^2,\varpi^2q^2,\varpi^{-2}q^2,
\varpi^3q^2,\varpi^{-3}q^2}{q^2}{\infty}}\\
\+=\+\frac{(-q;q^2)_\infty^5(q^2;q^2)^4_\infty}
        {(q^{14};q^{14})_\infty}. \enm

Consider
\[(q;q)^3_\infty\ffat{q^2,q,q}{q^2}{\infty}
=\sum_{i=0}^\infty\sum_{j=-\infty}^\infty
(-1)^{i+j}(2i+1)q^{\zbnm{i+1}{2}+j^2},\]
which can be deduced by
Jacobi identity \eqref{jacobi} and \eqref{modi-Jacobi}.

Replacing $q$ by $-q$ in the last identity, we have the
following series representation
\[(-q;q^2)^5_\infty(q^2;q^2)^4_{\infty}
=\sum_{i=0}^\infty\sum_{j=-\infty}^\infty
(-1)^{\zbnm{i}{2}}(2i+1)q^{\zbnm{i+1}{2}+j^2}.\]
If and only if $i\equiv_73$, we have
$2i+1\equiv_70$.
We see that in the $q$-expansion on the right side of the last
equation the coefficient of $q^n$ is a multiple of $7$
when $n\equiv6\pmod7$ referring to~\eqref{1221}. The
proof of Theorem~\ref{nu} has been finished.
\end{proof}
Let
\[C_{1^52^3}=\{(\!\lam_1,\!\lam_2,\!\lam_3,\!\lam_4,
\!\lam_5,2\!\lam_6,2\!\lam_7,2\!\lam_8)|
                 \lam_1,\lam_2,\lam_3,\lam_4\in\mc{P},
                 \lam_5,
\lam_6,\lam_7,\lam_8\in\mc{P^*}\}.\] We can say them as
partitions into 8-colors. For $\lam\in C_{1^52^3}$, we denote
the sum of parts $s_{1^52^3}(\lam)$, a weight $wt_{1^52^3}(\lam)$ and a
crank analog $c_{1^52^3}(\lam)$ by \bnm
s_{1^52^3}(\lam)&=&\sigma(\lam_1)+\sigma(\lam_2)+\sigma(\lam_3)+\sigma(\lam_4)
+\sigma^*(\lam_5)+2\sigma^*(\lam_6)+2\sigma^*(\lam_7)+2\sigma^*(\lam_8)\\
wt_{1^52^3}(\lam)&=&wt(\lam_5)wt(\lam_6)wt(\lam_7)wt(\lam_8)\\
c_{1^52^3}(\lam)&=&\#(\lam_1)\!-\!\#(\lam_2)\!+\!2\#(\lam_3)\!-\!2\#(\lam_4)
\!+\!3c^*(\lam_5)\!+\!c^*(\lam_6)\!+\!2c^*(\lam_7)\!+\!3c^*(\lam_8).
\enm
The number of 8-colored partitions of $n$ if $s_{1^52^3}(\lam)\!=\!n$
with crank analog $c_{1^52^3}(\lam)=m$
counted according to the weight $wt_{\!1^5\!2^3\!}(\lam)$
is denoted by
$M_{\!C_{\!1^5\!2^3\!}}(m,n)$, so that
\[M_{C_{1^52^3}}(m,n)
=\sum_{\double{\lam\in
C_{1^52^3},s_{1^52^3}(\lam)=n,}{c_{1^52^3}(\lam)=m}}
    wt_{1^52^3}(\lam).\]
The number of 8-colored partitions of $n$
with crank analog $c_{1^52^3}(\lam)$ congruent
to $m \pmod t$ is denoted by $M_{C_{1^52^3}}(m,t,n)$.
The
following generating function for $M_{C_{1^52^3}}(m,n)$ is
\bmn
&&\nnm\sum_{m\in
\mb{Z}}\sum_{n=0}^{\infty}M_{C_{1^52^3}}(m,n)z^mq^n\\
&=&\label{MC1^52^3gf}
\frac{(q;q)_\infty(q^2;q^2)^3_\infty}
{\ffat{zq,z^{\!-\!1}q,z^{2}q,z^{\!-\!2}q,
z^{3}q,z^{\!-\!3}q}{q}{\infty}
\ffat{zq^2,z^{\!-\!1}q^2,z^{2}q^2,z^{\!-\!2}q^2,
z^{3}q^2,z^{\!-\!3}q^2}{q^2}{\infty}}.
\emn
By setting $z=1$ in the identity~\eqref{MC1^52^3gf}, we find
\[\sum_{m=-\infty}^{\infty}M_{C_{1^52^3}}(m,n)=\mu(n),\]
where $\mu(n)$ is defined by
$\sum_{n=0}^\infty\mu(n)q^n
=\frac{1}{(q;q)^5_\infty(q^2;q^2)^3_\infty}$.
\begin{theorem}~\label{mu}
For $n\geq0$,
\[M_{\!C_{1^5\!2^3}\!}(0,7,7n\!+\!6)
\!\equiv\!M_{\!C_{1^5\!2^3}\!}(1,7,7n\!+\!6)
\equiv\cdots
\!\equiv\!M_{\!C_{1^5\!2^3}\!}(6,7,7n\!+\!6)
\pmod7.\]
\end{theorem}
It can also prove the identity
$\mu(7n+6)\equiv0\pmod7$.
\begin{proof}
%The proof of Theorem~\ref{mu} is similar to Theorem~\ref{nu}.
By letting $z=\varpi$ in \eqref{MC1^52^3gf}, we get
\bnm
\+\+\sum_{m\in
\mb{Z}}\sum_{n=0}^{\infty}M_{C_{1^52^3}}(m,n)\varpi^mq^n\\
\+=\+\frac{(q;q)_\infty(q^2;q^2)^3_\infty}
{\ffat{\varpi q,\varpi^{\!-\!1}q,\varpi^{2}q,\varpi^{\!-\!2}q,
\varpi^{3}q,\varpi^{\!-\!3}q}{q}{\infty}
\ffat{\varpi q^2,\varpi^{\!-\!1}q^2,\varpi^{2}q^2,\varpi^{\!-\!2}q^2,
\varpi^{3}q^2,\varpi^{\!-\!3}q^2}{q^2}{\infty}}\\
\+=\+\frac{(q;q)^2_\infty(q^2;q^2)^4_\infty}
        {(q^7;q^{7})_\infty(q^{14};q^{14})_\infty}.
\enm

Investigating product identity~\eqref{winq-1},
splitting the bilateral sum with respect to $i$
into two unilateral sums,
the numerator infinite products on the last line
of the last formula have the following series expression
\bmn
(q;q)_{\infty}^2(q^2;q^2)_{\infty}^4
=\label{1523}\sum_{i=0}^{\infty}\sum_{j={-\infty}}^{+\infty}
q^{3\zbnm{i+1}{2}+3\zbnm{j}{2}+j}
\frac{(1+3i+3j)(2+3i-3j)}{2}. \emn

We check that
the residues of q-exponent in the formal power series displayed
$3\binm{i+1}{2}+3\binm{j}{2}+j$ modulo $7$ are presented by the following table:
\centro{\begin{tabular}{||c|c|c|c|c|c|c|c||} \hline
\hline
$j\backslash i$&$0$&$1$&$2$&$3$&$4$&$5$&$6$\\
\hline
$0$&$0$&$3$&$2$&$4$&$2$&$3$&$0$\\
\hline
$1$&$1$&$4$&$3$&$5$&$3$&$4$&$1$\\
\hline
$2$&$5$&$1$&$0$&$2$&$0$&$1$&$5$\\
\hline
$3$&$5$&$1$&$0$&$2$&$0$&$1$&$5$\\
\hline
$4$&$1$&$4$&$3$&$5$&$3$&$4$&$1$\\
\hline
$5$&$0$&$3$&$2$&$4$&$2$&$3$&$0$\\
\hline
$6$&$2$&$5$&$4$&$6$&$4$&$5$&$2$\\
\hline \hline
\end{tabular}}
If and only if $3\binm{i+1}{2}+3\binm{j}{2}+j\equiv6\pmod7$,
we have $i\equiv_73$ and $j\equiv_76$.
Since the coefficient of $q^n$ on the right side of the last
identity is a multiple of $7$ when $n\equiv6\pmod7$,
and $1+\varpi+\varpi^2+\varpi^3+\varpi^4+\varpi^5+\varpi^6$ is a
minimal polynomial in $\mf{Z}[\varpi]$,
we finish the proof of Theorem~\ref{mu}.
\end{proof}
Consider
\[C_{1^24^22^{\!-\!3}}=\{(\lam_1,\lam_2,2\lam_3,2\lam_4,2\lam_5,2\lam_6,2\lam_7)|
                 \lam_1,\lam_2\in\mc{D}\mc{O},
                 \lam_3,\lam_4\in\mc{P},
                 \lam_5,\lam_6,\lam_7\in\mc{P^*},\}.\]
We call them as partitions into 7-colors. For $\lam\in
C_{1^24^22^{\!-\!3}}$, we define the sum of parts $s_{1^24^22^{\!-\!3}}(\lam)$, a
weight $wt_{1^24^22^{\!-\!3}}(\lam)$ and a crank analog
$c_{1^24^22^{\!-\!3}}(\lam)$ by
\bnm
s_{1^24^22^{\!-\!3}}(\lam)&=&\sigma(\lam_1)+\sigma(\lam_2)+2\sigma(\lam_3)+2\sigma(\lam_4)
+2\sigma^*(\lam_5)+2\sigma^*(\lam_6)+2\sigma^*(\lam_7)\\
wt_{1^24^22^{\!-\!3}}(\lam)
&=&(-1)^{\#(\lam_3)+\#(\lam_4)}
   wt(\lam_5)wt(\lam_6)wt(\lam_7)\\
c_{1^24^22^{\!-\!3}}(\lam)&=&c^*(\lam_5)+2c^*(\lam_6)+3c^*(\lam_7).
\enm
Let $M_{\!C_{1^24^22^{\!-\!3}}\!}(m,n)$ denote the number
of 7-colored partitions of $n$ if $s_{1^24^22^{\!-\!3}}(\lam)\!=\!n$ (counted according
to the weight $wt_{1^24^22^{\!-\!3}}(\lam)$)
with crank analog $c_{1^24^22^{\!-\!3}}(\lam)=m$, so
that
\[M_{C_{1^24^22^{\!-\!3}}}(m,n)
=\sum_{\double{\lam\in
C_{1^24^22^{\!-\!3}},s_{1^24^22^{\!-\!3}}(\lam)=n,}{c_{1^24^22^{\!-\!3}}(\lam)=m}}
    wt_{1^24^22^{\!-\!3}}(\lam).\]
The number of 7-colored partitions of $n$ with crank analog
$c_{1^24^22^{\!-\!3}}(\lam)\equiv
m\pmod t$ is denoted by $M_{\!C_{1^24^22^{\!-\!3}}\!}(m,t,n)$.
Then the two
variable generating function for
$M_{\!C_{1^24^22^{\!-\!3}}\!}(m,n)$ is
\eqn{\sum_{m\in
\mb{Z}}\sum_{n=0}^{\infty}M_{C_{1^24^22^{\!-\!3}}}(m,n)z^mq^n
=\label{MC1^24^22-3gf}
\frac{(-q;q^2)^2_\infty(q^2;q^2)_\infty^5}
{\ffat{zq^2,z^{\!-\!1}q^2,z^{2}q^2,z^{\!-\!2}q^2,z^{3}q^2,z^{\!-\!3}q^2}
{q^2}{\infty}}.}
If we simply put $z=1$ in the identity~\eqref{MC1^24^22-3gf}, and
read off the coefficients of like powers of $q$, we find
\[\sum_{m=-\infty}^{\infty}M_{C_{1^24^22^{\!-\!3}}}(m,n)=\beta(n),\]
where $\beta(n)$ is defined by
$\sum_{n=0}^\infty\beta(n)q^n
=\frac{(-q;q^2)^2_\infty}{(q^2;q^2)_\infty}$.

Putting $z=\varpi$ in \eqref{MC1^24^22-3gf} gives
\bnm \+\+\sum_{m\in
\mb{Z}}\sum_{n=0}^{\infty}M_{C_{1^24^22^{\!-\!3}}}(m,n)\varpi^mq^n
=\frac{(-q;q^2)^2_\infty(q^2;q^2)_\infty^5}
{\ffat{\varpi q^2,\varpi^{\!-\!1}q^2,\varpi^{2}q^2,
\varpi^{\!-\!2}q^2,\varpi^{3}q^2,\varpi^{\!-\!3}q^2}
{q^2}{\infty}}\\
\+=\+\frac{(-q;q^2)^2_\infty(q^2;q^2)_\infty^6}
        {(q^{14};q^{14})_\infty}.
\enm

Substituting $q\to-q$ into identity \eqref{1523},
the numerator infinite products have the
following series expression
\[(-q;q^2)_{\infty}^2(q^2;q^2)_{\infty}^6
=\sum_{i=0}^{\infty}\sum_{j={-\infty}}^{+\infty}
(-1)^{\zbnm{i+1}{2}+\zbnm{j+1}{2}}
q^{3\zbnm{i+1}{2}+3\zbnm{j}{2}+j}
\frac{(1+3i+3j)(2+3i-3j)}{2}. \]
It is easy to find that the power of $q$
is congruent to $6$ modulo $7$ if and only if
$i\equiv_73$ and $j\equiv_76$
considering the congruence relations in the proof of
Theorem~\ref{mu}.
Since the
coefficient of $q^n$ on the right side of the last identity is a multiple of $7$
when $n\equiv_76$, and
$1+\varpi+\varpi^2+\varpi^3+\varpi^4+\varpi^5+\varpi^6$ is a
minimal polynomial in $\mf{Z}[\varpi]$, our main result is as follows:
\begin{theorem}
For $n\geq0$ and $0\leq i< j\leq6$, we obtain
\[M_{C_{1^24^22^{\!-\!3}}}(i,7,7n+6)
\equiv M_{C_{1^24^22^{\!-\!3}}}(j,7,7n+6)
\pmod7.\]
\end{theorem}
It can also prove the identity
$\beta(7n+6)\equiv0\pmod7$.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\subsection*{Acknowledgements}
%Thanks to Professor Querty for suggesting the proof of
%Lemma~\ref{lem:Technical}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \bibliographystyle{plain}
% \bibliography{myBibFile}
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\end{document}
