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\begin{center}
\vskip 1cm{\LARGE\bf 
 On a Ramanujan-type Congruence for  \\
\vskip .1in
 Bipartitions with $5$-Cores
}
\vskip 1cm
\large
Ranganatha Dasappa\\
Department of Studies in Mathematics\\
University of Mysore, Manasagangotri\\
Mysuru-570006 \\
Karnataka \\
India \\
\href{mailto:ddranganatha@gmail.com}{\tt ddranganatha@gmail.com}
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\begin{abstract}
In this short note, we prove a Ramanujan-type congruence modulo $5^{\alpha}$ $(\alpha\geq1)$ for $A_{5}(n)$,
which counts the number of $5$-core bipartitions of $n$.
\end{abstract}

\section{Introduction}
A {\it partition\/} of a positive integer $n$ is a finite non-increasing sequence of positive integers $\lambda_{1}, \lambda_{2},\ldots, \lambda_{r}$ such that $\sum_{i=1}^{r}\lambda_{i}= n$. The $\lambda_{i}$ are called the parts of the partition. For example, the partitions of $4$ are $4, 3+1, 2+2, 2+1+1, 1+1+1+1$.

Given a partition $[\lambda]=\lambda_{1}+\lambda_{2}+\cdots+\lambda_{r}$ of $n$, where $\lambda_{1}\geq\lambda_{2}\geq\cdots \geq\lambda_{r}$,
the {\it Ferrers-Young diagram\/} of $[\lambda]$ is an array of nodes with $\lambda_{i}$ nodes in the $i^{th}$ row. The $(i,j)$ hook is
the set of nodes directly below, together with the set of nodes directly to the right of the $(i,j)$ nodes, as well as the $(i,j)$ node itself.
The hook number of $(i, j)$, denoted  by $H(i,j)$, is the total number of nodes in the $(i,j)$ hook.
For a positive integer $t\geq2$,  a partition of $n$ is said to be  
{\it $t$-core} if it has no hook numbers that are multiples of
 $t$.
\begin{example}
The Ferrers-Young diagram of the partition $\lambda=5+3+2$ of $10$ is
\begin{align*}
&\bullet\qquad\bullet\qquad\bullet\qquad\bullet\qquad\bullet\\
&\bullet\qquad\bullet~\qquad\bullet\\
&\bullet\qquad\bullet
\end{align*}
The nodes $(1,1), (1,2), (1,3), (1,4), (1,5), (2,1), (2,2), (2,3), (3,1)$ and $(3,2)$ have hook numbers $7, 6, 4, 2, 1, 4, 3, 1, 2$ and $1$, respectively. Therefore $\lambda$ is $5$-core. Note that $\lambda$ is $t$-core for $t\geq8$.
\end{example}

A {\it bipartition} of $n$ is a pair of partitions $(\lambda,\mu)$ such that the sum of all the parts of $\lambda$ and $\mu$ equals $n$. A bipartition with $t$-core of $n$ is a bipartition $(\lambda,\mu)$ of $n$ such that $\lambda$ and $\mu$ are both $t$-cores. Let $A_{t}(n)$ denote the number of bipartitions with $t$-cores of $n$. The generating function for $A_{t}(n)$ is given by
\begin{equation}\label{Df}
\sum_{n=0}^{\infty}A_{t}(n)q^{n}=\frac{(q^{t};q^{t})_{\infty}^{2t}}{(q;q)^{2}_{\infty}}.
\end{equation}
Here and throughout this note,  we assume that $|q|<1$  and we follow standard $q$-series notation:
\begin{equation*}
(a;q)_{\infty}:=\prod_{n=0}^{\infty}(1-aq^{n}).
\end{equation*}
 Motivated by the work of Ramanujan on congruences for unrestricted partition function $p(n)$, many mathematicians considered the function $A_{t}(n)$ and studied its congruence properties. For example, Lin \cite{Lin} established several congruences for $A_{3}(n)$. Soon after, Xia \cite{Xia} and Yao \cite{Yao} extended the list of congruences for $A_{3}(n)$.

The main aim of this note is to prove the following  Ramanujan-type congruence modulo $5^{\alpha}$ $(\alpha\geq1)$ for $A_{5}(n)$:
\begin{equation*}
 A_{5}(5^{\alpha}n+5^{\alpha}-2)\equiv0\pmod{5^{\alpha}}, ~~\alpha\geq1.
 \end{equation*}
 The following $5$-dissection formula for $(q;q)_{\infty}$ was first stated by Ramanujan \cite[p.\ 212]{Rama} without proof.
\begin{lemma}\cite[p.\ 212]{Rama} We have
\begin{equation}\label{5D}
(q;q)_{\infty}=(q^{25};q^{25})_{\infty}\Bigl(\frac{1}{R(q^{5})}-q-q^{2}R(q^{5})\Bigr),
\end{equation}
where $R(q)=\frac{(q;q^{5})_{\infty}(q^{4};q^{5})_{\infty}}{(q^{2};q^{5})_{\infty}(q^{3};q^{5})_{\infty}}$.
 \end{lemma}
 Watson \cite{Wat} presented a proof of \eqref{5D} using the quintuple product identity.
\begin{lemma}\cite[eq.\ (7.4.14), p.\ 165 ]{Ber} We have
\begin{align}\label{5D1}
\frac{1}{(q;q)_{\infty}}=&\frac{(q^{25};q^{25})_{\infty}^{5}}{(q^{5};q^{5})_{\infty}^{6}}\Bigl(\frac{1}{R(q^{5})^{4}}+\frac{q}{R(q^{5})^{3}}
+\frac{2q^{2}}{R(q^{5})^{2}}+\frac{3q^{3}}{R(q^{5})}
+5q^{4}-3q^{5}R(q^{5})\nonumber\\&+2q^{6}R(q^{5})^{2}-q^{7}R(q^{5})^{3}+q^{8}R(q^{5})^{4}\Bigr).
\end{align}
\end{lemma}
The following lemmas are useful to prove our main congruence for $A_{5}(n)$:
\begin{lemma}\label{Le1}Let $\sum_{n=0}^{\infty}a(n)q^{n}=\frac{q}{(q;q)^{2}_{\infty}}$. Then
\begin{equation}
\sum_{n=0}^{\infty}a(5n+4)q^{n}=125q\frac{(q^{5};q^{5})^{10}_{\infty}}{(q;q)_{\infty}^{12}}+10\frac{(q^{5};q^{5})_{\infty}^{4}}{(q;q)_{\infty}^{6}}.
\end{equation}
\end{lemma}
\begin{proof}
In view of \eqref{5D1}, we have
\begin{align}\label{G1}
\sum_{n=0}^{\infty}a(n)q^{n}=&\frac{(q^{25};q^{25})_{\infty}^{10}}{(q^{5};q^5)_{\infty}^{12}}\Bigl(\frac{q}{R(q^5)^{8}}+\frac{2q^{2}}{R(q^{5})^{7}}
+\frac{5q^{3}}{R(q^{5})^{6}}+\frac{10q^{4}}{R(q^{5})^{5}}+\frac{20q^{5}}{R(q^{5})^{4}}+\frac{16q^{6}}{R(q^{5})^{3}}\nonumber\\&+\frac{27q^{7}}{R(q^{5})^{2}}
+\frac{20q^{8}}{R(q^{5})}+15q^{9}-20q^{10}R(q^{5})+27q^{11}R(q^{5})^{2}-16q^{12}R(q^{5})^{3}\nonumber\\&+20q^{13}R(q^{5})^{4}-10q^{14}R(q^{5})^{5}
+5q^{15}R(q^{5})^{6}-2q^{16}R(q^{5})^{7}+q^{17}R(q^{5})^{8}\Bigr).
\end{align}
Extracting the terms involving $q^{5n+4}$ in \eqref{G1}, dividing by $q^4$ and replacing $q^5$ by $q$, we obtain
\begin{align}\label{G2}
\sum_{n=0}^{\infty}a(5n+4)q^{n}=&\frac{(q^{5};q^{5})_{\infty}^{10}}{(q;q)_{\infty}^{12}}
\Bigl(\frac{10}{R(q)^{5}}+15q-10q^2R(q)^{5}\Bigr)\nonumber\\
=&\frac{(q^{5};q^{5})_{\infty}^{10}}{(q;q)_{\infty}^{12}}\Bigl(10\Bigl(\frac{1}{R(q)^{5}}-11q-q^{2}R(q)^{5}\Bigr)+125q\Bigr).
\end{align}
Berndt \cite[Thm.\ 7.4.4]{Ber} proved the following identity:
\begin{equation}\label{G3}
\frac{1}{R(q)^{5}}-11q-q^{2}R(q)^{5}=\frac{(q;q)_{\infty}^{6}}{(q^{5};q^{5})_{\infty}^{6}}.
\end{equation}
Employing \eqref{G3} in \eqref{G2}, we obtain \eqref{Le1}.
\end{proof}
In a similar way, we have the following:
\begin{lemma}\label{Le2}
 Let $\sum\limits_{n=0}^{\infty}b(n)q^{n}=\frac{1}{(q;q)^{2}_{\infty}}$. Then
 \begin{equation*}
  \sum\limits_{n=0}^{\infty}b(5n+3)q^{n}
  =125q\frac{(q^{5};q^{5})^{10}_{\infty}}{(q;q)_{\infty}^{12}}+10\frac{(q^{5};q^{5})_{\infty}^{4}}{(q;q)_{\infty}^{6}}.
\end{equation*}
\end{lemma}
\begin{lemma}\label{Le3}Let  $\sum\limits_{n=0}^{\infty}c(n)q^{n}=(q;q)^{4}_{\infty}$. Then
\begin{equation*}
 \sum\limits_{n=0}^{\infty}c(5n+4)q^{n}=-5(q^{5};q^{5})^{4}_{\infty}.
 \end{equation*}
\end{lemma}
\begin{theorem}Let $\alpha$ be a integer $\geq1$. Then
\begin{equation}\label{G4}
\sum\limits_{n=0}^{\infty}\textnormal{A}_{5}(5^{\alpha}n+5^{\alpha}-2)q^{n}=5^{3\alpha}q\frac{(q^{5};q^{5})_{\infty}^{10}}{(q;q)_{\infty}^{2}}
+5^{\alpha}\Bigl(\frac{25^{\alpha}-(-1)^{\alpha}}{13}\Bigr)(q;q)_{\infty}^{4}(q^5;q^5)_{\infty}^{4}.
\end{equation}
\end{theorem}
\begin{proof}
From \eqref{Df} and Lemma \ref{Le2}, we have
\begin{align*}
\sum\limits_{n=0}^{\infty}\textnormal{A}_{5}(5n+3)q^{n}=&(q;q)_{\infty}^{10}
\Bigl(125q\frac{(q^{5};q^{5})^{10}_{\infty}}{(q;q)_{\infty}^{12}}+10\frac{(q^{5};q^{5})_{\infty}^{4}}{(q;q)_{\infty}^{6}}\Bigr)\\
=&10(q;q)_{\infty}^{4}(q^5;q^5)_{\infty}^{4}+125q\frac{(q^{5};q^{5})_{\infty}^{10}}{(q;q)_{\infty}^{2}},
\end{align*}
which is same as \eqref{G4} with $\alpha=1$. Suppose that \eqref{G4} holds for some $\alpha\geq1$.
From \eqref{G4}, Lemmas \ref{Le1} and \ref{Le3}, we deduce
\begin{align*}
&\sum\limits_{n=0}^{\infty}\textnormal{A}_{5}(5^{\alpha+1}n+5^{\alpha+1}-2)q^{n}\\&
=5^{3\alpha}(q;q)_{\infty}^{10}\Bigl(125q\frac{(q^{5};q^{5})^{10}_{\infty}}{(q;q)_{\infty}^{12}}+10\frac{(q^{5};q^{5})_{\infty}^{4}}{(q;q)_{\infty}^{6}} \Bigr)-5^{\alpha+1}\Bigl(\frac{25^{\alpha}-(-1)^{\alpha}}{13}\Bigr)(q;q)_{\infty}^{4}(q^5;q^5)_{\infty}^{4}\\&
=5^{3\alpha+3}q\frac{(q^{5};q^{5})_{\infty}^{10}}{(q;q)_{\infty}^{2}}
+\Bigl(10\cdot5^{3\alpha}-5^{\alpha+1}\Bigl(\frac{25^{\alpha}-(-1)^{\alpha}}{13}\Bigr)\Bigr)(q;q)_{\infty}^{4}(q^5;q^5)_{\infty}^{4}\\&
=5^{3\alpha+3}q\frac{(q^{5};q^{5})_{\infty}^{10}}{(q;q)_{\infty}^{2}}
+5^{\alpha+1}\Bigl(\frac{25^{\alpha+1}-(-1)^{\alpha+1}}{13}\Bigr)(q;q)_{\infty}^{4}(q^5;q^5)_{\infty}^{4}.
\end{align*}
That is, \eqref{G4} holds for $\alpha+1$. This completes the proof by induction of \eqref{G4}.
\end{proof}
From \eqref{G4}, we have the following congruence relation:
\begin{theorem}For all integers $n\geq0$ and $\alpha\geq1$,
\begin{equation}
\textnormal{A}_{5}(5^{\alpha}n+5^{\alpha}-2)\equiv 0\pmod{5^{\alpha}}.
\end{equation}
\end{theorem}

\section{Acknowledgments}
The author would like to thank the referee for his/her valuable comments. The
author would also like to thank Prof. Chadrashekar Adiga for his advice and guidance.

\begin{thebibliography}{9}

\bibitem{Sir}C. Adiga, B. C. Berndt, S. Bhargava, and G. N. Watson, {\it Chapter 16 of Ramanujan's
Second Notebook: Theta Functions and $q$-Series},  Mem. Amer. Math. Soc., 1985.
\bibitem{Ber}B. C. Berndt, {\it Number Theory in the Spirit of Ramanujan}, Amer. Math. Soc., 2006.
\bibitem{Lin}B. L. S. Lin, Some results on bipartitions with 3-core, {\it J. Number Theory} {\bf 139} (2014), 44--52.
\bibitem{Rama}S. Ramanujan, Collected Papers, Cambridge University Press, Cambridge, 1927.
\bibitem{Wat}G. N. Watson, Ramanujan's Vermutung \"uber Zerf\"allungsanzahlem, {\it J. reine angew. Math.} {\bf 179} (1938), 97--128.
\bibitem{Wat1}G. N. Watson, Theorems stated by Ramanujan (VII): theorems on contuined fractions, { \it J. London Math. Soc.} {\bf 4} (1929), 39--48.
\bibitem{Wat2}G. N. Watson, Theorems stated by Ramanujan (IX): two continued fractions, {\it J. London Math. Soc.} {\bf 4} (1929), 231--237.
\bibitem{Xia}E. X. W. Xia, Arithmetic properties of bipartitions with 3-cores, {\it Ramanujan J.} {\bf 38} (2015), 529--548.
\bibitem{Yao}O. Y. M. Yao, Infinite families of congruences modulo 3 and 9 for bipartitions with 3-cores, {\it Bull. Aust. Math.
Soc.} {\bf 91} (2015), 47--52.
\end{thebibliography}

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\noindent 2010 {\it Mathematics Subject Classification}:
Primary 05A17; Secondary 11P83.

\noindent \emph{Keywords: } congruence, partition,
bipartition,  core partition.

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\vspace*{+.1in}
\noindent
Received  September 5 2016;
revised versions received September 15 2016; September 16 2016;
September 19 2016.
Published in {\it Journal of Integer Sequences},
October 1 2016.

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