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\begin{center}
\vskip 1cm{\LARGE\bf 
On a Ramanujan-Type Congruence for  \\
\vskip .1in
Partition Triples with $5$-Cores
}
\vskip 1cm
\large
N. V. Majid and S. N. Fathima\\
Ramanujan School of Mathematical Science\\
Department of Mathematics\\
Pondicherry University\\
Puducherry \\
India \\
\href{mailto:nvmajidtgi@gmail.com}{\tt nvmajidtgi@gmail.com}\\ 
\href{mailto:dr.fathima.sn@gmail.com}{\tt dr.fathima.sn@gmail.com}
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\vskip .2 in

\begin{abstract}
Let $B_5^{(3)}(n)$ denote the number of partition triples with
$5$-cores. In this short note, we prove a Ramanujan-type congruence
modulo $5^\alpha$ $(\alpha\geq 1)$ for $B_5^{(3)}(n)$ by using recurrence
relations for the coefficients of $B_5^{(3)}(n)$.
\end{abstract}

\section{Introduction}

A {\it partition\/} of a positive integer $n$ is a finite non-increasing sequence of positive integers whose sum equals $n$. Let $p(n)$ be the number of partitions of $n$. The generating function for $p(n)$ is given by
\begin{equation*}
\displaystyle\sum_{n=0}^{\infty} p(n)q^n=\dfrac{1}{(q;q)_\infty},
\end{equation*}
where
$(a;q)_0=1$, $ \displaystyle(a;q)_{n}=\prod_{k=0}^{n-1}(1-aq^{k})$, and $\displaystyle(a;q)_{\infty}=\prod_{k=0}^{\infty}(1-aq^{k})$.

The most inspiring congruences of $p(n)$ discovered by Ramanujan for $n\geq 0$ are
\begin{align*}
&amp;p(5n+4)\equiv 0\pmod{5},\\
&amp;p(7n+5)\equiv 0\pmod{7},\\
&amp;p(11n+6)\equiv 0\pmod{11}.
\end{align*}
Ramanujan's discovery inspired researchers to study the arithmetic properties of the restricted partitions, such as the {\it t-core partition\/}. A {\it Ferrers-Young diagram\/} of a partition $n=\displaystyle \sum_{i=1}^k \lambda_i$, where $\lambda_i\geq\lambda_k$, $i\leq k$, is a pattern of dots, with $\lambda_i$ dots in the $i^{th}$ row. The hook number of a dot is defined by the number of dots directly below, together with the number of dots directly to the right, as well as the dot itself. For a positive integer $t\geq2$, a partition 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 $\lambda=4+3+2$ of $9$ with the  corresponding hook number is as follows:
\begin{align*}
&amp;\bullet^6 \qquad\bullet^5 \qquad\bullet^3 \qquad\bullet^1\\
&amp;\bullet^4 \qquad\bullet^3~\qquad\bullet^1\\
&amp;\bullet^2\qquad\bullet^1
\end{align*}
Here $\lambda$ is $7$-core and $\lambda$ is $t$-core for $t\geq7$.
\end{example}

We let $B_t(n)$ denote the number of {\it $t$-core partitions\/} of $n$. The generating function for the $B_t(n)$ is given by \cite[ Eq. (2.1)]{Gar}  
\begin{equation*}
\sum_{n=0}^{\infty} B_t(n)q^{n}=\dfrac{f_t^{t}}{f_1},
\end{equation*}
where $f_t=(q^t;q^t)_\infty$, for any integer $t\geq 2$. Arithmetic properties of $t$-core partitions have been studied by several mathematicians (see, for example \cite{BN,BOY,CHE,DAI,HS,Lin,Xia,Yao}). 

A {\it partition triple} or a {\it tripartition} of $n$ is a triple of partitions $(\lambda_1,\lambda_2,\lambda_3)$ such that the sum of all the parts of $\lambda_1$, $\lambda_2$, and $\lambda_3$ equals $n$. A tripartition with $t$-core of $n$ is a partition triples $(\lambda_1,\lambda_2,\lambda_3)$ of $n$ such that $\lambda_1$, $\lambda_2$, and $\lambda_3$ are $t$-cores. Let $B_t^{(3)}(n)$ denote the number of partition triples  with $t$-cores of $n$. The generating function for $B_t^{(3)}(n)$ is given by
\begin{equation}\label{Df}
\sum_{n=0}^{\infty} B_t^{(3)}(n)q^{n}=\frac{f_t^{3t}}{f_1^{3}}.
\end{equation}

 Dasappa \cite{DAS} proved a Ramanujan-type congruence modulo $5^\alpha$ for bipartition with $5$-cores, which motivated establishing a Ramanujan-type congruence for tripartitions with $5$-core.

The ultimate aim of this note is to prove the following Ramanujan-type congruence for $B_5^{(3)}(n)$:

\begin{equation*}
 B_5^{(3)}(5^{\alpha}n+5^{\alpha}-3)\equiv0\pmod{5^{\alpha}}, ~~\alpha\geq1.
 \end{equation*}

 The following are crucial lemmas that help prove our main congruence for $B_5^{(3)}(n)$:

\begin{lemma}\label{l1}
Let $\displaystyle\sum_{n=0}^\infty P_4(n)q^n=q f_5^9 f_1^3 $. Then 
\begin{equation}\label{4}
\displaystyle\sum_{n=0}^\infty P_4(5n+4)q^n=5 f_5^3 f_1^9.
\end{equation}
\end{lemma}

\begin{proof}
The following $5$-dissection formula was first stated by Ramanujan \cite[p.\ 212]{Rama} without any proof:
\begin{equation}\label{5D}
f_1=f_{25}\Bigl(\frac{1}{R(q^{5})}-q-q^{2}R(q^{5})\Bigr),
\end{equation}
where $R(q)=\frac{(q;q^{5})_{\infty}(q^{4};q^{5})_{\infty}}{(q^{2};q^{5})_{\infty}(q^{3};q^{5})_{\infty}}$.
Watson \cite{Wat} provided a proof of \eqref{5D}.

Using (\ref{5D}), we obtain
\begin{equation}\label{9}
\displaystyle \sum_{n=0}^\infty P_4(n)q^n= f_5^9 f_{25}^3\Bigl(\frac{q}{R(q^5)^3}-\frac{3q^2}{R(q^5)^2}+5q^4-3q^6 R(q^5)^2-q^7 R(q^5)^3\Bigr).
\end{equation}

Extracting the terms involving $q^{5n+4}$ from (\ref{9}), dividing by $q^4$, and replacing $q^5$ by $q$, we obtain (\ref{4}). 
\end{proof}

\begin{lemma}\label{l2}
Let $\displaystyle\sum_{n=0}^\infty P_5(n)q^n=f_5^3 f_1^9$. Then 
\begin{equation*}
\displaystyle\sum_{n=0}^\infty P_5(5n+4)q^n=-90 f_5^3 f_1^9-625 q f_5^9 f_1^3. 
\end{equation*}
\end{lemma}
\begin{proof}
Using (\ref{5D}), we obtain
\begin{align}
\displaystyle \sum_{n=0}^\infty P_5(n)q^n&amp;= f_5^3 f_{25}^9\Bigl(\frac{1}{R(q^5)^{9}}-\frac{9q}{R(q^5)^{8}}+\frac{27q^2}{R(q^5)^{7}}-\frac{12q^3}{R(q^5)^{6}}-\frac{90q^4}{R(q^5)^{5}}+\frac{126q^5}{R(q^5)^{4}}+\frac{126q^6}{R(q^5)^{3}}\nonumber\\&amp;-\frac{288q^7}{R(q^5)^{2}}-\frac{117q^8}{R(q^5)}+365q^6+117q^{10} R(q^5)-288q^{11}R(q^5)^{2}-126q^{12}R(q^5)^{3}\nonumber\\&amp;+126q^{13}R(q^5)^{4}+90q^{14}R(q^5)^{5}-12q^{15}R(q^5)^{6}-27q^{16}R(q^5)^{7}-9q^{17}R(q^5)^{8}\nonumber\\&amp;-q^{18}R(q^5)^{9}\Bigr).\label{5}
\end{align}

Extracting the terms involving $q^{5n+4}$ from (\ref{5}), dividing by $q^4$, and replacing $q^5$ by $q$, we obtain 
\begin{equation}\label{6}
\displaystyle \sum_{n=0}^\infty P_5(5n+4)q^n= f_5^9 f_1^3 \Bigl(90 q^2 {R(q)}^5+365q-\frac{90}{{R(q)}^5}\Bigr).
\end{equation}

Berndt \cite[Thm.\ 7.4.4]{Ber} gave 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 (\ref{G3}) in (\ref{6}), we obtain
\begin{equation*}
\displaystyle \sum_{n=0}^\infty P_5(5n+4)q^n=f_5^9 f_1^3\Bigl(-90\frac{f_1^6}{f_5^6}-625q\Bigr)=-90f_5^3f_1^9-625q f_5^9 f_1^3.
\end{equation*}
\end{proof}

\begin{lemma}\label{l3} Let $\displaystyle\sum_{n=0}^\infty P_3(n)q^n=\dfrac{1}{f_1^3}$. Then
\begin{equation*}
\displaystyle\sum_{n=0}^\infty P_3(5n+2)q^n=9\dfrac{f_5^3}{f_1^6}+375 q\dfrac{f_5^9}{f_1^{12}}+3125 q^2 \dfrac{f_5^{15}}{f_1^{18}}.
\end{equation*} 
\end{lemma}

\begin{proof}
 \cite[Eq.\ (7.4.14), p.\ 165]{Ber} We have
\begin{align}\label{5D1}
\dfrac{1}{f_1}=&amp;\dfrac{f_{25}^5}{f_5^6}\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\\&amp;+2q^{6}R(q^{5})^{2}-q^{7}R(q^{5})^{3}+q^{8}R(q^{5})^{4}\Bigr).
\end{align}

Using (\ref{5D1}), we obtain
\begin{align}\label{2}
\sum_{n=0}^\infty P_3(n)q^n&amp;= \frac{f_{25}^{15}}{f_5^{18}}\Bigl(\frac{1}{{R(q^5)}^{12}}+\frac{3q}{{R(q^5)}^{11}}+\frac{9q^2}{{R(q^5)}^{10}}+\frac{22q^3}{{R(q^5)}^{9}}+\frac{51q^4}{{R(q^5)}^{8}}+\frac{78q^5}{{R(q^5)}^{7}}+\frac{134 q^6}{{R(q^5)}^{6}}\nonumber\\&amp;+\frac{177q^7}{{R(q^5)}^{5}}+\frac{216q^8}{{R(q^5)}^{4}}+\frac{153q^9}{{R(q^5)}^{3}}+\frac{219q^{10}}{{R(q^5)}^{2}}+\frac{57q^{11}}{R(q^5)}+71q^{12}-57q^{13}R(q^5)\nonumber\\&amp;+219q^{14}{R(q^5)}^{2}-153q^{15}R(q^5)^{3}+216q^{16}R(q^5)^{4}-177q^{17}R(q^5)^{5}+134q^{18}R(q^5)^{6}\nonumber\\&amp;-78q^{19}R(q^5)^{7}+51q^{20}R(q^5)^{8}-22q^{21}R(q^5)^{9}+9q^{22}R(q^5)^{10}-3q^{23}R(q^5)^{11}\nonumber\\&amp;+q^{24}R(q^5)^{12}\Bigr).
\end{align}

Extracting the terms involving $q^{5n+2}$ from (\ref{2}), dividing by $q^2$, and replacing $q^5$ by $q$, we obtain
\begin{equation}\label{3}
\displaystyle\sum_{n=0}^\infty P_3(5n+2)q^n= \frac{f_5^{15}}{f_1^{18}}\Bigl(9q^4 R(q)^{10}-177q^3 R(q)^5+71q^2+177\frac{q}{R(q)^5}+\frac{9}{R(q)^{10}}\Bigr). 
\end{equation}

Employing (\ref{G3}) in (\ref{3}), we deduce that
\begin{align*}
\displaystyle\sum_{n=0}^\infty P_3(5n+2)q^n&amp;= \frac{f_5^{15}}{f_1^{18}}\Bigl( 9\frac{f_1^{12}}{f_5^{12}}+375q\frac{f_1^6}{f_5^6}+3125q^2\Bigr)\\
&amp;=9\frac{f_5^3}{f_1^6}+375q\frac{f_5^9}{f_1^{12}}+3125q^2\frac{f_5^{15}}{f_1^{18}}.
\end{align*}
\end{proof}

\begin{theorem}
For all integers $\alpha\geq 0$, we have
\begin{equation}\label{8}
\sum_{n=0}^\infty B_5^{(3)}(5^{\alpha+1}n+5^{\alpha+1}-3)q^n=A_\alpha f_5^3 f_1^9+ B_\alpha q f_5^9 f_1^3+C_\alpha q^2 \sum_{n=0}^\infty B_5^{(3)}(n)q^n,
\end{equation}
where $A_0=9$, $B_0=375$, $C_0=3125$, and for any integer $n\geq 1$
\begin{align}
A_n &amp;= -90 A_{n-1}+5 B_{n-1}+9 C_{n-1},\label{a}\\
B_n &amp;= -625 A_{n-1}+5 B_{n-1}+375 C_{n-1},\label{b}\\
C_n &amp;= C_0^{n+1}.\label{c}
\end{align}
\end{theorem}

\begin{proof}
From (\ref{Df}), we have
\begin{equation*}
\displaystyle\sum_{n=0}^\infty B_5^{(3)} (n)=\frac{f_5^{15}}{f_1^3}.
\end{equation*}

Employing Lemma \ref{l3}, we obtain
\begin{equation}\label{7}
\displaystyle\sum_{n=0}^\infty B_5^{(3)}(5n+2)q^n=9 f_5^3 f_1^9+375 q f_5^9f_1^3+3125 q^2 \displaystyle\sum_{n=0}^\infty B_5^{(3)}(n)q^n.
\end{equation}

Eq.~(\ref{7}) is the $\alpha=0$ case of Eq.~(\ref{8}). Now assume for $\alpha\geq 0$. Replacing $n$ by $5n+4$ in (\ref{8}), using Lemmas (\ref{l1}) and (\ref{l2}), and (\ref{7}), we obtain
\begin{align*}
&amp;\sum_{n=0}^\infty B_5^{(3)}(5^{\alpha+2}n+5^{\alpha+2}-3)q^n\\&amp;=A_\alpha \Bigl(-90f_5^3f_1^9-625qf_5^9f_1^3\Bigr)+B_\alpha 5f_5^3f_1^9+C_\alpha\Bigl(9f_5^3f_1^9+375qf_5^9f_1^3+3125q^2\sum_{n=0}^\infty B_5^{(3)}(n)q^n\Bigr)\\
&amp;=\bigl(-90A_\alpha+5B_\alpha+9C_\alpha\Bigr)f_5^3 f_1^9+\Bigl(-625A_\alpha+375C_\alpha\Bigr)f_5^9 f_1^3+3125C_\alpha q^2 \sum_{n=0}^\infty B_5^{(3)}(n)q^n\\
&amp;=A_{\alpha+1} f_5^3 f_1^9+ B_{\alpha+1} q f_5^9 f_1^3+C_{\alpha+1} q^2 \sum_{n=0}^\infty B_5^{(3)}(n)q^n.
\end{align*} That is, (\ref{8}) holds for $\alpha+1$. This completes the proof by induction of (\ref{8}).
\end{proof}

\begin{theorem} For all integers $\alpha\geq 1$ and $n\geq 0$, we have
\begin{equation}\label{M}
B_5^{(3)}(5^{\alpha}n+5^{\alpha}-3)\equiv0\pmod{5^{\alpha}}.
 \end{equation}
\end{theorem}

\begin{proof}
From (\ref{a}), (\ref{b}), and (\ref{c}), we see that
\begin{center}
\begin{tabular}{ccc}
$A_1\equiv0 \pmod{5}$, &amp; $B_1\equiv0 \pmod{5^3}$, &amp; $C_1\equiv0 \pmod{5^5}$,\\
$A_2\equiv0 \pmod{5^2}$, &amp; $B_2\equiv0 \pmod{5^4}$, &amp; $C_2\equiv0 \pmod{5^{10}}$,\\
\vdots &amp; \vdots &amp; \vdots\\
$A_\alpha\equiv0 \pmod{5^\alpha}$, &amp; $B_\alpha\equiv0 \pmod{5^{\alpha+2}}$, &amp; $C_\alpha\equiv0 \pmod{5^{5\alpha}}$.
\end{tabular}
\end{center}

That is, $A_\alpha$, $B_\alpha$, and $C_\alpha$ are multiples of $5^\alpha$. Then (\ref{8}) implies (\ref{M}).
\end{proof} 

\section{Acknowledgments}
The first author is thankful to CSIR-UGC Junior Research Fellowship, Government of India, Council of Scientific \&amp; Industrial Research, CSIR Complex, Library Avenue, Pusa, New Delhi-110012. We would also like to extend our gratitude to the referee from the \textit{Journal of Integer Sequences} for careful reading and comments on the manuscript.

\begin{thebibliography}{99}

\bibitem{BN} N. D. Baruah and K. Nath, Some results on 3-cores, {\it Proc. Amer. Math. Soc.} {\bf 142} (2014), 441--448.

\bibitem{Ber} B. C. Berndt, {\it Number Theory in the Spirit of Ramanujan}, Amer. Math. Soc., 2016.

\bibitem{BOY} M. Boylan, Congruences for $2^t$-core partition functions, {\it J. Number Theory} {\bf 92} (2002), 131--138.

\bibitem{CHE} S. C. Chen, Congruences for $t$-core partitions functions, {\it J. Number Theory} {\bf 133} (2013), 4036--4046.

\bibitem{DAI} H. B. Dai, Arithmetic of $3^t$-core partition functions, {\it Integers} {\bf 15} (2015), A7.

\bibitem{DAS} R. Dasappa, On a Ramanujan-type congruence for bipartition with 5-cores, {\it J. Integer Sequences} {\bf 19} (2016), Article 16.8.1.

\bibitem{Gar} F. Garvan, D. Kim, and D. Stanton, Cranks and $t$-cores, {\it Invent. Math.} {\bf 101} (1990), 1--17.

\bibitem{HS} M. D. Hirschhorn and J. A. Sellers, Elementary proofs of facts about 3-cores, {\it Bull. Aust. Math. Soc} {\bf 79} (2009), 507--512.

\bibitem{Lin} B. S. Lin, Some results on bipartitions with $t$-core, {\it J. Number Theory} {\bf 139} (2014), 44--52.

\bibitem{Rama} S. Ramanujan, {\it 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{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 2020 {\it Mathematics Subject Classification}:
Primary 05A17; Secondary 11P83.

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

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\vspace*{+.1in}
\noindent
Received December 29 2021;
revised versions received  December 30 2021; June 21 2022; June 23 2022.
Published in {\it Journal of Integer Sequences}, June 23 2022.

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