\documentclass[12pt,reqno]{article} \usepackage[usenames]{color} \usepackage{amssymb} \usepackage{graphicx} \usepackage{amscd} \usepackage[colorlinks=true, linkcolor=webgreen, filecolor=webbrown, citecolor=webgreen]{hyperref} \definecolor{webgreen}{rgb}{0,.5,0} \definecolor{webbrown}{rgb}{.6,0,0} \usepackage{color} \usepackage{fullpage} \usepackage{float} \usepackage{psfig} \usepackage{graphics,amsmath,amssymb} \usepackage{amsthm} \usepackage{amsfonts} \usepackage{latexsym} \usepackage{epsf} \setlength{\textwidth}{6.5in} \setlength{\oddsidemargin}{.1in} \setlength{\evensidemargin}{.1in} \setlength{\topmargin}{-.1in} \setlength{\textheight}{8.4in} \newcommand{\seqnum}[1]{\href{http://oeis.org/#1}{\underline{#1}}} \begin{document} \begin{center} \epsfxsize=4in \leavevmode\epsffile{logo129.eps} \end{center} \theoremstyle{plain} \newtheorem{theorem}{Theorem} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \newtheorem{conjecture}[theorem]{Conjecture} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \begin{center} \vskip 1cm{\LARGE\bf New Congruences for Broken $k$-Diamond \\ \vskip .05in Partitions } \vskip 1cm \large Dazhao Tang\\ College of Mathematics and Statistics\\ Huxi Campus\\ Chongqing University\\ Chongqing---401 331\\ PR China\\ \href{mailto:dazhaotang@cqu.edu.cn}{\tt dazhaotang@cqu.edu.cn} \\ \end{center} \vskip .2 in \begin{abstract} Andrews and Paule introduced a new class of directed graphs, called broken $k$-diamond partitions. Let $\Delta_{k}(n)$ denote the number of broken $k$-diamond partitions of $n$ for a fixed positive integer $k$. In this paper, we establish new infinite families of congruences modulo 5, 7, 25 and 49 for $\Delta_{k}(n)$ via a standard $q$-series technique and modular forms. \end{abstract} \section{Introduction}\label{sec1} In 2007, Andrews and Paule \cite{AP2007} introduced a new class of directed graphs, called \textit{broken $k$-diamond partitions}. They proved that the generating function of $\Delta_{k}(n)$, the number of broken $k$-diamond partitions of $n$, is given by \begin{align*} \sum_{n=0}^{\infty}\Delta_{k}(n)q^{n}=\dfrac{(q^{2};q^{2})_{\infty}(q^{2k+1};q^{2k+1})_{\infty}}{(q;q)_{\infty}^3(q^{2(2k+1)};q^{2(2k+1 )})_{\infty}}. \end{align*} Here and in the sequel, we assume $|q|<1$ and adopt the following customary notation on $q$-series and partitions: \begin{align*} (a;q)_{\infty} &:=\prod_{n=0}^{\infty}(1-aq^{n}). \end{align*} The following two congruences modulo 5 were subsequently proved by Chan \cite{Chan2008}, Radu \cite{Rad2015}, and Hirschhorn \cite{Hir2017}: \begin{align*} \Delta_{2}(25n+14) &\equiv0\pmod{5},\\ \Delta_{2}(25n+24) &\equiv0\pmod{5}. \end{align*} Moreover, a number of infinite families of congruences modulo 5 satisfied by $\Delta_{2}(n)$ have been proved. See, for example, Chan \cite{Chan2008}, Hirschhorn \cite{Hir2017}, Radu \cite{Rad2015}, and Xia\cite{Xia2017}. On the other hand, Jameson\cite{Jam2013} and Xia\cite{Xia2015} proved the following congruences modulo 7 enjoyed by $\Delta_{3}(n)$, which were conjectured by Paule and Radu \cite{PR2010}: \begin{align*} \Delta_{3}(343n+82) &\equiv0\pmod{7},\\ \Delta_{3}(343n+229) &\equiv0\pmod{7},\\ \Delta_{3}(343n+278) &\equiv0\pmod{7},\\ \Delta_{3}(343n+327) &\equiv0\pmod{7}. \end{align*} Quite recently, a variety of infinite families of congruences modulo 7 enjoyed by $\Delta_{3}(n)$ also have been found. See, for example, Xia \cite{Xia2015}, Yao, and Wang \cite{YW2016}. In this paper, we establish two infinite families of congruences modulo 5 and 25 for $\Delta_{k}(n)$ as follows: \begin{theorem}\label{THM:mod5} For all $n\geq0$, \begin{align} \Delta_{k}(25n+24) &\equiv0\pmod{5}, \quad \emph{if}\quad k\equiv12\pmod{25}.\label{con:mod5} \end{align} \end{theorem} \begin{theorem}\label{THM:mod25} For all $n\geq0$, \begin{align} \Delta_{k}(125n+99) &\equiv0\pmod{25}, \quad \emph{if}\quad k\equiv62\pmod{125}.\label{con:mod25} \end{align} \end{theorem} Moreover, we obtain the following infinite families of congruences modulo 7 and 49 for $\Delta_{k}(n)$. \begin{theorem}\label{THM:mod7} For all $n\geq0$, \begin{align} \Delta_{k}(49n+s)\equiv0\pmod{7}, \quad \emph{if}\quad k\equiv24\pmod{49}\label{con:mod7}, \end{align} where $s=19, 33, 40$, and $47$. \end{theorem} \begin{theorem}\label{THM:mod49} For all $n\geq0$, \begin{align} \Delta_{k}(343n+t)\equiv 0\pmod{49}, \quad \emph{if}\quad k\equiv171\pmod{343},\label{con:mod49} \end{align} where $t=96, 292$, and $341$. \end{theorem} \section{Proofs of Theorems \ref{THM:mod5}--\ref{THM:mod49}} \subsection{The case mod 5} To prove \eqref{con:mod5}, we collect some useful identities. Recall that the Ramanujan theta function $f(a,b)$ is defined by \begin{align} f(a,b):=\sum_{n=-\infty}^{\infty}a^{n(n+1)/2}b^{n(n-1)/2},\label{Ramanujan theta func} \end{align} where $|ab|<1$. The Jacobi triple product identity can be stated as \begin{align} f(a,b)=(-a,-b,ab;ab)_{\infty}.\label{JTP} \end{align} One specialization of \eqref{Ramanujan theta func} is given by \cite{Ber}: \begin{align*} \psi(q) &:=f(q,q^3)=\sum_{n=0}^{\infty}q^{n(n+1)/2}. \end{align*} According to \eqref{JTP}, we have \begin{align*} \psi(q)=\dfrac{(q^{2};q^{2})_{\infty}^2}{(q;q)_{\infty}}. \end{align*} The following two identities are given by Hirschhorn \cite{Hir2017,Hir} and Berndt \cite{BerIII}. \begin{lemma} \begin{align} \psi(q) &=a+qb+q^3c,\label{psi dissec 1}\\ \psi^{2}(q^{5}) &=ab+q^{5}c^{2}.\label{psi dissec 2} \end{align} where \begin{align*} a=f(q^{10},q^{15}),\quad b=f(q^{5},q^{20}),\quad c=\psi(q^{25}). \end{align*} \end{lemma} Eq.~\eqref{psi dissec 1} comes from \cite[Eq.\ (2.1)]{Hir2017}, \cite[Entry 10(i), p.\ 262]{BerIII} and Eq.~\eqref{psi dissec 2} comes from \cite[Eq.\ (34.1.21), p.\ 313]{Hir}. Employing the binomial theorem, we can easily establish the following congruence, which will be frequently used without explicit mention. \begin{lemma}\label{cong lemma} If $p$ is a prime, $\alpha$ is a positive integer, then \begin{align*} (q^{\alpha};q^{\alpha})_{\infty}^{p} &\equiv(q^{p\alpha};q^{p\alpha})\pmod{p},\\ (q;q)_{\infty}^{p^{\alpha}} &\equiv(q^{p};q^{p})^{p^{\alpha-1}}\pmod{p^{\alpha}}. \end{align*} \end{lemma} Now, we are ready to state the proof of \eqref{con:mod5}. Notice that $k\equiv12\pmod{25}$, then $2k+1\equiv0\pmod{25}$. Let $2k+1=25j$, where $j$ is a positive integer, we get, (all the following congruences are modulo 5) \begin{align*} \sum_{n=0}^{\infty}\Delta_{k}(n)q^{n} &=\dfrac{(q^{2};q^{2})_{\infty}(q^{25j};q^{25j})_{\infty}}{(q;q)_{\infty}^{3}(q^{50j};q^{50j})_{\infty}}=\dfrac{\psi^{3}(q)}{(q^{2};q^{2})^{5}} \dfrac{(q^{25j};q^{25j})_{\infty}}{(q^{50j};q^{50j})_{\infty}}\\ &\equiv\dfrac{\psi^{3}(q)}{(q^{10};q^{10})_{\infty}}\dfrac{(q^{25j};q^{25j})_{\infty}}{(q^{50j};q^{50j})_{\infty}}. \end{align*} Invoking \eqref{psi dissec 1}, we obtain \begin{align*} \sum_{n=0}^{\infty}\Delta_{k}(n)q^{n} &\equiv\dfrac{(q^{25j};q^{25j})_{\infty}}{(q^{10};q^{10})_{\infty}(q^{50j};q^{50j})_{\infty}}\times\Bigg(a^{3}+3qa^{2}b+3q^{2}ab^{2}\\ &\quad\quad+q^{3}b^{3}+3q^{3}a^{2}c+6q^{4}abc+3q^{5}b^{2}c+3q^{6}ac^{2}+3q^{7}bc^{2}+q^{9}c^{3}\Bigg). \end{align*} Extracting those terms involving the powers $q^{5n+4}$ and combining \eqref{psi dissec 2}, we have \begin{align*} \sum_{n=0}^{\infty}\Delta_{k}(5n+4)q^{5n} &\equiv\dfrac{\left(abc+q^{5}c^{3}\right)(q^{25j};q^{25j})_{\infty}}{(q^{10};q^{10})_{\infty}(q^{50j};q^{50j})_{\infty}} =\dfrac{c(ab+q^{5}c^{2})(q^{25j};q^{25j})_{\infty}}{(q^{10};q^{10})_{\infty}(q^{50j};q^{50j})_{\infty}}\\ &=\psi(q^{25})\psi^{2}(q^{5})\dfrac{(q^{25j};q^{25j})_{\infty}}{(q^{10};q^{10})_{\infty}(q^{50j};q^{50j})_{\infty}}, \end{align*} then \begin{align*} \sum_{n=0}^{\infty}\Delta_{k}(5n+4)q^{n} &\equiv\psi(q^{5})\psi^{2}(q)\dfrac{(q^{5j};q^{5j})_{\infty}}{(q^{2};q^{2})_{\infty}(q^{10j};q^{10j})_{\infty}}=\psi(q^{5})\dfrac{(q^{2};q^{2})_{\infty}^{3}}{(q;q)_{\infty} ^{2}}\dfrac{(q^{5j};q^{5j})_{\infty}}{(q^{10j};q^{10j})_{\infty}}\\ &\equiv\psi(q^{5})\dfrac{(q^{5j};q^{5j})_{\infty}}{(q^{10j};q^{10j})_{\infty}}\dfrac{(q;q)_{\infty}^{3}(q^{2};q^{2})_{\infty}^{3} }{(q^{5};q^{5})_{\infty}}:=\sum_{n=0}^{\infty}a(n)q^{n}, \end{align*} say. Thanks to Jacobi's identity \cite[Theorem 1.3.9, p.\ 14]{Ber} \begin{equation*} (q;q)_{\infty}^3=\sum_{n=0}^\infty(-1)^n(2n+1)q^{n(n+1)/2}, \end{equation*} it follows that \begin{align} (q;q)_{\infty}^3 &=J_{0}+J_{1}+J_{3},\nonumber\\ (q^{2};q^{2})_{\infty}^{3} &=J^{*}_{0}+J^{*}_{1}+J^{*}_{2},\label{2 JT} \end{align} where $J_{i}$ (resp.\ $J^{*}_{i}$) consists of those terms in which the power of $q$ is congruent to $i$ modulo 5. Furthermore, we see that $J_{3}\equiv 0\pmod{5}$ and $J^{*}_{1}\equiv0\pmod{5}$, so \begin{align*} (q;q)_\infty^3 &\equiv J_0+J_1 \pmod{5},\\ (q^{2};q^{2})_{\infty}^{3} &\equiv J^{*}_{0}+J^{*}_{2}\pmod{5}. \end{align*} Therefore, \begin{align*} (q;q)_{\infty}^3(q^{2};q^{2})_{\infty}^{3} \equiv\left(J_{0}+J_{1}\right)\left(J^{*}_{0}+J^{*}_{2}\right)\pmod{5}, \end{align*} which contains no terms of the form $q^{5n+4}$. Hence $a(5n+4)\equiv0\pmod{5}$, equivalently, \begin{align*} \Delta_{k}(25n+24)\equiv0\pmod{5}, \end{align*} as desired. To prove the remaining congruences \eqref{con:mod25}--\eqref{con:mod49}, we need to use a result of Radu and Sellers \cite[Lemma 2.4]{RS2011}. Before introducing the result of Radu and Sellers, we will briefly interpret some definitions and notation. For a positive integer $M$, let $R(M)$ be the set of integer sequences $\{r:r=(r_{\delta_{1}},\ldots,r_{\delta_{k}})\}$ indexed by the positive divisors $1=\delta_{1}<\cdots<\delta_{k}=M$ of $M$. For some $r\in R(M)$, define \begin{align*} f_r(q):=\prod_{\delta\mid M}(q^{\delta};q^{\delta})_{\infty}^{r_\delta}=\sum_{n=0}^{\infty}c_r(n)q^n. \end{align*} Given a positive integer $m$. Let $\mathbb{Z}^{*}_{m}$ be the set of all invertible elements in $\mathbb{Z}_{m}$, and $\mathbb{S}_{m}$ be the set of all squares in $\mathbb{Z}^{*}_{m}$. We also define the set \begin{align*} P_{m,r}(t):=\left\{t'\ \bigg|\ t'\equiv ts+\frac{s-1}{24}\sum_{\delta\mid M}\delta r_{\delta}\pmod{m},\quad0\le t'\le m-1,\quad[s]_{24m}\in\mathbb{S}_{24m}\right\}, \end{align*} where $t \in \{0,\ldots,m-1\}$ and $[s]_{m}=s+m\mathbb{Z}$. Let $\Gamma:=SL_2(\mathbb{Z})$ and $\Gamma_\infty:=\left\{\left.\begin{pmatrix}1 & h \\0 & 1 \end{pmatrix}\ \right|\ h\in\mathbb{Z}\right\}$. For a positive integer $N$, we define the \textit{congruence subgroup} of level $N$ as \begin{align*} \Gamma_0(N):=\left\{\left.\begin{pmatrix}a & b\\c & d\end{pmatrix}\in\Gamma\ \right|\ c\equiv 0\pmod{N}\right\}. \end{align*} The index of $\Gamma_0(N)$ in $\Gamma$ is given by \begin{align*} [\Gamma:\Gamma_0(N)]=N\prod_{p\mid N}\left(1+\dfrac{1}{p}\right), \end{align*} where the product runs through the distinct primes dividing $N$. Let $\kappa=\kappa(m)=\gcd(m^2-1,24)$ and denote $\Delta^*$ by the set of tuples $(m,M,N,t,r=(r_\delta))$ satisfying conditions given in \cite[p.\ 2255]{RS2011}, we set \begin{align*} p_{m,r}(\gamma)=\min_{\lambda\in\{0,\ldots,m-1\}}\frac{1}{24}\sum_{\delta\mid M}r_\delta\frac{\gcd^2(\delta(a+\kappa\lambda c),mc)}{\delta m} \end{align*} and \begin{align*} p_{r'}^*(\gamma)=\frac{1}{24}\sum_{\delta\mid N} \frac{r'_\delta\gcd^2(\delta,c)}{\delta}, \end{align*} where $\gamma=\begin{pmatrix}a & b \\c & d \end{pmatrix}$, $r\in R(M)$, and $r'\in R(N)$. The lemma of Radu and Sellers is stated as follows: \begin{lemma}\label{le:RS} Let $u$ be a positive integer, $(m,M,N,t,r=(r_\delta))\in\Delta^*$, $r'=(r'_\delta)\in R(N)$, $n$ be the number of double cosets in $\Gamma_0(N)\backslash\Gamma/\Gamma_\infty$ and $\{\gamma_1,\ldots,\gamma_n\}$ $\subset\Gamma$ be a complete set of representatives of the double coset $\Gamma_0(N)\backslash\Gamma/\Gamma_\infty$. Assume that $p_{m,r}(\gamma_i)+p_{r'}^*(\gamma_i)\ge 0$ for all $i=1,\ldots,n$. Let $t_{\min} := \min_{t'\in P_{m,r}(t)}t'$ and \begin{align*} v:=\frac{1}{24}\left(\left(\sum_{\delta\mid M}r_\delta+\sum_{\delta\mid N}r'_\delta\right)[\Gamma:\Gamma_0(N)]-\sum_{\delta\mid N}\delta r'_\delta\right)-\frac{1}{24m}\sum_{\delta\mid M}\delta r_\delta-\frac{t_{\min}}{m}. \end{align*} Then if \begin{align*} \sum_{n=0}^{\lfloor v \rfloor}c_r(mn+t')q^n\equiv 0 \pmod{u} \end{align*} for all $t'\in P_{m,r}(t)$, then \begin{align*} \sum_{n=0}^{\infty}c_r(mn+t')q^n\equiv 0 \pmod{u} \end{align*} for all $t'\in P_{m,r}(t)$. \end{lemma} \subsection{The cases mod 25}\label{subsec:mod25} Let \begin{align*} \sum_{n=0}^{\infty}b(n)q^n=\frac{(q^{2};q^{2})_{\infty}}{(q;q)_{\infty}^{3}}. \end{align*} By Lemma \ref{cong lemma}, we obtain \begin{align} \sum_{n=0}^{\infty}b(n)q^n =\frac{(q^{2};q^{2})_{\infty}}{(q;q)_{\infty}^{3}}\equiv \frac{(q;q)_{\infty}^{22}(q^{2};q^{2})_{\infty}}{(q^{5};q^{5})_{\infty}^5}\pmod{25}.\label{cong:mod25equlity} \end{align} In this case, we may take \begin{align*} (m,M,N,t,r=(r_1,r_2,r_{5},r_{10}))=(125,10,10,99,(22,1,-5,0))\in\Delta^*. \end{align*} By the definition of $P_{m,r}(t)$, we have $P_{m,r}(t)=\{99\}$. Now we can choose \begin{align*} r'=(r'_1,r'_2,r'_5,r'_{10})=(13, 0, 0, 0). \end{align*} Let \begin{align*} \gamma_\delta=\begin{pmatrix}1 & 0 \\ \delta & 1 \end{pmatrix}. \end{align*} Radu and Sellers \cite[Lemma 2.6]{RS2011} also proved that $\{\gamma_\delta:\delta\mid N\}$ contains a complete set of representatives of the double coset $\Gamma_0(N)\backslash\Gamma/\Gamma_\infty$. One readily verifies that all assumptions of Lemma \ref{le:RS} are satisfied. Furthermore we obtain the upper bound $\lfloor v \rfloor=21$. For all congruences in \eqref{cong:mod25}, we check that they hold for $n$ from 0 to their corresponding upper bound $\lfloor v \rfloor$ via \emph{Mathematica}. It follows by Lemma \ref{le:RS} and \eqref{cong:mod25equlity} that \begin{align} b(125n+99) &\equiv0\pmod{25}\label{cong:mod25} \end{align} holds for all $n\geq0$. When $k\equiv62\pmod{125}$, i.e., $2k+1\equiv0\pmod{125}$, \eqref{con:mod25} is a direct consequence of \eqref{cong:mod25}. \subsection{The cases mod 7} Firstly, we have \begin{align*} \sum_{n=0}^{\infty}b(n)q^n=\frac{(q^{2};q^{2})_{\infty}}{(q;q)_{\infty}^{3}}\equiv \frac{(q;q)_{\infty}^{4}(q^{2};q^{2})_{\infty}}{(q^{7};q^{7})_{\infty}}\pmod{7}. \end{align*} To prove \eqref{con:mod7}, it suffices to show \begin{align} b(49n+s) &\equiv0\pmod{7}\label{cong:mod7} \end{align} for $s=19, 33, 40$, and 47. We first show the cases $s=19, 33$, and 40. Taking \begin{align*} (m,M,N,t,r=(r_1,r_2,r_{7},r_{14}))=(49,14,14,33,(4,1,-1,0))\in\Delta^*. \end{align*} We compute that $P_{m,r}(t)=\{19,33,40\}$. Now we can choose \begin{align*} r'=(r'_1,r'_2,r'_7,r'_{14})=(3, 0, 0, 0). \end{align*} and taking $\gamma$ as in Subsection \ref{subsec:mod25}, we verify that all these constants satisfy the assumption of Lemma \ref{le:RS}. We thus obtain $\lfloor v \rfloor=6$. With the help of \emph{Mathematica}, we see that \eqref{cong:mod7} holds up to the bound $\lfloor v \rfloor$ with $t\in\{19, 33, 40\}$, and therefore it holds for all $n\geq0$ by Lemma \ref{le:RS}. Now we will turn to the case $s=47$. Again we take \begin{align*} (m,M,N,t,r=(r_1,r_2,r_{7},r_{14}))=(49,14,14,47,(4,1,-1,0))\in\Delta^* \end{align*} and \begin{align*} r'=(r'_1,r'_2,r'_7,r'_{14})=(3, 0, 0, 0). \end{align*} In this case we obtain $P_{m,r}(t)=\{47\}$. One readily computes that $\lfloor v \rfloor=6$. Thus we verify the first 7 terms of \eqref{cong:mod7} via \emph{Mathematica}. It follows by Lemma \ref{le:RS} that it holds for all $n\geq0$. \subsection{The case mod 49} Similarly, we have \begin{align*} \sum_{n=0}^{\infty}b(n)q^n=\frac{(q^{2};q^{2})_{\infty}}{(q;q)_{\infty}^{3}}\equiv \frac{(q;q)_{\infty}^{46}(q^{2};q^{2})_{\infty}}{(q^{7};q^{7})_{\infty}^{7}}\pmod{49}. \end{align*} To prove \eqref{con:mod49}, it needs to show \begin{align} b(343n+t) &\equiv0\pmod{49}\label{cong:mod49} \end{align} for $t=96$, 292, and 341. In these cases, we may set \begin{align*} (m,M,N,t,r=(r_1,r_2,r_{7},r_{14}))=(343,14,14,96,(46,1,-7,0))\in\Delta^*. \end{align*} We compute that $P_{m,r}(t)=\{96, 292, 341\}$. Now we can choose \begin{align*} r'=(r'_1,r'_2,r'_7,r'_{14})=(18, 0, 0, 0). \end{align*} We thus obtain $\lfloor v \rfloor=56$. Similarly we can verify the first 57 terms of \eqref{cong:mod49} through \emph{Mathematica}. Therefore, \begin{align*} \Delta_{k}(343n+96)\equiv\Delta_{k}(343n+292)\equiv\Delta_{k}(343n+341)\equiv 0\pmod{49} \end{align*} holds for all $n\geq0$ if $k\equiv171\pmod{343}$. \section{Final remarks} We would like to emphasize that unlike most congruences of $\Delta_k(n)$ studied in the literature, where $k$ is usually fixed, the congruences \eqref{con:mod5}--\eqref{con:mod49} are valid for infinitely many $k$. Moreover, despite the universality of Radu and Sellers' lemma, we should point out that their method is not elementary because it highly relies on modular forms. On the other hand, the proofs of \eqref{con:mod25}--\eqref{con:mod49} are routine to some extent. It would be interesting to find elementary proofs of these congruences. \section{Acknowledgements} The author would like to thank Shishuo Fu, Shane Chern and Michael D. Hirschhorn for their helpful comments and suggestions on the original manuscript. The author would like to acknowledge the referee for his/her careful reading and helpful comments on an earlier version of the paper. This work was supported by the National Natural Science Foundation of China (No.~11501061). \begin{thebibliography}{HD} \normalsize \baselineskip=17pt \bibitem{AP2007}G.~E. Andrews and P. Paule, MacMahon's partition analysis XI, broken diamonds and modular forms, \textit{Acta Arith.} {\bf 126} (2007), 281--294. \bibitem{Ber} B. C. Berndt, \textit{Number Theory in the Spirit of Ramanujan}, American Mathematical Society, 2006. \bibitem{BerIII} B. C. Berndt, \textit{Ramanujan Notebooks Part III}, Springer--Verlag, 1991. \bibitem{Chan2008}S. H. Chan, Some congruences for Andrews-Paule's broken 2-diamond partitions, \textit{Discrete Math.} \textbf{308} (2008), 5735--5741. \bibitem{Hir2017}M. D. Hirschhorn, Broken 2-diamond partitions modulo 5, \textit{Ramanujan J.} {\bf 45} (2018), 517--520. \bibitem{Hir}M. D. Hirschhorn, \textit{The Power of $q$}, Developments in Mathematics, Vol.~49, Springer, 2017. \bibitem{Jam2013}M. Jameson, Congruences for broken $k$-diamond partitions, \textit{Ann. Comb.} \textbf{17} (2013), 333--338. \bibitem{PR2010}P. Paule and C. S. Radu, Infinite families of strange partition congruences for broken 2-diamonds, \textit{Ramanujan. J} \textbf{23} (2010), 409--416. \bibitem{Rad2015}C. S. Radu, An algorithmic approach to Ramanujan-Kolberg identities, \textit{J. Symb. Comput.} \textbf{68} (2015), 225--253. \bibitem{RS2011}C. S. Radu and J. A. Sellers, Congruence properties modulo $5$ and $7$ for the $\mathrm{pod}$ function, \textit{Int. J. Number Theory} \textbf{7} (2011), 2249--2259. \bibitem{Xia2015}E. X. W. Xia, Infinite families of congruences modulo 7 for broken 3-diamond partitions, \textit{Ramanujan J.} {\bf 40} (2016), 389--403. \bibitem{Xia2017}E. X. W. Xia, More infinite families of congruences modulo 5 for broken 2-diamond partitions, \textit{J. Number Theory} \textbf{170} (2017), 250--262. \bibitem{YW2016}O. X. M. Yao and Y. J. Wang, Newman's identity and infinite families of congruences modulo 7 for broken 3-diamond partitions, \textit{Ramanujan J.} {\bf 43} (2017), 619--631. \end{thebibliography} \bigskip \hrule \bigskip \noindent 2010 {\it Mathematics Subject Classification}: Primary 05A17; Secondary 11P83. \noindent \emph{Keywords: } partition, broken $k$-diamond partition, congruences. \bigskip \hrule \bigskip \vspace*{+.1in} \noindent Received September 11 2017; revised versions received June 26 2018; June 29 2018. Published in {\it Journal of Integer Sequences}, July 1 2018. \bigskip \hrule \bigskip \noindent Return to \htmladdnormallink{Journal of Integer Sequences home page}{http://www.cs.uwaterloo.ca/journals/JIS/}. \vskip .1in \end{document}