Congruences modulo 11 for broken 5-diamond partitions

Abstract

The notion of broken k-diamond partitions was introduced by Andrews and Paule in 2007. For a fixed positive integer k, let $\Delta _k(n)$ denote the number of broken k-diamond partitions of n. Recently, Paule and Radu conjectured two relations on $\Delta _5(n)$ which were proved by Xiong and Jameson, respectively. In this paper, employing these relations, we prove that, for any prime p with $p\equiv 1\ (\mathrm{mod}\ 4)$, there exists an integer $\lambda (p)\in \{2,\ 3,\ 5,\ 6,\ 11\}$ such that, for $n, \alpha \ge 0$, if $p\not \mid (2n+1)$, then

$$\begin{aligned} \Delta _5\left( 11p^{\lambda (p)(\alpha +1)-1} n+\frac{11p^{\lambda (p)(\alpha +1)-1}+1}{2}\right) \equiv 0\ (\mathrm{mod}\ 11). \end{aligned}$$

Moreover, some non-standard congruences modulo 11 for $\Delta _5(n)$ are deduced. For example, we prove that, for $\alpha \ge 0$, $\Delta _5\left( \frac{11\times 5^{5\alpha }+1}{2}\right) \equiv 7\ (\mathrm{mod}\ 11)$.