1-Shell totally symmetric plane partitions (TSPPs) modulo powers of 5

Abstract

Let $s(n)$ be the number of 1-shell totally symmetric plane partitions (TSPPs) of $n$. In this paper, an infinite family of congruences modulo powers of $5$ for $s(n)$ will be deduced through an elementary approach. Namely, $$s\left(2\cdot 5^{2\hat{I}\pm -1}n+5^{2\hat{I}\pm -1}\right)\equiv 0(\mathrm{mod}{5}^{\hat{I}}\pm ).$$