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

TL;DR

This paper establishes an infinite family of congruences modulo powers of 5 for the number of 1-shell totally symmetric plane partitions, using an elementary approach to advance understanding of their divisibility properties.

Contribution

It introduces a new elementary method to derive infinite congruences modulo powers of 5 for the enumeration of 1-shell TSPPs.

Findings

Proves that s(2*5^{2α-1}n + 5^{2α-1}) ≡ 0 mod 5^α for all α ≥ 1.

Establishes an infinite family of divisibility properties for 1-shell TSPPs.

Provides a new elementary proof technique for partition congruences.

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, $$s\left(2\cdot 5^{2\alpha-1}n+5^{2\alpha-1}\right)\equiv 0 \pmod{5^{\alpha}}.$$