Congruences for 1-shell totally symmetric plane partitions

TL;DR

This paper proves new congruences for the number of 1-shell totally symmetric plane partitions using modular forms, extending previous results to mod 125 and 11.

Contribution

It introduces novel congruences for $f(n)$ modulo 125 and 11, expanding the understanding of partition congruences with modular form techniques.

Findings

Proves $f(1250n+125) \\equiv 0 \\pmod{125}$ for all $n$.

Establishes $f(2750n+825) \\equiv 0 \\pmod{11}$ for all $n$.

Extends previous congruence results to new moduli.

Abstract

Let $f(n)$ denote the number of 1-shell totally symmetric plane partitions of weight $n$. Recently, Hirschhorn and Sellers, Yao, and Xia established a number of congruences modulo 2 and 5, 4 and 8, and 25 for $f(n)$, respectively. In this note, we shall prove several new congruences modulo 125 and 11 by using some results of modular forms. For example, for all $n\ge 0$, we have \begin{align*} f(1250n+125)&\equiv 0 \pmod{125},\\ f(1250n+1125)&\equiv 0 \pmod{125},\\ f(2750n+825)&\equiv 0 \pmod{11},\\ f(2750n+1925)&\equiv 0 \pmod{11}. \end{align*}