Congruences modulo powers of $7$ for $k$-elongated plane partitions

TL;DR

This paper discovers infinite families of congruences modulo powers of 7 for the enumeration functions of k-elongated plane partitions, extending known results for primes other than 7.

Contribution

It introduces new infinite congruence families for d_3(n) and d_5(n) modulo powers of 7, expanding the understanding of partition congruences.

Findings

Infinite congruences for d_3(n) modulo powers of 7

Infinite congruences for d_5(n) modulo powers of 7

Extension of known prime-based congruences to prime 7

Abstract

The enumeration $d_k(n)$ of $k$-elongated plane partition diamonds has emerged as a generalization of the classical integer partition function $p(n)$. Congruences for $d_k(n)$ modulo certain powers of primes have been proven via elementary means and modular forms by many authors. Recently, Banerjee and Smoot established an infinite family of congruences for $d_5(n)$ modulo powers of 5. In this paper we have discovered an infinite congruence family for $d_3(n)$ and $d_5(n)$ modulo powers of 7.