Congruences modulo $7$ and $11$ for certain two restricted partition functions

TL;DR

This paper establishes infinite families of congruences modulo 7 and 11 for two specialized partition functions, extending recent results and employing elementary q-series methods rather than modular forms.

Contribution

It introduces new infinite congruence families for generalized cubic and diamond partition functions using elementary techniques, broadening previous modular form-based results.

Findings

Proves infinite families of congruences modulo 7 and 11 for a_c(n) and d_c(n).

Extends prior specific congruences to more general cases.

Uses elementary q-series techniques instead of modular forms.

Abstract

For an integer $c\geq 1$, let $a_c(n)$ count the number of generalized cubic partitions of $n$, which are partitions of $n$ whose even parts may appear in $c$ different colors, and $d_c(n)$ count the number of partitions obtained by adding the links of the $c$-elongated plane partition diamonds of length $n$. We prove in this note infinite families of congruences modulo $7$ and $11$ for $a_c(n)$ and $d_c(n)$ by employing elementary $q$-series techniques. These results generalize particular congruences modulo $7$ and $11$ for $a_c(n)$ and $d_c(n)$ recently found by Dockery, and Baruah, Das, and Talukdar, respectively, using modular forms.