A Family of Congruences Modulo 7 for Partitions with Monochromatic Even Parts and Multi--Colored Odd Parts

TL;DR

This paper generalizes a partition function counting partitions with monochromatic even parts and multi-colored odd parts, establishing an infinite family of congruences modulo 7 through elementary $q$-series techniques.

Contribution

It introduces a broad family of related partition functions and proves infinitely many modulo 7 congruences using elementary methods.

Findings

Established an infinite family of congruences modulo 7.

Extended previous specific congruence results to a general family.

Used elementary generating functions and classical $q$-series identities.

Abstract

In recent work, Amdeberhan and Merca considered the integer partition function $a(n)$ which counts the number of integer partitions of weight $n$ wherein even parts come in only one color (i.e., they are monochromatic), while the odd parts may appear in one of three colors. One of the results that they proved was that, for all $n\geq 0$, $a(7n+2) \equiv 0 \pmod{7}$. In this work, we generalize this function $a(n)$ by naturally placing it within an infinite family of related partition functions. Using elementary generating function manipulations and classical $q$--series identities, we then prove infinitely many congruences modulo 7 which are satisfied by members of this family of functions.