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

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.