Additional Congruences for generalized Color Partitions of Hirschhorn and Sellers

TL;DR

This paper establishes new congruences for a generalized color partition function, extending Ramanujan's classical results, using elementary methods like generating functions, theta functions, and q-dissection techniques.

Contribution

It introduces novel congruences for the partition function $a_k(n)$ modulo powers of 2 and 11, expanding the understanding of partition congruences with elementary proofs.

Findings

Congruences modulo powers of 2 for infinitely many k

Infinite family of congruences modulo 11

Elementary proof techniques used

Abstract

Let $a_k(n)$ denote the number of partitions of $n$ wherein even parts come in only one color, while the odd parts may be ``colored" with one of $k$ colors, for fixed $k$. In this note, we find some congruences for $a_k(n)$ in the spirit of Ramanujan's congruences. We prove a number of results for $a_k(n)$ modulo powers of $2$ for infinitely many values of $k$. Our approach is truly elementary, relying on generating function manipulations, theta functions and $q$-dissection techniques. We then close by demonstrating an infinite family of congruences modulo 11 which is proven using a result of Ahlgren.