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arXiv · 2510.25250

Additional Congruences for generalized Color Partitions of Hirschhorn and Sellers

Anjelin Mariya Johnson·James A. Sellers·S. N. Fathima
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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.

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Anjelin Mariya Johnson, James A. Sellers, S. N. Fathima. 2025-10-29. Additional Congruences for generalized Color Partitions of Hirschhorn and Sellers. https://arxiv.org/abs/2510.25250

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