Arithmetic properties of $(\ell,m)$-regular colored partitions

TL;DR

This paper investigates the arithmetic properties of $( ext{ell}, ext{m})$-regular colored partitions, establishing congruences for their counts, which enhances understanding of their modular behavior.

Contribution

The paper introduces new congruence relations for the counts of $( ext{ell}, ext{m})$-regular colored partitions, expanding the theoretical framework of partition arithmetic.

Findings

Proved that $b^{2}_{4,5}(8n+7) ot ext{is divisible by } 40$ for all $n$.

Established several new congruences for $b_{ ext{ell,m}}(n)$.

Enhanced understanding of the modular properties of colored partitions.

Abstract

Let $b^{k}_{\ell,m}(n)$ denotes the number of $k-$colored partitions of $n$ into parts that are not multiples of $\ell$ or $m$. We establish several congruence relations for $b_{\ell,m}(n)$. For instance, for any nonnegative integer $n$ $$b^{2}_{4,5}(8n+7) \equiv 0 \pmod{40}.$$