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

TL;DR

This work studies the arithmetic properties of $(\ell,m)$-regular partitions in $k$ colors, whose generating function is $\sum_{n\ge0} b^{k}_{\ell,m}(n) q^n = \frac{f_{\ell}^k f_m^k}{f_1^k f_{\ell m}^k}$. By leveraging Ramanujan theta functions and a suite of $2$- and $3$-dissection identities, the authors derive numerous infinite families of congruences for specific $(\ell,m)$-pairs and values of $k$, including substantial results modulo powers of two and various odd moduli. Key contributions include congruences such as $b^{2}_{4,5}(8n+7) \equiv 0 \pmod{40}$, a detailed description of $b^{2}_{3,4}(4n+1)$ in terms of triangular numbers, and parallel families for $(3,2t)$, $(3,4t)$, $(2,5)$, $(5,4t)$, and $(2,3t)$. The methods combine $q$-series dissections with modular-forms-style arguments to systematize divisibility properties of restricted partitions, advancing the understanding of how color and regularity constraints influence arithmetic structure.

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 $n$ $$b^{2}_{4,5}(8n+7) \equiv 0 \pmod{40}.$$

Theorems & Definitions (53)

• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• Lemma 2.5
• Lemma 2.6
• Lemma 2.7
• Lemma 2.8
• Lemma 2.9
• Theorem 3.1