Extending recent work of Nath, Saikia, and Sarma on $k$-tuple $\ell$-regular partitions

TL;DR

The paper studies $T_{\ell,k}(n)$, the number of $k$-tuple $\ell$-regular partitions, extending the work of Nath, Saikia, and Sarma. Using elementary $q$-series techniques and generating function manipulations, it proves a stronger version of the conjectured infinite family of congruences for $T_{2}(n)$ and derives new congruence families, including infinite families modulo $8$ and $24$ and related prime-power results. Key contributions include a general congruence framework based on $q$-series identities and a parity analysis that yields vanishing coefficients on arithmetic progressions. The results push forward the understanding of Ramanujan-type partition congruences for $\ell$-regular partitions and offer a tractable approach for discovering further congruences in this area.

Abstract

Let $T_{\ell,k}(n)$ denote the number of $\ell$-regular $k$-tuple partitions of $n$. In a recent work, Nath, Saikia, and Sarma derived several families of congruences for $T_{\ell,k}(n)$, with particular emphasis on the cases $T_{2,3}(n)$ and $T_{4,3}(n)$. In the concluding remarks of their paper, they conjectured that $T_{2,3}(n)$ satisfies an infinite set of congruences modulo 6. In this paper, we confirm their conjecture by proving a much more general result using elementary $q$-series techniques. We also present new families of congruences satisfied by $T_{\ell,k}(n)$.

Theorems & Definitions (29)

• Conjecture 1.1: Nath, Saikia, Sarma Nath-Saikia-Sarma
• Theorem 1.2
• Corollary 1.3
• proof
• Theorem 1.4
• Corollary 1.5
• proof
• Lemma 2.1
• proof
• Lemma 2.2