Arithmetic properties of $k$-tuple $\ell$-regular partitions

TL;DR

This work investigates the arithmetic properties of $k$-tuple $\ell$-regular partitions with generating function $\sum_{n\ge0} T_{\ell,k}(n)q^n = \dfrac{f_\ell^k}{f_1^k}$. It establishes infinite families of Ramanujan-type congruences and density (lacunarity) results for various parameter choices, notably $(\ell,k)=(2,3)$, $(4,3)$, $(\ell,p)$, and more general cases. The authors combine elementary $q$-series techniques with modular-form theory, including eta-quotients, Hecke operators, and Serre/Ono frameworks, to produce congruences and prove lacunarity phenomena for the generalized partition functions $T_{\ell,k}(n)$. These results extend the landscape of Ramanujan-type congruences in partition theory and provide concrete infinite families and density statements, highlighting the deep interplay between combinatorial generating functions and modular forms in generalized partition settings.

Abstract

In this paper, we study arithmetic properties satisfied by the $k$-tuple $\ell$-regular partitions. A $k$-tuple of partitions $(ξ_1, ξ_2, \ldots, ξ_k)$ is said to be $\ell$-regular if all the $ξ_i$'s are $\ell$-regular. We study the cases $(\ell, k)=(2,3), (4,3), (\ell, p)$, where $p$ is a prime, and even the general case when both $\ell$ and $k$ are unrestricted. Using elementary means as well as the theory of modular forms we prove several infinite family of congruences and density results for these family of partitions.

Theorems & Definitions (39)

• Theorem 1.1
• proof
• Theorem 1.2
• Theorem 1.3
• Remark 1.1
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Remark 1.2