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

TL;DR

This paper investigates the arithmetic properties of $k$-tuple $ ext{l}$-regular partitions, establishing congruences and density results through elementary methods and modular form theory.

Contribution

It introduces new congruences and density results for $k$-tuple $ ext{l}$-regular partitions, covering specific cases and the general case with both parameters unrestricted.

Findings

Proved infinite families of congruences for these partitions.

Established density results for the distribution of such partitions.

Analyzed specific cases like $( ext{l},k)=(2,3)$ and $(4,3)$.

Abstract

In this paper, we study arithmetic properties satisfied by the $k$-tuple $\ell$-regular partitions. A $k$-tuple of partitions $(\xi_1, \xi_2, \ldots, \xi_k)$ is said to be $\ell$-regular if all the $\xi_i$'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.