Arithmetic properties of $k$-tuple $\mathrm{\ell }$-regular partitions

In this paper, we study arithmetic properties satisfied by the $k$-tuple $\mathrm{\ell }$-regular partitions. A $k$-tuple of partitions $({\xi }_1,{\xi }_2,\dots ,{\xi }_k)$ is said to be $\mathrm{\ell }$-regular if all the ${\xi }_i$’s are $\mathrm{\ell }$-regular. We study the cases $(\mathrm{\ell },k)=(2,3),(4,3),(\mathrm{\ell },p)$, where $p$ is a prime, and even the general case when both $\mathrm{\ell }$ 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.