Distribution of certain $\ell$-regular partitions and triangular numbers

Abstract

Let $po{d}_{\ell }(n)$ be the number of $\ell$-regular partitions of $n$ with distinct odd parts. In this article, prove that for any positive integer $k$, the set of non-negative integers $n$ for which $po{d}_{\ell }(n)\equiv 0(\mathrm{mod}{p}^k)$ has density one under certain conditions on $\ell$ and $p$. For $p\in \{3,5,7\}$, we also exhibit multiplicative identities for $po{d}_p(n)$ modulo $p.$