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

TL;DR

This paper investigates the divisibility properties of $ ext{pod}_ ext{ extl}$(n) for $ ext{ extl}$-regular partitions with distinct odd parts, establishing density results and identities for specific primes.

Contribution

It proves that for certain $ ext{ extl}$ and primes $p$, the set of integers where $ ext{pod}_ ext{ extl}(n)$ is divisible by $p^k$ has density one, and provides new identities for $p=3,5,7$.

Findings

Density one of integers with divisible $ ext{pod}_ ext{ extl}(n)$ by $p^k$ under certain conditions.

Explicit multiplicative identities for $ ext{pod}_p(n)$ modulo $p$ for $p=3,5,7$.

Enhanced understanding of divisibility patterns in $ ext{ extl}$-regular partitions.

Abstract

Let $pod_{\ell}(n)$ 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 $pod_{\ell}(n)\equiv 0 \pmod{p^{k}}$ has density one under certain conditions on $\ell$ and $p$. For $p \in \{3,5,7\}$, we also exhibit multiplicative identities for $pod_{p}(n)$ modulo $p.$