$4$-Regular partitions and the pod function

TL;DR

This paper explores properties of the partition function $pod(n)$, focusing on partitions with specific restrictions, and introduces new inequalities and identities related to $pod(n)$, enhancing understanding of its combinatorial structure.

Contribution

The paper introduces new properties, inequalities, and Watson-type identities for the partition function $pod(n)$, specifically for partitions with certain modular restrictions.

Findings

Derived two new infinite families of inequalities involving $pod(n)$

Established new identities of Watson type for $pod(n)$

Connected $pod(n)$ properties with partitions into distinct parts not congruent to $k$ modulo 4

Abstract

The partition function $pod(n)$ enumerates the partitions of $n$ wherein odd parts are distinct and even parts are unrestricted. Recently, a number of properties for $pod(n)$ have been established. In this paper, for $k\in\{0,2\}$ we consider the partitions of $n$ into distinct parts not congruent to $k$ modulo $4$ and the $4$-regular partitions of $n$ in order to obtain new properties for $pod(n)$. In this context, we derive two new infinite families of linear inequalities involving the function $pod(n)$ and obtain new identities of Watson type.