Arithmetic of 2-regular partitions with distinct odd parts

TL;DR

This paper investigates the properties of 2-regular partitions with distinct odd parts, deriving congruences modulo 2 and 8 through generating functions and Hecke eigenform theory.

Contribution

It introduces new congruences for the partition function $pod_2(n)$ using advanced mathematical tools, expanding understanding of partition arithmetic.

Findings

Established congruences for $pod_2(n)$ mod 2 and 8.

Applied generating function techniques to partition problems.

Utilized Hecke eigenform theory to derive number theoretic results.

Abstract

Let $pod_2(n)$ denote the number of $2$-regular partitions of $n$ with distinct odd parts (even parts are unrestricted). In this article, we obtain congruences for $pod_2(n)$ mod $2$ and mod $8$ using some generating function manipulations and the theory of Hecke eigenform.