Arithmetic Properties of Partition Quadruples With Odd Parts Distinct

TL;DR

This paper investigates the arithmetic properties of a specific partition function counting quadruples with odd parts distinct, revealing infinite families of congruences and internal relations involving various moduli.

Contribution

It introduces new congruences and internal relations for the partition quadruple function with odd parts distinct, expanding understanding of its arithmetic structure.

Findings

Infinite family of congruences modulo 9 for $ ext{pod}_{-4}(n)$

Internal congruences satisfied by $ ext{pod}_{-4}(n)$

Congruences modulo 2, 5, and 8 for $ ext{pod}_{-4}(n)$

Abstract

Let $\mathrm{pod}_{-4}(n)$ denote the number of partition quadruples of $n$ where the odd parts in each partition are distinct. We find many arithmetic properties of $\mathrm{pod}_{-4}(n)$ involving the following infinite family of congruences: for any integers $\alpha \ge 1$ and $n \ge 0$, \[\mathrm{pod}_{-4}\Big({{3}^{\alpha +1}}n+\frac{5\cdot {{3}^{\alpha }}+1}{2}\Big)\equiv 0 \pmod{9}.\] We also establish some internal congruences and some congruences modulo 2, 5 and 8 satisfied by $\mathrm{pod}_{-4}(n)$.