Explicit Formulas for Partition Pairs and Triples with 3-Cores

TL;DR

This paper derives explicit formulas for counting partition pairs and triples with 3-core partitions using advanced q-series identities, confirming a conjecture and revealing new arithmetic properties.

Contribution

It provides the first explicit formulas for $A_3(n)$ and $B_3(n)$, advancing the understanding of 3-core partition enumeration.

Findings

Explicit formulas for $A_3(n)$ and $B_3(n)$ derived.

Confirmed Xia's conjecture on 3-core partitions.

Discovered new arithmetic identities for these partition counts.

Abstract

Let $A_{3}(n)$ (resp. ${{B}_{3}}(n)$) denote the number of partition pairs (resp. triples) of $n$ where each partition is 3-core. By applying Ramanujan's ${}_{1}\psi_{1}$ formula and Bailey's ${}_{6}\psi_{6}$ formula, we find the explicit formulas for $A_{3}(n)$ and $B_{3}(n)$. Using these formulas, we confirm a conjecture of Xia and establish many arithmetic identities satisfied by $A_{3}(n)$ and $B_{3}(n)$.