Linear identities for partition pairs with $4$-cores

TL;DR

This paper derives linear identities for counting partition pairs with 4-cores using elementary q-series techniques, leading to new infinite congruences for these counts.

Contribution

It introduces a novel approach to establish linear identities and congruences for partition pairs with 4-cores, expanding understanding in partition theory.

Findings

Derived an infinite family of linear identities for A_4(n)

Discovered an infinite family of congruences for A_4(n)

Applied elementary q-series and 3-dissection formulas

Abstract

We determine an infinite family of linear identities for the number $A_4(n)$ of partition pairs of $n$ with $4$-cores by employing elementary $q$-series techniques and certain $3$-dissection formulas. We then discover an infinite family of congruences for $A_4(n)$ as a consequence of these linear identities.