Congruence modulo 4 for Andrews' integer partition with even parts below odd parts

TL;DR

This paper establishes new congruences modulo 4 for Andrews' partition with specific quadratic form conditions, analyzes their distribution, and proves that these congruences hold for almost all integers, extending previous related results.

Contribution

The paper introduces new modulo 4 congruences for a class of Andrews' partitions and analyzes their distribution, providing proofs that these congruences hold for almost all integers.

Findings

Proved congruences modulo 4 for Andrews' partition with certain quadratic form.

Analyzed the distribution of $ar{ ext{EO}}(n)$ and proved it is divisible by 4 for almost all n.

Extended previous work on similar modulo 4 congruences.

Abstract

We find and prove a class of congruences modulo 4 for Andrews' partition with certain ternary quadratic form. We also discuss distribution of $\overline{\mathcal{EO}}(n)$ and further prove that $\overline{\mathcal{EO}}(n)\equiv0\pmod4$ for almost all $n$. This study was inspired by similar congruences modulo 4 in the work by the second author and Garvan.