Multiplicative congruences for Andrews's even parts below odd parts function and related infinite products

TL;DR

This paper establishes new multiplicative congruences modulo 2^12 for Andrews's specialized partition function, using advanced modular form techniques, extending Atkin's classical results to more general infinite product settings.

Contribution

It introduces novel multiplicative congruences for Andrews's partition function and related infinite products, employing Fricke involutions and half-integer weight Hecke operators.

Findings

Proved congruences mod 2^12 for Andrews's partition function

Derived analogous congruences for broader classes of infinite products

Utilized modular form techniques like Fricke involutions and Hecke operators

Abstract

We prove multiplicative congruences mod $2^{12}$ for George Andrews's partition function, $\overline{\mathcal{EO}}(n)$, the number of partitions of $n$ in which every even part is less than each odd part and only the largest even part occurs an odd number of times. We find analogous congruences for more general infinite products. These congruences are obtained using Fricke involutions and Newman's approach to half integer weight Hecke operators on eta quotients, and were inspired by Atkin's multiplicative congruences for the partition function.