Ranks For Two Partition Quadruple Functions

TL;DR

This paper introduces a rank-type statistic for two new integer partition quadruple functions, providing a combinatorial refinement and reproof of their Ramanujan-type congruences modulo 3, 5, and 7.

Contribution

It defines a new combinatorial rank statistic that refines and reestablishes known congruences for two novel partition quadruple functions.

Findings

Reproves congruences modulo 3, 5, and 7

Introduces a new rank-type statistic

Provides combinatorial refinement of partition congruences

Abstract

Recently the author introduced two new integer partition quadruple functions, which satisfy Ramanujan-type congruences modulo $3$, $5$, $7$, and $13$. Here we reprove the congruences modulo $3$, $5$, and $7$ by defining a rank-type statistic that gives a combinatorial refinement of the congruences.