The part-frequency matrices of a partition

TL;DR

This paper introduces the part-frequency matrix sequence of a partition, a simple combinatorial tool inspired by Glaisher's bijection, which reveals new insights into partition congruences and mock theta functions.

Contribution

It defines the part-frequency matrix sequence, demonstrates its usefulness in partition theory, and generalizes a theorem on mock theta functions.

Findings

Existence of a statistic realizing all Ramanujan-type congruences.

Simple description of the part-frequency matrix sequence.

Generalization of a theorem on mock theta functions.

Abstract

A new combinatorial object is introduced, the part-frequency matrix sequence of a partition, which is elementary to describe and is naturally motivated by Glaisher's bijection. We prove results that suggest surprising usefulness for such a simple tool, including the existence of a related statistic that realizes every possible Ramanujan-type congruence for the partition function. To further exhibit its research utility, we give an easy generalization of a theorem of Andrews, Dixit and Yee on the mock theta functions. Throughout, we state a number of observations and questions that can motivate an array of investigations.