The part-frequency matrices of a partition

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 [1] on the mock
theta functions. Throughout, we state a number of observations and questions that can motivate an
array of investigations.

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