The part-frequency matrices of a partition
Keywords:
Partitions, Partition rank, Glaisher’s bijectionAbstract
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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References
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