Dyson's Crank and the Mex of Integer Partitions

TL;DR

This paper explores the relationship between Dyson's crank statistic and the mex of partitions, introducing a new generalization of mex and establishing properties that connect these two partition statistics.

Contribution

It extends the connection between the crank and mex, introduces a new generalization of mex, and proves properties linking these partition statistics.

Findings

Established a strengthened relationship between crank and mex.

Introduced a new generalization of the mex of partitions.

Proved properties that relate crank and mex statistics.

Abstract

Andrews and Newman have recently introduced the notion of the mex of a partition, the smallest positive integer that is not a part. The concept has been used since at least 2011, though, with connections to Frobenius symbols. Recently the parity of the mex has been associated to the crank statistic named by Dyson in 1944. In this note, we extend and strengthen the connection between the crank and mex (along with a new generaliztion of the mex) by proving a number of properties that naturally relate these partition statistics.