Combinatorial Perspectives on the Crank and Mex Partition Statistics

TL;DR

This paper explores the combinatorial relationships between Dyson's crank, the mex statistic, and Frobenius symbols, providing new generating functions and interpretations for partitions with bounded crank.

Contribution

It introduces a new generating function for partitions with bounded crank and offers combinatorial interpretations involving Durfee rectangles, extending previous connections between these statistics.

Findings

Derived a generating function for bounded crank partitions

Provided a combinatorial interpretation involving Durfee rectangles

Extended the connection between crank, mex, and Frobenius symbols

Abstract

Several authors have recently considered the smallest positive part missing from an integer partition, known as the minimum excludant or mex. In this work, we revisit and extend connections between Dyson's crank statistics, the mex, and Frobenius symbols, with a focus on combinatorial proof techniques. One highlight is a generating function expression for the number of partitions with a bounded crank that does not include an alternating sum. This leads to a combinatorial interpretation involving types of Durfee rectangles.