Combinatorial Perspectives on the Crank and Mex Partition Statistics

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 statistic, 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. A recurring combinatorial technique uses sign reversing involutions on certain triples of partitions to establish a result of Fine and other identities.

Dyson's crank and the mex of integer partitions

“…The j-Durfee rectangle was instrumental in proving a new generating function for partitions with bounded crank [9,Theorem 8]. For completeness, we repeat that result here.…”

“…Finally, it has been fruitful to separate the odd mex partitions into those with mex congruent to 1 modulo 4 and those with mex congruent to 3 modulo 4 [8,9]. Are there interesting ways to refine the partitions with (generalized) fixed points?…”

A Bijective Proof of a Generalization of the Non-Negative Crank-Odd Mex Identity

“…The bijective proof of Theorem 6 that we provide in this paper, was deeply inspired by the work of Hopkins, Sellers, and Yee, who described in [8] combinatorial relations that link the crank and the Durfee decomposition of a partition. Our work is based on a simple yet subtle definition related the very notion of Durfee decomposition.…”

“…Before constructing the bijection for Theorem 12, we first state the key result given by Hopkins, Sellers and Yee in [8], and that provides a combinatorial link between the crank and the Durfee decomposition. Recall that, for all partitions λ, ω(λ) = #{i ∈ {1, .…”