Combinatorial Perspectives on the Crank and Mex Partition Statistics

Brian Hopkins, James A. Sellers, Ae Ja Yee

Research output: Contribution to journalArticlepeer-review

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 Frobe-nius 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 interpreta-tion 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.

Original languageEnglish (US)
Article numberP2.11
JournalElectronic Journal of Combinatorics
Volume29
Issue number2
DOIs
StatePublished - 2022
Externally publishedYes

Bibliographical note

Funding Information:
We appreciate the helpful comments of two anonymous referees. The third named author was partially supported by a grant (#633963) from the Simons Foundation.

Publisher Copyright:
© The authors.

Fingerprint

Dive into the research topics of 'Combinatorial Perspectives on the Crank and Mex Partition Statistics'. Together they form a unique fingerprint.

Cite this