English

Combinatorial Perspectives on the Crank and Mex Partition Statistics

Combinatorics 2022-05-27 v1

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.

Keywords

Cite

@article{arxiv.2108.09414,
  title  = {Combinatorial Perspectives on the Crank and Mex Partition Statistics},
  author = {Brian Hopkins and James A. Sellers and Ae Ja Yee},
  journal= {arXiv preprint arXiv:2108.09414},
  year   = {2022}
}

Comments

15 pages, 2 figures, 1 table

R2 v1 2026-06-24T05:17:59.648Z