A bijective proof of a generalization of the non-negative crank--odd mex identity

DOI10.37236/11472arXiv2203.04267WikidataQ121924749 ScholiaQ121924749MaRDI QIDQ6393160

Publication date: 8 March 2022

Abstract: Recent works of Andrews--Newman and Hopkins--Sellers unveil an interesting relation between two partition statistics, the crank and the mex. They state that, for a positive integer

n

, there are as many partitions of

n

with non-negative crank as partitions of

n

with odd mex. In this paper, we give a bijective proof of a generalization of this identity provided by Hopkins, Sellers, and Stanton. Our method uses an alternative definition of the Durfee decomposition, whose combinatorial link to the crank was recently studied by Hopkins, Sellers, and Yee.

Mathematics Subject Classification ID

Combinatorial identities, bijective combinatorics (05A19) Combinatorial aspects of partitions of integers (05A17) Partitions; congruences and congruential restrictions (11P83) Partition identities; identities of Rogers-Ramanujan type (11P84)

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