Combinatorial Perspectives on the Crank and Mex Partition Statistics Article Swipe

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Crank Bounded function Partition (number theory) Mathematics Combinatorics Partition function (quantum field theory) Integer (computer science) Combinatorial analysis Combinatorial proof Focus (optics) Generating function Discrete mathematics Computer science Physics Mathematical analysis Geometry Cylinder Quantum mechanics Optics Programming language

Brian Hopkins , James A. Sellers , Ae Ja Yee ·

YOU? · · 2022 · Open Access · · DOI: https://doi.org/10.37236/10776 · OA: W3193301600

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.