Geodesic
Combinatorial perspectives on the crank and mex partition statistics
Brian Hopkins1 ; James Sellers ; Ae Ja Yee
1Saint Peter's University
The electronic journal of combinatorics, Tome 29 (2022) no. 2

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Zbl DOI arXiv
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.
DOI : 10.37236/10776
Classification : 11P81, 05A17
Mots-clés : mex partition statistics, Durfee rectangles, Frobenius symbols

Brian Hopkins  1   ; James Sellers    ; Ae Ja Yee 

1 Saint Peter's University 
Brian Hopkins; James Sellers; Ae Ja Yee. Combinatorial perspectives on the crank and mex partition statistics. The electronic journal of combinatorics, Tome 29 (2022) no. 2. doi: 10.37236/10776
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