Interlaced Rectangular Parking Functions

Jean-Christophe Aval   ; François Bergeron

Séminaire lotharingien de combinatoire, Tome 81 (2020)

Résumé

The aim of this work is to extend the Grossman-Bizley [Scripta Math. 16 (1950), 207-212; J. Inst. Actuar. 80 (1954), 55-62] paradigm that allows the enumeration of Dyck paths in an m x n-rectangle to a general S_m x S_n-module context. We obtain an explicit formula for the the "bi-Frobenius" characteristic of what we call interlaced rectangular parking functions in an m x n-rectangle. These are obtained by labeling the n vertical steps of an m x n-Dyck path by the numbers from 1 to n, together with an independent labeling of its horizontal steps by integers from 1 to m. Our formula specializes to give the Frobenius characteristic of the S_n-module of m x n-parking functions in the general situation. Hence, it subsumes the result of Armstrong, Loehr and Warrington of [Ann. Combin. 20 (2016), 21-58], which furnishes such a formula for the special case where m and n are coprime integers.