Séminaire Lotharingien de Combinatoire, B81h (2020), 15 pp.
Jean-Christophe Aval and François Bergeron
Interlaced Rectangular Parking Functions
Abstract.
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 Sm x Sn-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 Sn-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.
Received: February 17, 2019.
Accepted: June 6, 2019.
The following versions are available: