2015/03/13 by Jean-Christophe Aval, François Bergeron, Aval, Jean-Christophe +1 · 1 citation
Computer Science · Mathematics · #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Topological and Geometric Data Analysis #math.CO #msc:05A15 #msc:05A19 #msc:05E05 #msc:05E10
paper · pdf · doi:10.48550/arxiv.1503.03991
arxiv created 2015/03/13 · arxiv updated 2015/03/16
The aim of this work is to extend to a general Sm× Sn-module context the Grossman-Bizley paradigm that allows the enumeration of Dyck paths in a m× n-rectangle. We obtain an explicit formula for the the "bi-Frobenius" characteristic of what we call \em interlaced rectangular parking functions in an m× n-rectangle. These are obtained by labelling the n vertical steps of an m× n-Dyck path by the numbers from 1 to n, together with an independent labelling of its horizontal steps by integers from 1 to m. Our formula specializes to give the Frobenius characteristic of the Sn-module of m× n-parking functions in the general situation. Hence, it subsumes the result of Armstrong-Loehr-Warrington which furnishes such a formula for the special case when m and n are coprime integers.