Geodesic
Interlaced Rectangular Parking Functions
Séminaire lotharingien de combinatoire, Tome 81 (2020)

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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. 

@article{SLC_2020_81_a7,
     author = {Jean-Christophe Aval and Fran\c{c}ois Bergeron},
     title = {Interlaced {Rectangular} {Parking} {Functions}},
     journal = {S\'eminaire lotharingien de combinatoire},
     publisher = {mathdoc},
     volume = {81},
     year = {2020},
     url = {http://geodesic.mathdoc.fr/item/SLC_2020_81_a7/}
}
TY  - JOUR
AU  - Jean-Christophe Aval
AU  - François Bergeron
TI  - Interlaced Rectangular Parking Functions
JO  - Séminaire lotharingien de combinatoire
PY  - 2020
VL  - 81
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/SLC_2020_81_a7/
ID  - SLC_2020_81_a7
ER  - 
%0 Journal Article
%A Jean-Christophe Aval
%A François Bergeron
%T Interlaced Rectangular Parking Functions
%J Séminaire lotharingien de combinatoire
%D 2020
%V 81
%I mathdoc
%U http://geodesic.mathdoc.fr/item/SLC_2020_81_a7/
%F SLC_2020_81_a7
Jean-Christophe Aval; François Bergeron. Interlaced Rectangular Parking Functions. Séminaire lotharingien de combinatoire, Tome 81 (2020). http://geodesic.mathdoc.fr/item/SLC_2020_81_a7/