eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2020-06-10
8:1
8:14
10.4230/LIPIcs.AofA.2020.8
article
More Models of Walks Avoiding a Quadrant
Bousquet-Mélou, Mireille
1
Wallner, Michael
2
3
https://orcid.org/0000-0001-8581-449X
CNRS, Université de Bordeaux, Laboratoire Bordelais de Recherche en Informatique, UMR 5800, 351 cours de la Libération, 33405 Talence Cedex, France
Université de Bordeaux, Laboratoire Bordelais de Recherche en Informatique, UMR 5800, 351 cours de la Libération, 33405 Talence Cedex, France
TU Wien, Institute for Discrete Mathematics and Geometry, Wiedner Hauptstraße 8 - 10, 1040 Wien, Austria
We continue the enumeration of plane lattice paths avoiding the negative quadrant initiated by the first author in [Bousquet-Mélou, 2016]. We solve in detail a new case, the king walks, where all 8 nearest neighbour steps are allowed. As in the two cases solved in [Bousquet-Mélou, 2016], the associated generating function is proved to differ from a simple, explicit D-finite series (related to the enumeration of walks confined to the first quadrant) by an algebraic one. The principle of the approach is the same as in [Bousquet-Mélou, 2016], but challenging theoretical and computational difficulties arise as we now handle algebraic series of larger degree.
We also explain why we expect the observed algebraicity phenomenon to persist for 4 more models, for which the quadrant problem is solvable using the reflection principle.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol159-aofa2020/LIPIcs.AofA.2020.8/LIPIcs.AofA.2020.8.pdf
Enumerative combinatorics
lattice paths
non-convex cones
algebraic series
D-finite series