License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.AofA.2020.8
URN: urn:nbn:de:0030-drops-120383
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2020/12038/
Go to the corresponding LIPIcs Volume Portal

Bousquet-Mélou, Mireille ; Wallner, Michael

More Models of Walks Avoiding a Quadrant

pdf-format:
LIPIcs-AofA-2020-8.pdf (0.5 MB)

Abstract

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.

BibTeX - Entry

@InProceedings{bousquetmlou_et_al:LIPIcs:2020:12038,
  author =	{Mireille Bousquet-M{\'e}lou and Michael Wallner},
  title =	{{More Models of Walks Avoiding a Quadrant}},
  booktitle =	{31st International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2020)},
  pages =	{8:1--8:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-147-4},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{159},
  editor =	{Michael Drmota and Clemens Heuberger},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2020/12038},
  URN =		{urn:nbn:de:0030-drops-120383},
  doi =		{10.4230/LIPIcs.AofA.2020.8},
  annote =	{Keywords: Enumerative combinatorics, lattice paths, non-convex cones, algebraic series, D-finite series}
}

Keywords: Enumerative combinatorics, lattice paths, non-convex cones, algebraic series, D-finite series
Collection: 31st International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2020)
Issue Date: 2020
Date of publication: 10.06.2020

DROPS-Home | Fulltext Search | Imprint | Privacy Published by LZI