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More Models of Walks Avoiding a Quadrant

Authors Mireille Bousquet-Mélou, Michael Wallner

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Subject Classification

ACM Subject Classification
  • Theory of computation → Random walks and Markov chains
  • Mathematics of computing → Enumeration
  • Mathematics of computing → Generating functions
  • Mathematics of computing → Computations on polynomials
Keywords
  • Enumerative combinatorics
  • lattice paths
  • non-convex cones
  • algebraic series
  • D-finite series

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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.

Cite As Get BibTex

Mireille Bousquet-Mélou and Michael Wallner. More Models of Walks Avoiding a Quadrant. In 31st International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 159, pp. 8:1-8:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020) https://doi.org/10.4230/LIPIcs.AofA.2020.8

Author Details

Mireille Bousquet-Mélou
  • CNRS, Université de Bordeaux, Laboratoire Bordelais de Recherche en Informatique, UMR 5800, 351 cours de la Libération, 33405 Talence Cedex, France
Michael Wallner
  • 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

Funding

  • Wallner, Michael: Supported by the Erwin Schrödinger Fellowship of the Austrian Science Fund (FWF): J 4162-N35.

Acknowledgements

We thank our referees for their careful reading.

References

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