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More Models of Walks Avoiding a Quadrant
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Michael Wallner 
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31st International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2020)
Series: Leibniz International Proceedings in Informatics (LIPIcs)
Conference: International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA) - License:
Creative Commons Attribution 3.0 Unported license
- Publication Date: 2020-06-10
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Acknowledgements
We thank our referees for their careful reading.
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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
https://doi.org/10.4230/LIPIcs.AofA.2020.8
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.
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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References
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