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arXiv · 2109.14307

More models of walks avoiding a quadrant (extended abstract)

Mireille Bousquet-Melou·Michael Wallner
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Abstract

We continue the enumeration of plane lattice paths avoiding the negative quadrant initiated by the first author in [Bousquet-M{\'e}lou, 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-M{\'e}lou, 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-M{\'e}lou, 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.

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BibTeXRIS

Mireille Bousquet-Melou, Michael Wallner. 2021-09-29. More models of walks avoiding a quadrant (extended abstract). https://doi.org/10.4230/lipics.aofa.2020.13

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