More models of walks avoiding a quadrant (extended abstract)
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.
Keywords
Cite
@article{arxiv.2109.14307,
title = {More models of walks avoiding a quadrant (extended abstract)},
author = {Mireille Bousquet-Melou and Michael Wallner},
journal= {arXiv preprint arXiv:2109.14307},
year = {2025}
}
Comments
Analysis of Algorithms, 2020, Klagenfurt, Austria