Séminaire Lotharingien de Combinatoire, 89B.30 (2023), 12 pp.

Andrei Asinowski, Cyril Banderier and Sarah J. Selkirk

From Kreweras to Gessel: A Walk through Patterns in the Quarter Plane

Abstract. We initiate a study of pattern avoidance in quarter-plane lattice walks. First we demonstrate surprising links between Kreweras excursions (avoiding a pattern of length 2) and some famous lattice walk models, such as Gessel, Gouyou-Beauchamps, and P\'olya excursions. Next we explore the nature (algebraic, hypergeometric, D-finite) of the corresponding generating functions. In particular, we show that pattern avoidance does not necessarily preserve algebraicity or D-finiteness.

Received: November 15, 2022. Accepted: February 20, 2023. Final version: April 1, 2023.

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