Mathematics > Combinatorics
[Submitted on 9 Oct 2018 (v1), last revised 17 Feb 2020 (this version, v2)]
Title:Bijections Between Łukasiewicz Walks and Generalized Tandem Walks
View PDFAbstract:In this article, we study the enumeration by length of several walk models on the square lattice. We obtain bijections between walks in the upper half-plane returning to the $x$-axis and walks in the quarter plane. A recent work by Bostan, Chyzak, and Mahboubi has given a bijection for models using small north, west, and south-east steps. We adapt and generalize it to a bijection between half-plane walks using those three steps in two colours and a quarter-plane model over the symmetrized step set consisting of north, north-west, west, south, south-east, and east. We then generalize our bijections to certain models with large steps: for given $p\geq1$, a bijection is given between the half-plane and quarter-plane models obtained by keeping the small south-east step and replacing the two steps north and west of length 1 by the $p+1$ steps of length $p$ in directions between north and west. This model is close to, but distinct from, the model of generalized tandem walks studied by Bousquet-Mélou, Fusy, and Raschel.
Submission history
From: Frédéric Chyzak [view email][v1] Tue, 9 Oct 2018 16:46:06 UTC (58 KB)
[v2] Mon, 17 Feb 2020 13:34:25 UTC (78 KB)
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