Bijections between Łukasiewicz walks and generalized tandem walks

Abstract : 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 southeast 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, northwest , west, south, southeast , and east. We then generalize our bijections to certain models with large steps: for given p ≥ 1, a bijection is given between the half-plane and quarter-plane models obtained by keeping the small southeast 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.
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Submitted on : Wednesday, October 10, 2018 - 12:08:06 AM
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Frédéric Chyzak, Karen Yeats. Bijections between Łukasiewicz walks and generalized tandem walks. 2018. ⟨hal-01891792⟩

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