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.