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In this article, we study the enumeration by length of several walk models onthe square lattice. We obtain bijections between walks in the upper half-planereturning to the $x$-axis and walks in the quarter plane. A recent work byBostan, Chyzak, and Mahboubi has given a bijection for models using smallnorth, west, and south-east steps. We adapt and generalize it to a bijectionbetween half-plane walks using those three steps in two colours and aquarter-plane model over the symmetrized step set consisting of north,north-west, west, south, south-east, and east. We then generalize ourbijections to certain models with large steps: for given $p\geq1$, a bijectionis given between the half-plane and quarter-plane models obtained by keepingthe small south-east step and replacing the two steps north and west of length1 by the $p+1$ steps of length $p$ in directions between north and west. Thismodel is close to, but distinct from, the model of generalized tandem walksstudied by Bousquet-M\'elou, Fusy, and Raschel.

Bijections Between {\L}ukasiewicz Walks and Generalized Tandem Walks