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

TL;DR

This paper develops bijections between various lattice walk models in the half-plane and quarter-plane, generalizing previous work to include colored steps and larger step sizes, advancing combinatorial enumeration methods.

Contribution

It introduces new bijections connecting half-plane and quarter-plane walks with small and large steps, extending prior bijections to more complex models with colored and larger steps.

Findings

Established bijections for models with small steps in two colors.

Generalized bijections to models with large steps of arbitrary length.

Connected new models to existing generalized tandem walk frameworks.

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 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-M\'elou, Fusy, and Raschel.