Section: New Results

Bijections between Łukasiewicz walks and generalized tandem walks

In [9], Frédéric Chyzak, together with Karen Yeats (University of Waterloo, Canada), have studied the enumeration by length of several walk models on the square lattice. They have obtained bijections between walks in the upper half-plane returning to the $x$-axis and walks in the quarter plane. An ongoing work by Bostan, Chyzak, and Mahboubi has given a bijection for models using small north, west, and south-east steps. The work in [9] has adapted and generalized 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. They have then generalized their bijections to certain models with large steps: for given $p\ge 1$, a bijection has been 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. An article was submitted this year.