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-Mélou, Fusy, and Raschel.
@article{10_37236_8261,
author = {Fr\'ed\'eric Chyzak and Karen Yeats},
title = {Bijections between {{\L}ukasiewicz} walks and generalized tandem walks},
journal = {The electronic journal of combinatorics},
year = {2020},
volume = {27},
number = {2},
doi = {10.37236/8261},
zbl = {1442.05012},
url = {http://geodesic.mathdoc.fr/articles/10.37236/8261/}
}
TY - JOUR
AU - Frédéric Chyzak
AU - Karen Yeats
TI - Bijections between Łukasiewicz walks and generalized tandem walks
JO - The electronic journal of combinatorics
PY - 2020
VL - 27
IS - 2
UR - http://geodesic.mathdoc.fr/articles/10.37236/8261/
DO - 10.37236/8261
ID - 10_37236_8261
ER -
%0 Journal Article
%A Frédéric Chyzak
%A Karen Yeats
%T Bijections between Łukasiewicz walks and generalized tandem walks
%J The electronic journal of combinatorics
%D 2020
%V 27
%N 2
%U http://geodesic.mathdoc.fr/articles/10.37236/8261/
%R 10.37236/8261
%F 10_37236_8261
Frédéric Chyzak; Karen Yeats. Bijections between Łukasiewicz walks and generalized tandem walks. The electronic journal of combinatorics, Tome 27 (2020) no. 2. doi: 10.37236/8261
