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 -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 , 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 steps of length in directions between north and west.
An article was submitted this year.