Section: New Results
Counting walks with large steps in an orthant
In the past fifteen years, the enumeration of lattice walks with steps taken
in a prescribed set and confined to a given cone, especially the first
quadrant of the plane, has been intensely studied. As a result, the generating
functions of quadrant walks are now well-understood, provided the allowed
steps are small. In particular, having small steps is crucial for the
definition of a certain group of bi-rational transformations of the plane. It
has been proved that this group is finite if and only if the corresponding
generating function is D-finite. This group is also the key to the uniform
solution of 19 of the 23 small step models possessing a finite group. In
contrast, almost nothing was known for walks with arbitrary steps.
In [7], Alin Bostan together with Mireille
Bousquet-Mélou (CNRS, Bordeaux) and Stephen Melczer (U. Pennsylvania,
Philadelphia, USA), extended the definition of the group, or rather of the
associated orbit, to this general case, and generalized the above uniform
solution of small step models. When this approach works, it invariably yields
a D-finite generating function. They applied it to many quadrant problems,
including some infinite families.
After developing the general theory, the authors of [7]
considered the