By Arvind Ayyer and Doron Zeilberger
In How many ways can you walk in the discrete plane, with
arbitrary set of steps and interact with a boundary?
These questions are of interest in statistical mechanics,
but for general sets of steps is beyond the scope of
mere humans. But, once the algorithms in
this paper are taught to a computer, using Maple
(or any other computer algebra system), it can do amazing
symbol-crunching that gives exact and rigorous
results, that in turn, can be used to derive exact numerics,
more efficiently and much more accurately, than simulation.
Added March, 2007:
In this revised version write a revised version we
referenced Duchon's work appropriately.
.pdf
.ps
[Appeared in "Tapas in Experimental Mathematics", Contemporary Mathematics,
v. 457 (2008), 1-20, (Tewodros Amdeberhan and Victor Moll, eds.)].
Written: Jan. 29, 2007.
Important: This article is accompanied by Maple
package
POLYMER (written by Arvind Ayyer, with some minimal guidance
by Doron Zeilberger)
that automatically computes generating functions
and other quantities of interest regarding two dimensional
directed lattice walks with boundaries.
Added Feb. 28, 2007:
Philippe Duchon (LaBRI, Universite Bordeaux 1) has kindly pointed out that
the results about the "unbounded case" were already discovered by him in his
article:
"On the enumeration and generation of generalyzed Dyck words",
Discrete Mathematics vol. 225 (2000).
Doron Zeilberger's List of Papers