By Doron Zeilberger
Written: March 9, 2001.
This third sequel to
the Foundations article in this Umbral Transfer Matrix series ,
illustrates how to automatically derive Umbral Schemes
for 2D self-avoiding polygons and walks,
each of whose vertical cross-sections
has a bounded number of horizontal edges, but each
interval between consecutive horizontal edges can be
as long as you wish.
IMPORTANT: This article is accompanied by three Maple packages
USAP, for self-avoiding polygons,
USAW, for self-avoiding walks, and
MAYLIS, for the automatic counting of
classes of convex polyominoes by perimeter.
maple -q < inUSAP1 > oUSAP1
then, after a few seconds you should get the
first output file for USAP.
If you have package USAP in the same directory as
the second input file for USAP:
inUSAP2, and type:
maple -q < inUSAP2 > oUSAP2
then, after a few days you should get the
second output file for USAP.
If you have package USAW in the same directory as
the input file for USAW:
inUSAW, and type:
maple -q < inUSAW > oUSAW
then, after a few minutes you should get the
output file for USAW.
If you have package MAYLIS in the same directory as
the first input file for MAYLIS:
inMAYLIS1, and type:
maple -q < inMAYLIS1 > oMAYLIS1
then, after a few seconds you should get the
first output file for MAYLIS,
that gives a fully computer-generated proof to the
celebrated Delest-Viennot formula for the number of
convex polyominoes of a given perimeter.
If you have package MAYLIS in the same directory as
the second input file for MAYLIS:
inMAYLIS2, and type:
maple -q < inMAYLIS2 > oMAYLIS2
then, after a few minutes you should get the
second output file for MAYLIS,
that fully automatically, and rigorously (of course)
computes enumerating generating functions for 81
different classes of convex polyominoes by perimeter.
Doron Zeilberger's List of Papers
.pdf
.ps
.tex
Appeared in Elec. J. Comb. v. 8(1) (2001), R28.
SAMPLE Input and Output Files
If you have package USAP in the same directory as
the first input file for USAP:
inUSAP1, and type: