Untying The Gordian Knot via Experimental Mathematics
By Yukun Yao and Doron Zeilberger
.pdf
.ps
.tex
Written: Dec. 17, 2018. This vesion: Jan. 7, 2019.
[To appear in "Algorithmic CombinatoricsEnumerative Combinatorics, Special Functions, and Computer Algebra: In honor of Peter Paule's 60th birthday", Springer;
edited by Veronika Pillwein and Carsten Schneider]
Dedicated to Peter Paule on his 60th birthday
Peter Paule is one
of the pioneers of Symbolic computation and experimental mathematics, and the coauthor (with Manuel Kauers) of the
bible in our field.
This article is dedicated to him with friendship and admiration.
Maple packages

GFMatrix.txt,
a Maple package for automatically computing, both experimentally and rigorously rational generating
functions for sequences of determinants of "almost diagonal matrices"

SpanningTrees.txt,
a Maple package for automatically computing rational generating
functions for sequences enumerating spanning trees, with applications to joint resistance.

JointConductance.txt,
a Maple package for joint conductance .
Sample Input and Output for GFMatrix.txt

If you want to see rational generating functions for determinants of sequences of "almost diagonal matrices"
with up to 11 nonzero diagonals, for SYMBOLIC matrices the
input file yields the
output file
[Warning: very big file!]

If you want to see rational generating functions for determinants of sequences of "almost diagonal matrices"
with up to 11 nonzero diagonals, for random NUMERICAL matrices the
input file yields the
output file
Sample Input and Output for SpanningTrees.txt

If you want to see rational generating functions for enumerating the spanning trees of the graphs GxP_{n}
for various G
input file yields the
output file

If you want to see rational generating functions for enumerating the spanning trees of the grid graphs P_{m}xP_{n}
for m from 2 to 6
input file yields the
output file
Sample Input and Output for JointConductance.txt

If you want to see the rational generating functions enumerating the sequences enumerating the number of spanning
trees in an m by n rectangular grid, for m from 2 to 7
the input file generates the
output file.
Note that the generating functions for n ≤ 6 have been previously computed by Paul Raff and Faase, but m=7 seems to be new.
Suprisingly, the degree of the denominator is 48, rather than 64 (until m=6 the degree is 2^{m1})
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Articles of Doron Zeilberger
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