LinearTime and ConstantSpace Algorithms to compute MultiSequences that arise in Enumerative Combinatorics (and Elsewhere)
By Shalosh B. Ekhad and Doron Zeilberger
.pdf
.tex
Written: March 9, 2022
Abstract: How many ways, exactly, can a Chess King, always moving forward (i.e. with steps [1,0],[0,1],[1,1]) walk to [100000,200000]?
Thanks to the amazing ApagoduZeilberger extension of the AlmkvistZeilberger algorithm, adapted in this article for combinatorial applications,
this 104492digit number,
can be computed in less than 33 seconds. But not just this particular number. Many other numbers that come up in enumerative combinatorics, can be computed just as efficiently.
Maple packages

PureRec.txt,
a Maple package for generating fast schemes to compute multisequences that come up in enumerative combinatorics

PureRecRat.txt,
A superpackage to the above with applications to lattice walks.
Sample Input and Output for PureRec.txt

If you want to see several random examples in one variable
The
input file generates the output file

If you want to see several reandom examples in one variable of functions of the form exp(polynomial)
The
input file generates the output file

If you want to see several examples in two variable
The
input file generates the output file

If you want to see several random examples in two variable of functions of the form exp(polynomial)
The
input file generates the output file

If you want to see several examples in three variable
The
input file generates the output file

If you want to see several random examples in three variable of functions of the form exp(polynomial)
The
input file generates the output file
Sample Input and Output for PureRecRat.txt

If you want to see an article with 29 theorems about 2dimensional lattice walks with many difference sets of `atomic steps'
The
input file generates the output file
[Note that for six of them we encountered "singularities" but this should be easily fixed by picking other paths]
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