Linear-Time and Constant-Space Algorithms to compute Multi-Sequences 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 Apagodu-Zeilberger extension of the Almkvist-Zeilberger algorithm, adapted in this article for combinatorial applications, this 104492-digit 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 multi-sequences that come up in enumerative combinatorics

• PureRecRat.txt, A super-package 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 2-dimensional 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]