Linear-Time and Constant-Space Algorithms to compute Multi-Sequences that arise in Enumerative Combinatorics (and Elsewhere)

By Shalosh B. Ekhad and Doron Zeilberger

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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

Sample Input and Output for PureRec.txt

Sample Input and Output for PureRecRat.txt