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

TL;DR

This paper introduces linear-time, constant-space algorithms for efficiently computing multi-sequences in enumerative combinatorics, enabling rapid calculation of large combinatorial numbers previously difficult to compute quickly.

Contribution

The paper adapts the Apagodu-Zeilberger extension of the Almkvist-Zeilberger algorithm for combinatorial applications, providing a novel method for fast computation of complex multi-sequences.

Findings

Able to compute a 104492-digit number in less than 33 seconds

Efficiently computes various combinatorial numbers previously hard to calculate quickly

Demonstrates the algorithm's broad applicability in enumerative combinatorics

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