The computational challenge of enumerating high-dimensional rook walks

TL;DR

This paper presents guessed recurrence relations for counting high-dimensional rook paths on chessboards, proposes refined asymptotic formulas, and challenges readers to rigorously prove these conjectures.

Contribution

It introduces conjectured recurrence equations and asymptotic formulas for rook path counting in dimensions 2 to 12, advancing combinatorial enumeration methods.

Findings

Guessed recurrence equations for dimensions 2 to 12

Refined asymptotic formulas for rook path counts

Open challenges for rigorous proof of conjectures

Abstract

We provide guessed recurrence equations for the counting sequences of rook paths on d-dimensional chess boards starting at (0..0) and ending at (n..n), where d=2,3,...,12. Our recurrences suggest refined asymptotic formulas of these sequences. Rigorous proofs of the guessed recurrences as well as the suggested asymptotic forms are posed as challenges to the reader.