Explicit formula for the generating series of diagonal 3D rook paths

TL;DR

This paper derives an explicit formula for the generating series counting 3D rook paths on an n×n×n chessboard, using computer-aided discovery and proof involving hypergeometric functions.

Contribution

It provides the first explicit hypergeometric expression for the generating series of 3D rook paths, combining computational discovery with rigorous proof.

Findings

Explicit generating series formula involving hypergeometric functions

Computer-aided discovery and proof methodology

New insights into combinatorial path counting in 3D

Abstract

Let $a_n$ denote the number of ways in which a chess rook can move from a corner cell to the opposite corner cell of an $n \times n \times n$ three-dimensional chessboard, assuming that the piece moves closer to the goal cell at each step. We describe the computer-driven \emph{discovery and proof} of the fact that the generating series $G(x)= \sum_{n \geq 0} a_n x^n$ admits the following explicit expression in terms of a Gaussian hypergeometric function: \[ G(x) = 1 + 6 \cdot \int_0^x \frac{\,\pFq21{1/3}{2/3}{2} {\frac{27 w(2-3w)}{(1-4w)^3}}}{(1-4w)(1-64w)} \, dw.\]