Bijective enumeration of rook walks

TL;DR

This paper provides a bijective proof for counting rook walks of length k on an m by n chessboard, addressing a question posed by R. Stanley about enumeration methods.

Contribution

It introduces a novel bijective proof for rook walk enumeration, offering a new combinatorial approach to a problem posed by Stanley.

Findings

Established a bijective correspondence for rook walks

Derived an explicit enumeration formula

Addressed a question from Stanley's collection

Abstract

In this paper we answer a question posed by R. Stanley in his collection of Bijection Proof Problems (Problem 240). We present a bijective proof for the enumeration of walks of length $k$ a chess rook can move along on an $m\times n$ board starting and ending on the same square.