The q-Queens Problem: One-Move Riders on the Rectangular Board

TL;DR

This paper derives explicit formulas for counting nonattacking configurations of one-move riders on rectangular chessboards, using symmetric functions and generating functions, resolving conjectures and generalizing previous formulas.

Contribution

It provides two novel methods to compute the number of nonattacking one-move riders on rectangular boards, extending prior results and solving open problems.

Findings

Explicit formulas for nonattacking configurations

Two different computational methods demonstrated

Generalization of existing formulas and resolution of conjectures

Abstract

We generalize the recent results of Chaiken et al. to a rectangular $m\times n$ chessboard. An explicit formula for the number of nonattacking configurations of one-move riders on such a chessboard is calculated in two different ways, one utilizing the theory of symmetric functions and the other the theory of generating functions. With these newly found results, several conjectures and open problems are resolved, and various formulas found by Kotesovec are generalized.