Nonattacking Queens in a Rectangular Strip

TL;DR

This paper analyzes the counting function for nonattacking chess piece placements in a rectangular strip, revealing its piecewise polynomial nature, and provides asymptotic probabilities and exact counts for various pieces.

Contribution

It extends lattice point counting methods to nonattacking chess configurations, developing generating functions and detailed graph analysis for the problem.

Findings

Counting function is piecewise polynomial and eventually polynomial.

Asymptotic probability of nonattacking configurations for chess pieces.

Exact counts for small numbers of queens, bishops, knights, and nightriders.

Abstract

The function that counts the number of ways to place nonattacking identical chess or fairy chess pieces in a rectangular strip of fixed height and variable width, as a function of the width, is a piecewise polynomial which is eventually a polynomial and whose behavior can be described in some detail. We deduce this by converting the problem to one of counting lattice points outside an affinographic hyperplane arrangement, which Forge and Zaslavsky solved by means of weighted integral gain graphs. We extend their work by developing both generating functions and a detailed analysis of deletion and contraction for weighted integral gain graphs. For chess pieces we find the asymptotic probability that a random configuration is nonattacking, and we obtain exact counts of nonattacking configurations of small numbers of queens, bishops, knights, and nightriders.