A $q$-Queens Problem. I. General Theory

TL;DR

This paper develops a general mathematical framework using Ehrhart theory to count nonattacking placements of chess pieces with unbounded moves on convex polygonal boards, providing explicit formulas and studying combinatorial properties.

Contribution

It introduces a quasipolynomial counting formula for nonattacking placements based on inside-out polytopes and matroid theory, extending previous empirical results.

Findings

Quasipolynomial formula for placements of q pieces on variable-sized boards.

Exact formulas for small numbers of queens, bishops, and nightriders.

Analysis of the quasipolynomial's degree, coefficients, and period.

Abstract

By means of the Ehrhart theory of inside-out polytopes we establish a general counting theory for nonattacking placements of chess pieces with unbounded straight-line moves, such as the queen, on a polygonal convex board. The number of ways to place $q$ identical nonattacking pieces on a board of variable size $n$ but fixed shape is given by a quasipolynomial function of $n$, of degree $2q$, whose coefficients are polynomials in $q$. The number of combinatorially distinct types of nonattacking configuration is the evaluation of our quasipolynomial at $n=-1$. The quasipolynomial has an exact formula that depends on a matroid of weighted graphs, which is in turn determined by incidence properties of lines in the real affine plane. We study the highest-degree coefficients and also the period of the quasipolynomial, which is needed if the quasipolynomial is to be interpolated from data, and which is bounded by some function, not well understood, of the board and the piece's move directions. In subsequent parts we specialize to the square board and then to subsets of the queen's moves, and we prove exact formulas (most but not all already known empirically) for small numbers of queens, bishops, and nightriders. Each part concludes with open questions, both specialized and broad.