A $q$-Queens Problem. IV. Attacking Configurations and Their Denominators

TL;DR

This paper investigates the denominators of quasipolynomials counting nonattacking chess piece placements, deriving formulas for certain pieces and showing exponential growth for complex move sets, advancing understanding of combinatorial chess problems.

Contribution

It provides exact formulas for denominators of certain chess pieces and demonstrates exponential growth in complexity for pieces with multiple moves.

Findings

Exact denominator formulas for one-move and two-move pieces with horizontal moves.

Denominator growth is at least exponential for pieces with three or more moves.

Period of quasipolynomials is bounded by the least common denominator of vertices.

Abstract

In Parts I-III we showed that the number of ways to place $q$ nonattacking queens or similar chess pieces on an $n\times n$ chessboard is a quasipolynomial function of $n$ whose coefficients are essentially polynomials in $q$. In this part we focus on the periods of those quasipolynomials. We calculate denominators of vertices of the inside-out polytope, since the period is bounded by, and conjecturally equal to, their least common denominator. We find an exact formula for that denominator of every piece with one move and of two-move pieces having a horizontal move. For pieces with three or more moves, we produce geometrical constructions related to the Fibonacci numbers that show the denominator grows at least exponentially with $q$.