A $q$-Queens Problem. V. Some of Our Favorite Pieces: Queens, Bishops, Rooks, and Nightriders

TL;DR

This paper advances the understanding of counting nonattacking placements of various chess pieces, including queens, bishops, rooks, and nightriders, on an n×n board by analyzing their quasipolynomial formulas and coefficients.

Contribution

It extends previous work by deriving formulas for partial queens and nightriders, proving some of Kotěšovec's empirical formulas and conjectures, and exploring the properties of their counting quasipolynomials.

Findings

Derived formulas for partial queens and nightriders

Proved some of Kotěšovec's formulas and conjectures

Analyzed the periods and coefficients of the quasipolynomials

Abstract

Parts I-IV 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 $q$. For partial queens, which have a subset of the queen's moves, we proved complete formulas for these counting quasipolynomials for small numbers of pieces and other formulas for high-order coefficients of the general counting quasipolynomials. We found some upper and lower bounds for the periods of those quasipolynomials by calculating explicit denominators of vertices of the inside-out polytope. Here we discover more about the counting quasipolynomials for partial queens, both familiar and strange, and the nightrider and its subpieces, and we compare our results to the empirical formulas found by Kot\v{e}\v{s}ovec. We prove some of Kot\v{e}\v{s}ovec's formulas and conjectures about the quasipolynomials and their high-order coefficients, and in some instances go beyond them.