A $q$-Queens Problem. IV. Queens, Bishops, Nightriders (and Rooks)

Christopher R. H. Hanusa, Thomas Zaslavsky, Seth Chaiken

Published 2016-09-03Version 1

Parts I-III 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 $q$ and, for pieces with some of the queen's moves, proved formulas for these counting quasipolynomials for small numbers of pieces and high-order coefficients of the general counting quasipolynomials. In this part, we focus on the periods of those quasipolynomials by calculating explicit denominators of vertices of the inside-out polytope. We find an exact formula for the denominator when a piece has one move, give intuition for the denominator when a piece has two moves, and show that when a piece has three or more moves, geometrical constructions related to the Fibonacci numbers show that the denominators grow at least exponentially with the number of pieces. Furthermore, we provide the current state of knowledge about the counting quasipolynomials for queens, bishops, rooks, and pieces with some of their moves. We extend these results to the nightrider and its subpieces, and we compare our results with the empirical formulas of Kot\v{e}\v{s}ovec.