A $q$-Queens Problem. V. The Bishops' Period

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

Published 2014-05-12, updated 2016-09-08Version 2

Part I showed that the number of ways to place $q$ nonattacking queens or similar chess pieces on an $n\times n$ square chessboard is a quasipolynomial function of $n$. We prove the previously empirically observed period of the bishops quasipolynomial, which is exactly 2 for three or more bishops. The proof depends on signed graphs and the Ehrhart theory of inside-out polytopes.