A $q$Queens Problem. V. Some of Our Favorite Pieces: Queens, Bishops, Rooks, and Nightriders
Abstract
Parts IIV 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 highorder 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 insideout 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 Kotěšovec. We prove some of Kotěšovec's formulas and conjectures about the quasipolynomials and their highorder coefficients, and in some instances go beyond them.
 Publication:

arXiv eprints
 Pub Date:
 September 2016
 arXiv:
 arXiv:1609.00853
 Bibcode:
 2016arXiv160900853C
 Keywords:

 Mathematics  Combinatorics;
 Primary 05A15;
 Secondary 00A08;
 52C07;
 52C35
 EPrint:
 39 pp., many figures