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A $q$-Queens Problem. IV. Queens, Bishops, Nightriders (and Rooks) | | Christopher R. H. Hanusa
; Thomas Zaslavsky
; Seth Chaiken
; | | Date: |
3 Sep 2016 | | Abstract: | Parts I-III showed that the number of ways to place $q$ nonattacking queens
or similar chess pieces on an $n imes n$ 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 Kotv{e}v{s}ovec. | | Source: | arXiv, 1609.0853 | | Services: | Forum | Review | PDF | Favorites | |
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