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A $q$-Queens Problem | | Seth Chaiken
; Christopher R. H. Hanusa
; Thomas Zaslavsky
; | | Date: |
8 Mar 2013 | | Abstract: | We establish a general counting theory for nonattacking placements of chess
pieces with unbounded straight-line moves, such as the queen, and we apply the
theory to square boards. We show that the number of ways to place $q$
nonattacking queens on a chessboard of variable size $n$ but fixed shape is a
quasipolynomial function of $n$. The period of the quasipolynomial is bounded
by a function of the queen’s move directions. Similar conclusions hold for any
piece whose moves have unlimited length.
We apply our theory to the square board, to show that the highest-order
coefficients of the counting quasipolynomial do not depend on the size of the
board. On the other hand, we present simple pieces for which the fourth
quasipolynomial coefficient is periodic. | | Source: | arXiv, 1303.1879 | | Services: | Forum | Review | PDF | Favorites | |
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