Mathematics – Combinatorics
Scientific paper
2011-05-25
Annals of Combinatorics 14 (2010), 419-441
Mathematics
Combinatorics
21 pages, 3 figures, preprint of published version copyright Springer Basel AG 2011, Published online February 15, 2011, submi
Scientific paper
10.1007/s00026-011-0068-7
The function that counts the number of ways to place nonattacking identical chess or fairy chess pieces in a rectangular strip of fixed height and variable width, as a function of the width, is a piecewise polynomial which is eventually a polynomial and whose behavior can be described in some detail. We deduce this by converting the problem to one of counting lattice points outside an affinographic hyperplane arrangement, which Forge and Zaslavsky solved by means of weighted integral gain graphs. We extend their work by developing both generating functions and a detailed analysis of deletion and contraction for weighted integral gain graphs. For chess pieces we find the asymptotic probability that a random configuration is nonattacking, and we obtain exact counts of nonattacking configurations of small numbers of queens, bishops, knights, and nightriders.
Chaiken Seth
Hanusa Christopher R. H.
Zaslavsky Thomas
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