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It is shown how the placement of non-attacking bishops on a chessboard C is related to the matching polynomial of a bipartite graph. Reduction algorithms for finding the bishop polynomial of C are given. We interpret combinatorially the coefficients of this polynomial and construct some interesting boards. Some applications of the bishop polynomials are given.doi:10.15517/rmta.v8i1.197 fatcat:snp7zje32jgrbpurmfdxrlnrle