The bishop polynomial on a board rotated in an angle of 〖45〗^o is considered a special case of the rook polynomial. Rook polynomials are a powerful tool in the theory of restricted permutations. It is known that the rook polynomial of any board can be computed recursively, using a cell decomposition technique of Riordan. This independent study examines counting problems of non-attacking bishop placements in the game of chess and its movements in the direction of θ=〖45 〗^o to capture pieces in the same direction as the bishop with restricted positions. In this investigation, we developed the total number of ways to arrange n bishops among m positions (m≥n) and also constructed the general formula of a generating function for bishop polynomial that decomposes into n disjoint sub-boards B_1,B_2,…B_n by using an m×n array board. Furthermore, we applied it to combinatorial problems which involve permutation with forbidden positions to construct bishop polynomials in a combinatorial way.
M. Laisin (2019); Enumerative Techniques for Bishop Polynomials Generated by a θ^o Board With an m×n Array ; International Journal of Scientific and Research Publications (IJSRP)
9(1) (ISSN: 2250-3153), DOI: http://dx.doi.org/10.29322/IJSRP.9.01.2019.p8599