An Efficient Approach to Solve Complex Rook Polynomials in Strategic Games

A rook polynomial in combinatorial mathematics is a generating polynomial of the number of possible arrangements of non-attacking rooks on a checkerboard-like board, where no two rooks may be arranged in the same row or column. The board is any subset of the squares on a m by n rectangular board; we refer to it as the squares where a rook may be placed. There are various formulas like expansion formula and product formula that provides a polynomial related to a given circumstance i.e. for a particular number of rooks in a particular size of checkboard given but for generating functions like n x n arrays to construct rook polynomials for a board that is decomposes into n disjoint sub-boards C
1
, C
2
, C
3
,….., C
n
and a rook polynomials with R
n
, n(x) possible rook movement spaces in a combinatorial way is very tedious task with practical approach.Therefore, in this paper we attempt to automate the rook polynomial and apply them on a practical basis to solve strategic games. Further attempts have been made to formulate an algorithm to paint a picture of solving rook’s polynomial using backtracking and various other strategies like recursion and loops to combine or place the contents of the list such that no similar content will be placed in same row or same column.