Emilija Milosevic: Tornpolynom

Abstract.

A rook polynomial is a generating polynomial that calculates the number of ways to place rooks on a board structured like a chessboard, where the rooks do not attack each other. It takes into account the restriction that there should not be two rooks in the same row or column on the board and generates a polynomial whose terms represent different arrangements of the rooks satisfying these criteria. In combinatorics, a rook polynomial enables the method to calculate permutations with restricted positions. The aim of this research is to convey a fundamental understanding of rook polynomials and how they are calculated for various types of boards, such as quadratic, disjoint and Ferrers boards, including others. Furthermore, within the research, we will focus on three theorems: the theorem on the relationship between rook polynomials and hit numbers, the factorization theorem for Ferrers boards and the Rook Reciprocity Theorem. These theorems describe the relationship between rook polynomials and other mathematical concepts, primarily within combinatorics. All theorems will be presented with proofs and examples to deepen the understanding.