Enumeration of restricted positions, including the use of Rook polynominals

Understanding Restricted Positions and Rook Polynomials

In combinatorics and chessboard problems, restricted positions often come into play. These are conditions that restrict how objects can be arranged in a set or a chessboard.
The most common application of this principle can be understood through the Rook Polynomial, a generating function used to count arrangements.
Rook polynomials were introduced by Arthur Schensted in his research about the game of chess. They are used to calculate the number of ways a certain number of non-attacking rooks can be placed on a chess board.

The Rook Polynomial for a board B is calculated as R(B,x) = r0 + r1x + r2x^2 + … + rnx^n.
This polynomial represents the total number of ways non-attacking rooks could be positioned on the board, where rn denotes how many ways n rooks can be positioned and x is the variable.
A rook is non-attacking if no two rooks are in the same row or the same column.

The Rook Polynomial can be applied to any board of any dimension. The simplest example would be a 1x1 board, the rook polynomial of which is 1 + x.
For instance, a 2x2 board B would yield the Rook Polynomial R(B,x) = 1 + 4x + x^2. That means there are 1 way with 0 rooks, 4 ways with 1 rook, and 1 way with 2 rooks.

The Rook Polynomial is often used in the field of discrete mathematics, a branch of mathematics studying mathematical structures that are fundamentally discrete rather than continuous.
Its relevance and application span many areas such as computer science, physics, engineering, and numerous other fields where specific scenarios require calculated permutations and combinations.