Rook Polynomials: Rook Problem 1 and 2

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Rook polynomials is the number of ways to place k non-attacking rooks on an original chess board where no two rooks can be in the same row or column. The general formula to calculate the number of arrangements of non-attacking rooks is.

The formula for calculating non attacking rooks is The polynomials below show the arrangements of each rook. The notation Rn(x) indicates the number of rooks being used e.g r1(x) means 1. The powers of x indicate the number of rooks so for example the first row means 1 rook can be arranged in 1 way and zero rooks can be arranged in one way.

Rook problem 1

A famous problem called the “Eight rooks problem” by H.E Dudeney shows the maximum amount of non-attacking rooks on a chessboard is eight by arranging them on one diagonal of the board that covers 8 squares. The question of the problem “In how many ways can eight rooks be placed on an 8 × 8 chessboard so that neither of them attacks the other?”. The answer is eight factorial as it behaves as an injective function. On the first row of the chess board the rook has eight positions to be placed on. Then the rook has seven positions it can be on in the second row and so on until the eight row where the rook has only one position it can be on. As a result the different ways a rook can be placed on a chess board without them attacking each other is 8! which is equivalent to 40,320.

Rook problem 2

Another problem that relates to rooks is “In how many ways can one arrange k rooks on an m × n board in such a way that they do not attack each other?”. To approach this problem k would have to be less or equal to the number m and n. Since the number of rows is m of which k must be chosen the formula becomes mCk. Also the set of k columns on which to place rooks can be chosen is nCk ways. As the way to choose k from M and N are independent from each other then the formula becomes mCk multiplied by nCk ways to choose the square to place the rook. However to calculate the amount of non-attacking rook arrangements, the number of ways to choose the square on which to place the rook must be multiplied by k!, as that is the number of ways k rooks can be arranged to not attack each other. As a result the number of ways non-attacking rook arrangements is mCk multiplied nCk multiplied k!.