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How to compute ROOK Polynomials for NxM Matrices for k objects ?

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closed as off topic by Karolina Sołtys, Aaron Sterling, Lev Reyzin, Kaveh, Peter Shor Nov 3 '10 at 3:35

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Tuhin, the purpose of this site is not for asisting in homework. I'd say you may look at, but you already asked there ( – Diego de Estrada Nov 3 '10 at 1:44
Also, "rook" shouldn't be capitalized. It's a chess piece, not an abbreviation. – Jeffε Nov 3 '10 at 16:01
up vote 2 down vote accepted

See the answers of this question.

Your case is a lot easier, just choose $k$ of the $m$ columns and then you have $n (n-1)\ldots (n-k+1)$ ways to put the $k$ rooks. So the coefficient of $x^k$ is

$\displaystyle r_k = \binom{m}{k} n (n-1)\ldots (n-k+1) = \binom{m}{k}\binom{n}{k}k!$.

The provided link also gives a dynamic programming approach to compute $r_k$.

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