How to verify if a given polynomial is a rook polynomial? Let’s assume I have a chessboard 5x5 and those 5 polynomials:
$r(x) = 2 + 11x + 35x^2 + 50x^3 + 26x^4 + 5x^5$
$r(x) = 1 + 15x + 35x^2 + 50x^3 + 26x^4 + 5x^6$
$r(x) = 1 + 10x + 35x^2 + 50x^3 + 26x^4 + 5x^5$
$r(x) = 1 + 15x + 35x^2 + 50x^3 + 5x^5$
$r(x) = 1 + 26x + 35x^2 + 50x^3 + 26x^4 +5x^5$
I know the first one can not be rook polynomial because there is 2 instead of 1 on the first position.
Second is not valid because six rooks can't be placed on 5x5 board($x^6$).
4th is also not valid because if 4 rooks can not be placed on the board then it's not possible to place 5 rooks(missing $x^4$).
I do not have a clue how to verify if 3rd and 5th are valid or not, any ideas?
According to wikipedia the coefficient of $x^5$ is the number of ways to place 5 non-attacking rooks on a 5x5 board, and this is not $5$.
Every row and columm must have exactly one rook in such configurations, so you have 5 spots available for the first columm, 4 spots for the second and so on. So the $x^5$ coefficient must be $5!$. For this reason, none of these polynomials could be rook polynomials.