Counting monomials (Q2574986)

The author shows how enumeration techniques from computational commutative algebra can be applied to graph theoretical problems. In doing so, the author phrases the problems in the language of chess. The essential idea consists in reducing the problems to the question of enumeration of sets of monomials. Then, this enumeration is treated by means of Hilbert functions. An implementation in \texttt{Macaulay2} is provided. The chessboard questions are: In how many ways can we place \(k\) queens on a \(n\times n\) chessboard to get exactly \(u\) unattacked squares? How many squares can a knight, on an infinity chessboard, reach in \(d\) moves? and in \(d\) and no less?