The lex game and some applications

Let F be a field, V a finite subset of F. We introduce the lex game, which yields a combinatorial description of the lexicographic standard monomials of the ideal I(V ) of polynomials vanishing on V . As a consequence, we obtain a fast algorithm which computes the lexicographic standard monomials of I(V ). We apply the lex game to calculate explicitly the standard monomials for special types of subsets of {0, 1}n. For D ⊆ Z let VD denote the vectors y ∈ {0, 1}n in which the number of ones (the Hamming weight of y) is in D. We calculate the lexicographic standard monomials of VD, where