Stanley symmetric function

In mathematics and especially in algebraic combinatorics, the Stanley symmetric functions are a family of symmetric polynomials introduced by in his study of the symmetric group of permutations. 
Formally, the Stanley symmetric function "F""w"("x"1, "x"2, ...) indexed by a permutation "w" is defined as a sum of certain fundamental quasisymmetric functions. Each summand corresponds to a reduced decomposition of "w", that is, to a way of writing "w" as a product of a minimal possible number of adjacent transpositions. They were introduced in the course of Stanley's enumeration of the reduced decompositions of permutations, and in particular his proof that the permutation "w"0 = "n"("n" − 1)...21 (written here in one-line notation) has exactly
reduced decompositions. (Here formula_2 denotes the binomial coefficient "n"("n" − 1)/2 and ! denotes the factorial.)
Properties.
The Stanley symmetric function "F""w" is homogeneous with degree equal to the number of inversions of "w". Unlike other nice families of symmetric functions, the Stanley symmetric functions have many linear dependencies and so do not form a basis of the ring of symmetric functions. When a Stanley symmetric function is expanded in the basis of Schur functions, the coefficients are all non-negative integers.
The Stanley symmetric functions have the property that they are the stable limit of Schubert polynomials
where we treat both sides as formal power series, and take the limit coefficientwise.