Stanley symmetric function

In mathematics and especially in algebraic combinatorics, the Stanley symmetric functions are a family of symmetric functions introduced by Richard Stanley (1984) in his study of the symmetric group of permutations.

Formally, the Stanley symmetric function F_w(x₁, x₂, ...) 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 w₀ = n(n − 1)...21 (written here in one-line notation) has exactly

${\displaystyle {\frac {{\binom {n}{2}}!}{1^{n-1}\cdot 3^{n-2}\cdot 5^{n-3}\cdots (2n-3)^{1}}}}$

reduced decompositions. (Here ${\displaystyle {\binom {n}{2}}}$ denotes the binomial coefficient n(n − 1)/2 and ! denotes the factorial.)