: Mathématiques: Stanley symmetric function

mathematical analysis > combinatorics > permutation > Stanley symmetric function

algebra > combinatorics > permutation > Stanley symmetric function

mathematical analysis > function > symmetric function > Stanley symmetric function

mathematical analysis > combinatorics > algebraic combinatorics > Stanley symmetric function

algebra > combinatorics > algebraic combinatorics > Stanley symmetric function

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
${\\displaystyle {\ rac {{\inom {n}{2}}!}{1^{n-1}\\cdot 3^{n-2}\\cdot 5^{n-3}\\cdots (2n-3)^{1}}}}$
reduced decompositions. (Here ${\\displaystyle {\inom {n}{2}}}$ denotes the binomial coefficient n(n − 1)/2 and ! denotes the factorial.)
(Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/Stanley_symmetric_function)

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