# 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}(*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

- [math]\displaystyle{ \frac{\binom{n}{2}! }{1^{n - 1} \cdot 3^{n - 2} \cdot 5^{n - 3} \cdots (2n - 3)^1} }[/math]

reduced decompositions. (Here [math]\displaystyle{ \binom{n}{2} }[/math] 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

- [math]\displaystyle{ F_w(x) = \lim_{n \to \infty} \mathfrak{S}_{1^n \times w}(x) }[/math]

where we treat both sides as formal power series, and take the limit coefficientwise.

## References

- Stanley, Richard P. (1984), "On the number of reduced decompositions of elements of Coxeter groups",
*European Journal of Combinatorics***5**(4): 359–372, doi:10.1016/s0195-6698(84)80039-6, ISSN 0195-6698, http://dedekind.mit.edu/~rstan/pubs/pubfiles/56.pdf

Original source: https://en.wikipedia.org/wiki/Stanley symmetric function.
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