• Source: 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 Fw(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










(


n
2


)



!



1

n
−
1


⋅

3

n
−
2


⋅

5

n
−
3


⋯
(
2
n
−
3

)

1







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


reduced decompositions. (Here






(


n
2


)





{\displaystyle {\binom {n}{2}}}

denotes the binomial coefficient n(n − 1)/2 and ! denotes the factorial.)




Properties


The Stanley symmetric function Fw 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





F

w


(
x
)
=

lim

n
→
∞





S




1

n


×
w


(
x
)


{\displaystyle F_{w}(x)=\lim _{n\to \infty }{\mathfrak {S}}_{1^{n}\times w}(x)}


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" (PDF), European Journal of Combinatorics, 5 (4): 359–372, doi:10.1016/s0195-6698(84)80039-6, ISSN 0195-6698, MR 0782057

Kata Kunci Pencarian:

• Daftar masalah matematika yang belum terpecahkan
• Stanley symmetric function
• Ring of symmetric functions
• Symmetric polynomial
• Chromatic symmetric function
• Elementary symmetric polynomial
• Quasisymmetric function
• Power sum symmetric polynomial
• Complete homogeneous symmetric polynomial
• Newton's identities
• Schur polynomial