In mathematics and especially in

algebraic combinatorics Algebraic combinatorics is an area of mathematics that employs methods of abstract algebra, notably group theory and representation theory, in various combinatorial contexts and, conversely, applies combinatorial techniques to problems in alg ...

, the Stanley symmetric functions are a family of

symmetric functions In mathematics, a function of n variables is symmetric if its value is the same no matter the order of its arguments. For example, a function f\left(x_1,x_2\right) of two arguments is a symmetric function if and only if f\left(x_1,x_2\right) = f ...

introduced by in his study of the

symmetric group In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group ...

permutation In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or p ...

s. 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 In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or pr ...

) has exactly :

\frac

reduced decompositions. (Here

\binom

denotes the

binomial coefficient In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers and is written \tbinom. It is the coefficient of the t ...

''n''(''n'' − 1)/2 and ! denotes the

factorial In mathematics, the factorial of a non-negative denoted is the product of all positive integers less than or equal The factorial also equals the product of n with the next smaller factorial: \begin n! &= n \times (n-1) \times (n-2) ...

Properties

The Stanley symmetric function ''F''_''w'' is

homogeneous Homogeneity and heterogeneity are concepts often used in the sciences and statistics relating to the uniformity of a substance or organism. A material or image that is homogeneous is uniform in composition or character (i.e. color, shape, siz ...

with

degree Degree may refer to: As a unit of measurement * Degree (angle), a unit of angle measurement ** Degree of geographical latitude ** Degree of geographical longitude * Degree symbol (°), a notation used in science, engineering, and mathemati ...

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 Basis may refer to: Finance and accounting *Adjusted basis, the net cost of an asset after adjusting for various tax-related items *Basis point, 0.01%, often used in the context of interest rates * Basis trading, a trading strategy consisting o ...

of the

ring of symmetric functions In algebra and in particular in algebraic combinatorics, the ring of symmetric functions is a specific limit of the rings of symmetric polynomials in ''n'' indeterminates, as ''n'' goes to infinity. This ring serves as universal structure in which ...

. When a Stanley symmetric function is expanded in the basis of Schur functions, the coefficients are all

non-negative In mathematics, the sign of a real number is its property of being either positive, negative, or zero. Depending on local conventions, zero may be considered as being neither positive nor negative (having no sign or a unique third sign), or it ...

integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...

s. The Stanley symmetric functions have the property that they are the stable limit of Schubert polynomials :

F_w(x) = \lim_ \mathfrak_(x)

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

References

*{{Citation , last1=Stanley , first1=Richard P. , title=On the number of reduced decompositions of elements of Coxeter groups , url=http://dedekind.mit.edu/~rstan/pubs/pubfiles/56.pdf , mr=782057 , year=1984 , journal=European Journal of Combinatorics , issn=0195-6698 , volume=5 , issue=4 , pages=359–372 , doi=10.1016/s0195-6698(84)80039-6 Polynomials Symmetric functions