mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

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 algeb ...

, 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 grou ...

permutation In mathematics, a permutation of a set can mean one of two different things: * an arrangement of its members in a sequence or linear order, or * the act or process of changing the linear order of an ordered set. An example of the first mean ...

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 can mean one of two different things: * an arrangement of its members in a sequence or linear order, or * the act or process of changing the linear order of an ordered set. An example of the first meanin ...

) 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 (mathematics), 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 ...

Properties

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

homogeneous Homogeneity and heterogeneity are concepts relating to the uniformity of a substance, process or image. A homogeneous feature is uniform in composition or character (i.e., color, shape, size, weight, height, distribution, texture, language, i ...

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 Basis is a term used in mathematics, finance, science, and other contexts to refer to foundational concepts, valuation measures, or organizational names; here, it may refer to: Finance and accounting * Adjusted basis, the net cost of an asse ...

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 whi ...

. 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 0. Depending on local conventions, zero may be considered as having its own unique sign, having no sign, or having both positive and negative sign. ...

integer An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...

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 In mathematics, a power series (in one variable) is an infinite series of the form \sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots where ''a_n'' represents the coefficient of the ''n''th term and ''c'' is a co ...

, 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