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In [[mathematics]] and especially in [[algebraic combinatorics]], the '''Stanley symmetric functions''' are a family of [[symmetric polynomial]]s introduced by {{harvs|txt|authorlink=Richard P. Stanley|first=Richard|last=Stanley|year=1984}} in his study of the [[symmetric group]] of [[permutation]]s.
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Formally, the Stanley symmetric function ''F''<sub>''w''</sub>(''x''<sub>1</sub>, ''x''<sub>2</sub>, ...) indexed by a permutation ''w'' is defined as a sum of certain [[Quasisymmetric_function#Important_bases|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 [[Cycle_(mathematics)#Transpositions|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''<sub>0</sub> = ''n''(''n'' &minus; 1)...21 (written here in [[one-line notation]]) has exactly
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: <math>\frac{\binom{n}{2}! }{1^{n - 1} \cdot 3^{n - 2} \cdot 5^{n - 3} \cdots (2n - 3)^1}</math>
reduced decompositions.  (Here <math>\binom{n}{2}</math> denotes the [[binomial coefficient]] ''n''(''n'' &minus; 1)/2 and ! denotes the [[factorial]].)


==Properties==
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The Stanley symmetric function ''F''<sub>''w''</sub> is [[homogeneous polynomial|homogeneous]] with [[degree of a polynomial|degree]] equal to the number of [[inversion (discrete mathematics)|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 (linear algebra)|basis]] of the [[ring of symmetric functions]].  When a Stanley symmetric function is expanded in the basis of [[Schur polynomials|Schur functions]], the coefficients are all [[non-negative]] [[integer]]s.
 
==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}}
 
[[Category:Polynomials]]
[[Category:Symmetric functions]]

Latest revision as of 23:43, 26 February 2014

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