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94:. They were introduced in the course of Stanley's enumeration of the reduced decompositions of permutations, and in particular his proof that the permutation
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338:. Unlike other nice families of symmetric functions, the Stanley symmetric functions have many linear dependencies and so do not form a
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241:{\displaystyle {\frac {{\binom {n}{2}}!}{1^{n-1}\cdot 3^{n-2}\cdot 5^{n-3}\cdots (2n-3)^{1}}}}
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Stanley symmetric functions have the property that they are the stable limit of
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450:{\displaystyle F_{w}(x)=\lim _{n\to \infty }{\mathfrak {S}}_{1^{n}\times w}(x)}
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476:"On the number of reduced decompositions of elements of Coxeter groups"
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346:. When a Stanley symmetric function is expanded in the basis of
82:. Each summand corresponds to a reduced decomposition of
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90:as a product of a minimal possible number of
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51:Formally, the Stanley symmetric function
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464:, and take the limit coefficientwise.
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460:where we treat both sides as formal
301:− 1)/2 and ! denotes the
80:fundamental quasisymmetric functions
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483:European Journal of Combinatorics
109:− 1)...21 (written here in
74:, ...) indexed by a permutation
313:The Stanley symmetric function
283:{\displaystyle {\binom {n}{2}}}
251:reduced decompositions. (Here
86:, that is, to a way of writing
78:is defined as a sum of certain
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495:10.1016/s0195-6698(84)80039-6
474:Stanley, Richard P. (1984),
350:, the coefficients are all
344:ring of symmetric functions
26:Stanley symmetric functions
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330:equal to the number of
92:adjacent transpositions
22:algebraic combinatorics
451:
284:
242:
40:) in his study of the
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362:Schubert polynomials
292:binomial coefficient
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120:
532:Symmetric functions
34:Richard Stanley
30:symmetric functions
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20:and especially in
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111:one-line notation
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42:symmetric group
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489:(4): 359–372,
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462:power series
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352:non-negative
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290:denotes the
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46:permutations
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527:Polynomials
324:homogeneous
18:mathematics
521:Categories
468:References
332:inversions
309:Properties
503:0195-6698
431:×
407:∞
404:→
303:factorial
220:−
208:⋯
200:−
189:⋅
181:−
170:⋅
162:−
355:integers
511:0782057
342:of the
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328:degree
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340:basis
326:with
499:ISSN
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