Identifier
- St000570: Permutations ⟶ ℤ
Values
[1,2] => 1
[2,1] => 1
[1,2,3] => 1
[1,3,2] => 1
[2,1,3] => 1
[2,3,1] => 1
[3,1,2] => 1
[3,2,1] => 1
[1,2,3,4] => 1
[1,2,4,3] => 1
[1,3,2,4] => 1
[1,3,4,2] => 1
[1,4,2,3] => 1
[1,4,3,2] => 1
[2,1,3,4] => 1
[2,1,4,3] => 2
[2,3,1,4] => 1
[2,3,4,1] => 1
[2,4,1,3] => 1
[2,4,3,1] => 1
[3,1,2,4] => 1
[3,1,4,2] => 1
[3,2,1,4] => 1
[3,2,4,1] => 1
[3,4,1,2] => 1
[3,4,2,1] => 1
[4,1,2,3] => 1
[4,1,3,2] => 1
[4,2,1,3] => 1
[4,2,3,1] => 1
[4,3,1,2] => 1
[4,3,2,1] => 1
[1,2,3,4,5] => 1
[1,2,3,5,4] => 1
[1,2,4,3,5] => 1
[1,2,4,5,3] => 1
[1,2,5,3,4] => 1
[1,2,5,4,3] => 1
[1,3,2,4,5] => 1
[1,3,2,5,4] => 2
[1,3,4,2,5] => 1
[1,3,4,5,2] => 1
[1,3,5,2,4] => 1
[1,3,5,4,2] => 1
[1,4,2,3,5] => 1
[1,4,2,5,3] => 1
[1,4,3,2,5] => 1
[1,4,3,5,2] => 1
[1,4,5,2,3] => 1
[1,4,5,3,2] => 1
[1,5,2,3,4] => 1
[1,5,2,4,3] => 1
[1,5,3,2,4] => 1
[1,5,3,4,2] => 1
[1,5,4,2,3] => 1
[1,5,4,3,2] => 1
[2,1,3,4,5] => 1
[2,1,3,5,4] => 2
[2,1,4,3,5] => 2
[2,1,4,5,3] => 2
[2,1,5,3,4] => 2
[2,1,5,4,3] => 3
[2,3,1,4,5] => 1
[2,3,1,5,4] => 2
[2,3,4,1,5] => 1
[2,3,4,5,1] => 1
[2,3,5,1,4] => 1
[2,3,5,4,1] => 1
[2,4,1,3,5] => 1
[2,4,1,5,3] => 2
[2,4,3,1,5] => 1
[2,4,3,5,1] => 1
[2,4,5,1,3] => 1
[2,4,5,3,1] => 1
[2,5,1,3,4] => 1
[2,5,1,4,3] => 2
[2,5,3,1,4] => 1
[2,5,3,4,1] => 1
[2,5,4,1,3] => 1
[2,5,4,3,1] => 1
[3,1,2,4,5] => 1
[3,1,2,5,4] => 2
[3,1,4,2,5] => 1
[3,1,4,5,2] => 1
[3,1,5,2,4] => 2
[3,1,5,4,2] => 2
[3,2,1,4,5] => 1
[3,2,1,5,4] => 3
[3,2,4,1,5] => 1
[3,2,4,5,1] => 1
[3,2,5,1,4] => 2
[3,2,5,4,1] => 2
[3,4,1,2,5] => 1
[3,4,1,5,2] => 1
[3,4,2,1,5] => 1
[3,4,2,5,1] => 1
[3,4,5,1,2] => 1
[3,4,5,2,1] => 1
[3,5,1,2,4] => 1
[3,5,1,4,2] => 1
[3,5,2,1,4] => 1
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Description
The Edelman-Greene number of a permutation.
This is the sum of the coefficients of the expansion of the Stanley symmetric function $F_\omega$ in Schur functions. Equivalently, this is the number of semistandard tableaux whose column words - obtained by reading up columns starting with the leftmost - are reduced words for $\omega$.
This is the sum of the coefficients of the expansion of the Stanley symmetric function $F_\omega$ in Schur functions. Equivalently, this is the number of semistandard tableaux whose column words - obtained by reading up columns starting with the leftmost - are reduced words for $\omega$.
References
[1] Billey, S., Pawlowski, B. Permutation patterns, Stanley symmetric functions, and generalized Specht modules MathSciNet:3252657
Code
# very slow
def statistic(pi):
if pi.avoids([2,1,4,3]):
return 1
s = SymmetricFunctions(QQ).s()
G = SymmetricGroup(pi.size())
f = G(pi).stanley_symmetric_function()
return sum(s(f).coefficients())
Created
Aug 09, 2016 at 19:54 by Martin Rubey
Updated
Aug 10, 2016 at 09:47 by Martin Rubey
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