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Statistic identifier: St001811

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Collection: Permutations

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Description: The Castelnuovo-Mumford regularity of a permutation. 

The ''Castelnuovo-Mumford regularity'' of a permutation $\sigma$ is the ''Castelnuovo-Mumford regularity'' of the ''matrix Schubert variety'' $X_\sigma$. 
Equivalently, it is the difference between the degrees of the ''Grothendieck polynomial'' and the ''Schubert polynomial'' for $\sigma$. It can be computed by subtracting the ''Coxeter length'' [[St000018]] from the ''Rajchgot index'' [[St001759]].

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References: [1]   Pechenik, O., E Speyer, D., Weigandt, A. Castelnuovo-Mumford regularity of matrix Schubert varieties [[arXiv:2111.10681]]

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Code:
def statistic(x):
    return max(v.major_index() for v in x.permutohedron_smaller()) - x.length()

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Statistic values:

[1,2]       => 0
[2,1]       => 0
[1,2,3]     => 0
[1,3,2]     => 1
[2,1,3]     => 0
[2,3,1]     => 0
[3,1,2]     => 0
[3,2,1]     => 0
[1,2,3,4]   => 0
[1,2,4,3]   => 2
[1,3,2,4]   => 1
[1,3,4,2]   => 1
[1,4,2,3]   => 1
[1,4,3,2]   => 2
[2,1,3,4]   => 0
[2,1,4,3]   => 2
[2,3,1,4]   => 0
[2,3,4,1]   => 0
[2,4,1,3]   => 1
[2,4,3,1]   => 1
[3,1,2,4]   => 0
[3,1,4,2]   => 1
[3,2,1,4]   => 0
[3,2,4,1]   => 0
[3,4,1,2]   => 0
[3,4,2,1]   => 0
[4,1,2,3]   => 0
[4,1,3,2]   => 1
[4,2,1,3]   => 0
[4,2,3,1]   => 0
[4,3,1,2]   => 0
[4,3,2,1]   => 0
[1,2,3,4,5] => 0
[1,2,3,5,4] => 3
[1,2,4,3,5] => 2
[1,2,4,5,3] => 2
[1,2,5,3,4] => 2
[1,2,5,4,3] => 4
[1,3,2,4,5] => 1
[1,3,2,5,4] => 4
[1,3,4,2,5] => 1
[1,3,4,5,2] => 1
[1,3,5,2,4] => 3
[1,3,5,4,2] => 3
[1,4,2,3,5] => 1
[1,4,2,5,3] => 3
[1,4,3,2,5] => 2
[1,4,3,5,2] => 2
[1,4,5,2,3] => 2
[1,4,5,3,2] => 2
[1,5,2,3,4] => 1
[1,5,2,4,3] => 3
[1,5,3,2,4] => 2
[1,5,3,4,2] => 2
[1,5,4,2,3] => 2
[1,5,4,3,2] => 3
[2,1,3,4,5] => 0
[2,1,3,5,4] => 3
[2,1,4,3,5] => 2
[2,1,4,5,3] => 2
[2,1,5,3,4] => 2
[2,1,5,4,3] => 4
[2,3,1,4,5] => 0
[2,3,1,5,4] => 3
[2,3,4,1,5] => 0
[2,3,4,5,1] => 0
[2,3,5,1,4] => 2
[2,3,5,4,1] => 2
[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] => 3
[2,5,3,1,4] => 1
[2,5,3,4,1] => 1
[2,5,4,1,3] => 2
[2,5,4,3,1] => 2
[3,1,2,4,5] => 0
[3,1,2,5,4] => 3
[3,1,4,2,5] => 1
[3,1,4,5,2] => 1
[3,1,5,2,4] => 2
[3,1,5,4,2] => 3
[3,2,1,4,5] => 0
[3,2,1,5,4] => 3
[3,2,4,1,5] => 0
[3,2,4,5,1] => 0
[3,2,5,1,4] => 2
[3,2,5,4,1] => 2
[3,4,1,2,5] => 0
[3,4,1,5,2] => 1
[3,4,2,1,5] => 0
[3,4,2,5,1] => 0
[3,4,5,1,2] => 0
[3,4,5,2,1] => 0
[3,5,1,2,4] => 1
[3,5,1,4,2] => 2
[3,5,2,1,4] => 1
[3,5,2,4,1] => 1
[3,5,4,1,2] => 1
[3,5,4,2,1] => 1
[4,1,2,3,5] => 0
[4,1,2,5,3] => 2
[4,1,3,2,5] => 1
[4,1,3,5,2] => 1
[4,1,5,2,3] => 1
[4,1,5,3,2] => 2
[4,2,1,3,5] => 0
[4,2,1,5,3] => 2
[4,2,3,1,5] => 0
[4,2,3,5,1] => 0
[4,2,5,1,3] => 1
[4,2,5,3,1] => 1
[4,3,1,2,5] => 0
[4,3,1,5,2] => 1
[4,3,2,1,5] => 0
[4,3,2,5,1] => 0
[4,3,5,1,2] => 0
[4,3,5,2,1] => 0
[4,5,1,2,3] => 0
[4,5,1,3,2] => 1
[4,5,2,1,3] => 0
[4,5,2,3,1] => 0
[4,5,3,1,2] => 0
[4,5,3,2,1] => 0
[5,1,2,3,4] => 0
[5,1,2,4,3] => 2
[5,1,3,2,4] => 1
[5,1,3,4,2] => 1
[5,1,4,2,3] => 1
[5,1,4,3,2] => 2
[5,2,1,3,4] => 0
[5,2,1,4,3] => 2
[5,2,3,1,4] => 0
[5,2,3,4,1] => 0
[5,2,4,1,3] => 1
[5,2,4,3,1] => 1
[5,3,1,2,4] => 0
[5,3,1,4,2] => 1
[5,3,2,1,4] => 0
[5,3,2,4,1] => 0
[5,3,4,1,2] => 0
[5,3,4,2,1] => 0
[5,4,1,2,3] => 0
[5,4,1,3,2] => 1
[5,4,2,1,3] => 0
[5,4,2,3,1] => 0
[5,4,3,1,2] => 0
[5,4,3,2,1] => 0

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Created: Jul 04, 2022 at 22:16 by Oliver Pechenik

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Last Updated: Jul 05, 2022 at 10:54 by Martin Rubey