- FindStat

Identifier

St001811: Permutations ⟶ ℤ

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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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 St000018The number of inversions of a permutation. from the Rajchgot index St001759The Rajchgot index of a permutation..

References

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

Code

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

Created

Jul 04, 2022 at 22:16 by Oliver Pechenik

Updated

Jul 05, 2022 at 10:54 by Martin Rubey