Identifier
Values
[1] => [1,0,1,0] => [1,1,0,0] => [2,3,1] => 0
[2] => [1,1,0,0,1,0] => [1,0,1,1,0,0] => [3,1,4,2] => 1
[1,1] => [1,0,1,1,0,0] => [1,1,0,0,1,0] => [2,4,1,3] => 1
[3] => [1,1,1,0,0,0,1,0] => [1,1,0,1,1,0,0,0] => [4,3,1,5,2] => 1
[2,1] => [1,0,1,0,1,0] => [1,1,1,0,0,0] => [2,3,4,1] => 0
[1,1,1] => [1,0,1,1,1,0,0,0] => [1,1,1,0,0,1,0,0] => [2,5,4,1,3] => 2
[3,1] => [1,1,0,1,0,0,1,0] => [1,0,1,0,1,1,0,0] => [4,1,2,5,3] => 2
[2,2] => [1,1,0,0,1,1,0,0] => [1,0,1,1,0,0,1,0] => [3,1,5,2,4] => 2
[2,1,1] => [1,0,1,1,0,1,0,0] => [1,1,0,0,1,0,1,0] => [2,5,1,3,4] => 1
[3,2] => [1,1,0,0,1,0,1,0] => [1,0,1,1,1,0,0,0] => [3,1,4,5,2] => 1
[3,1,1] => [1,0,1,1,0,0,1,0] => [1,1,0,0,1,1,0,0] => [2,4,1,5,3] => 2
[2,2,1] => [1,0,1,0,1,1,0,0] => [1,1,1,0,0,0,1,0] => [2,3,5,1,4] => 2
[3,2,1] => [1,0,1,0,1,0,1,0] => [1,1,1,1,0,0,0,0] => [2,3,4,5,1] => 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..
Map
Ringel
Description
The Ringel permutation of the LNakayama algebra corresponding to a Dyck path.
Map
to Dyck path
Description
Sends a partition to the shortest Dyck path tracing the shape of its Ferrers diagram.
Map
peaks-to-valleys
Description
Return the path that has a valley wherever the original path has a peak of height at least one.
More precisely, the height of a valley in the image is the height of the corresponding peak minus $2$.
This is also (the inverse of) rowmotion on Dyck paths regarded as order ideals in the triangular poset.