Identifier
Values
[1,0] => [2,1] => [2,1] => [1,2] => 0
[1,0,1,0] => [3,1,2] => [2,3,1] => [1,2,3] => 0
[1,1,0,0] => [2,3,1] => [3,1,2] => [3,1,2] => 0
[1,0,1,0,1,0] => [4,1,2,3] => [2,3,4,1] => [1,2,3,4] => 0
[1,0,1,1,0,0] => [3,1,4,2] => [4,2,1,3] => [4,3,1,2] => 0
[1,1,0,0,1,0] => [2,4,1,3] => [3,1,2,4] => [3,4,2,1] => 0
[1,1,0,1,0,0] => [4,3,1,2] => [4,2,3,1] => [1,3,4,2] => 1
[1,1,1,0,0,0] => [2,3,4,1] => [3,4,1,2] => [4,1,2,3] => 0
[1,0,1,0,1,0,1,0] => [5,1,2,3,4] => [2,3,4,5,1] => [1,2,3,4,5] => 0
[1,0,1,0,1,1,0,0] => [4,1,2,5,3] => [5,2,3,1,4] => [5,3,4,1,2] => 0
[1,0,1,1,0,0,1,0] => [3,1,5,2,4] => [4,2,1,3,5] => [4,3,5,2,1] => 0
[1,0,1,1,0,1,0,0] => [5,1,4,2,3] => [2,5,3,4,1] => [1,2,4,5,3] => 2
[1,0,1,1,1,0,0,0] => [3,1,4,5,2] => [4,2,5,1,3] => [5,3,1,2,4] => 0
[1,1,0,0,1,0,1,0] => [2,5,1,3,4] => [3,1,2,4,5] => [3,4,2,5,1] => 0
[1,1,0,0,1,1,0,0] => [2,4,1,5,3] => [3,5,2,1,4] => [5,4,2,1,3] => 0
[1,1,0,1,0,0,1,0] => [5,3,1,2,4] => [4,2,3,5,1] => [1,3,4,2,5] => 1
[1,1,0,1,0,1,0,0] => [5,4,1,2,3] => [5,2,3,4,1] => [1,3,4,5,2] => 1
[1,1,0,1,1,0,0,0] => [4,3,1,5,2] => [5,4,2,1,3] => [5,4,1,3,2] => 1
[1,1,1,0,0,0,1,0] => [2,3,5,1,4] => [3,4,1,2,5] => [4,5,2,3,1] => 0
[1,1,1,0,0,1,0,0] => [2,5,4,1,3] => [3,1,5,2,4] => [3,5,2,1,4] => 1
[1,1,1,0,1,0,0,0] => [5,3,4,1,2] => [4,5,2,3,1] => [1,4,5,2,3] => 2
[1,1,1,1,0,0,0,0] => [2,3,4,5,1] => [3,4,5,1,2] => [5,1,2,3,4] => 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
Tanimoto
Description
Add 1 to every entry of the permutation (n becomes 1 instead of n+1), except that when n appears at the front or the back of the permutation, instead remove it and place 1 at the other end of the permutation.
Map
Kreweras complement
Description
Sends the permutation $\pi \in \mathfrak{S}_n$ to the permutation $\pi^{-1}c$ where $c = (1,\ldots,n)$ is the long cycle.
Map
Ringel
Description
The Ringel permutation of the LNakayama algebra corresponding to a Dyck path.