Identifier
Values
[[1,2]] => [1,2] => [2,1] => [2,1] => 0
[[1],[2]] => [2,1] => [1,2] => [1,2] => 0
[[1,2,3]] => [1,2,3] => [2,3,1] => [3,2,1] => 0
[[1,3],[2]] => [2,1,3] => [3,2,1] => [3,2,1] => 0
[[1,2],[3]] => [3,1,2] => [3,1,2] => [3,2,1] => 0
[[1],[2],[3]] => [3,2,1] => [1,3,2] => [1,3,2] => 1
[[1,2,3,4]] => [1,2,3,4] => [2,3,4,1] => [4,2,3,1] => 0
[[1,3,4],[2]] => [2,1,3,4] => [3,2,4,1] => [4,2,3,1] => 0
[[1,2,4],[3]] => [3,1,2,4] => [3,4,2,1] => [4,3,2,1] => 0
[[1,2,3],[4]] => [4,1,2,3] => [3,4,1,2] => [4,3,2,1] => 0
[[1,3],[2,4]] => [2,4,1,3] => [4,2,1,3] => [4,3,2,1] => 0
[[1,2],[3,4]] => [3,4,1,2] => [4,1,2,3] => [4,2,3,1] => 0
[[1,4],[2],[3]] => [3,2,1,4] => [4,3,2,1] => [4,3,2,1] => 0
[[1,3],[2],[4]] => [4,2,1,3] => [4,3,1,2] => [4,3,2,1] => 0
[[1,2],[3],[4]] => [4,3,1,2] => [4,1,3,2] => [4,2,3,1] => 0
[[1],[2],[3],[4]] => [4,3,2,1] => [1,4,3,2] => [1,4,3,2] => 2
[[1,2,3,4,5]] => [1,2,3,4,5] => [2,3,4,5,1] => [5,2,3,4,1] => 0
[[1,3,4,5],[2]] => [2,1,3,4,5] => [3,2,4,5,1] => [5,2,3,4,1] => 0
[[1,2,4,5],[3]] => [3,1,2,4,5] => [3,4,2,5,1] => [5,3,2,4,1] => 0
[[1,2,3,5],[4]] => [4,1,2,3,5] => [3,4,5,2,1] => [5,4,3,2,1] => 0
[[1,2,3,4],[5]] => [5,1,2,3,4] => [3,4,5,1,2] => [5,4,3,2,1] => 0
[[1,3,5],[2,4]] => [2,4,1,3,5] => [4,2,5,3,1] => [5,4,3,2,1] => 0
[[1,2,5],[3,4]] => [3,4,1,2,5] => [4,5,2,3,1] => [5,4,3,2,1] => 0
[[1,3,4],[2,5]] => [2,5,1,3,4] => [4,2,5,1,3] => [5,4,3,2,1] => 0
[[1,2,4],[3,5]] => [3,5,1,2,4] => [4,5,2,1,3] => [5,4,3,2,1] => 0
[[1,2,3],[4,5]] => [4,5,1,2,3] => [4,5,1,2,3] => [5,4,3,2,1] => 0
[[1,4,5],[2],[3]] => [3,2,1,4,5] => [4,3,2,5,1] => [5,3,2,4,1] => 0
[[1,3,5],[2],[4]] => [4,2,1,3,5] => [4,3,5,2,1] => [5,4,3,2,1] => 0
[[1,2,5],[3],[4]] => [4,3,1,2,5] => [4,5,3,2,1] => [5,4,3,2,1] => 0
[[1,3,4],[2],[5]] => [5,2,1,3,4] => [4,3,5,1,2] => [5,4,3,2,1] => 0
[[1,2,4],[3],[5]] => [5,3,1,2,4] => [4,5,3,1,2] => [5,4,3,2,1] => 0
[[1,2,3],[4],[5]] => [5,4,1,2,3] => [4,5,1,3,2] => [5,4,3,2,1] => 0
[[1,4],[2,5],[3]] => [3,2,5,1,4] => [5,3,2,1,4] => [5,4,3,2,1] => 0
[[1,3],[2,5],[4]] => [4,2,5,1,3] => [5,3,1,2,4] => [5,4,3,2,1] => 0
[[1,2],[3,5],[4]] => [4,3,5,1,2] => [5,1,3,2,4] => [5,2,4,3,1] => 1
[[1,3],[2,4],[5]] => [5,2,4,1,3] => [5,3,1,4,2] => [5,4,3,2,1] => 0
[[1,2],[3,4],[5]] => [5,3,4,1,2] => [5,1,3,4,2] => [5,2,4,3,1] => 1
[[1,5],[2],[3],[4]] => [4,3,2,1,5] => [5,4,3,2,1] => [5,4,3,2,1] => 0
[[1,4],[2],[3],[5]] => [5,3,2,1,4] => [5,4,3,1,2] => [5,4,3,2,1] => 0
[[1,3],[2],[4],[5]] => [5,4,2,1,3] => [5,4,1,3,2] => [5,4,3,2,1] => 0
[[1,2],[3],[4],[5]] => [5,4,3,1,2] => [5,1,4,3,2] => [5,2,4,3,1] => 1
[[1],[2],[3],[4],[5]] => [5,4,3,2,1] => [1,5,4,3,2] => [1,5,4,3,2] => 3
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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
Demazure product with inverse
Description
This map sends a permutation $\pi$ to $\pi^{-1} \star \pi$ where $\star$ denotes the Demazure product on permutations.
This map is a surjection onto the set of involutions, i.e., the set of permutations $\pi$ for which $\pi = \pi^{-1}$.
Map
reading word permutation
Description
Return the permutation obtained by reading the entries of the tableau row by row, starting with the bottom-most row in English notation.
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.