Identifier
Values
{{1,2}} => [2,1] => [1,2] => [1,2] => 0
{{1},{2}} => [1,2] => [1,2] => [1,2] => 0
{{1,2,3}} => [2,3,1] => [1,2,3] => [1,2,3] => 0
{{1,2},{3}} => [2,1,3] => [1,2,3] => [1,2,3] => 0
{{1,3},{2}} => [3,2,1] => [1,3,2] => [1,3,2] => 1
{{1},{2,3}} => [1,3,2] => [1,2,3] => [1,2,3] => 0
{{1},{2},{3}} => [1,2,3] => [1,2,3] => [1,2,3] => 0
{{1,2,3,4}} => [2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 0
{{1,2,3},{4}} => [2,3,1,4] => [1,2,3,4] => [1,2,3,4] => 0
{{1,2,4},{3}} => [2,4,3,1] => [1,2,4,3] => [1,2,4,3] => 2
{{1,2},{3,4}} => [2,1,4,3] => [1,2,3,4] => [1,2,3,4] => 0
{{1,2},{3},{4}} => [2,1,3,4] => [1,2,3,4] => [1,2,3,4] => 0
{{1,3,4},{2}} => [3,2,4,1] => [1,3,4,2] => [1,4,3,2] => 2
{{1,3},{2,4}} => [3,4,1,2] => [1,3,2,4] => [1,3,2,4] => 1
{{1,3},{2},{4}} => [3,2,1,4] => [1,3,2,4] => [1,3,2,4] => 1
{{1,4},{2,3}} => [4,3,2,1] => [1,4,2,3] => [1,4,2,3] => 1
{{1},{2,3,4}} => [1,3,4,2] => [1,2,3,4] => [1,2,3,4] => 0
{{1},{2,3},{4}} => [1,3,2,4] => [1,2,3,4] => [1,2,3,4] => 0
{{1,4},{2},{3}} => [4,2,3,1] => [1,4,2,3] => [1,4,2,3] => 1
{{1},{2,4},{3}} => [1,4,3,2] => [1,2,4,3] => [1,2,4,3] => 2
{{1},{2},{3,4}} => [1,2,4,3] => [1,2,3,4] => [1,2,3,4] => 0
{{1},{2},{3},{4}} => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
{{1,2,3,4,5}} => [2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,2,3,4},{5}} => [2,3,4,1,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,2,3,5},{4}} => [2,3,5,4,1] => [1,2,3,5,4] => [1,2,3,5,4] => 3
{{1,2,3},{4,5}} => [2,3,1,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,2,3},{4},{5}} => [2,3,1,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,2,4,5},{3}} => [2,4,3,5,1] => [1,2,4,5,3] => [1,2,5,4,3] => 4
{{1,2,4},{3,5}} => [2,4,5,1,3] => [1,2,4,3,5] => [1,2,4,3,5] => 2
{{1,2,4},{3},{5}} => [2,4,3,1,5] => [1,2,4,3,5] => [1,2,4,3,5] => 2
{{1,2,5},{3,4}} => [2,5,4,3,1] => [1,2,5,3,4] => [1,2,5,3,4] => 2
{{1,2},{3,4,5}} => [2,1,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,2},{3,4},{5}} => [2,1,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,2,5},{3},{4}} => [2,5,3,4,1] => [1,2,5,3,4] => [1,2,5,3,4] => 2
{{1,2},{3,5},{4}} => [2,1,5,4,3] => [1,2,3,5,4] => [1,2,3,5,4] => 3
{{1,2},{3},{4,5}} => [2,1,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,2},{3},{4},{5}} => [2,1,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,3,4,5},{2}} => [3,2,4,5,1] => [1,3,4,5,2] => [1,5,4,3,2] => 3
{{1,3,4},{2,5}} => [3,5,4,1,2] => [1,3,4,2,5] => [1,4,3,2,5] => 2
{{1,3,4},{2},{5}} => [3,2,4,1,5] => [1,3,4,2,5] => [1,4,3,2,5] => 2
{{1,3,5},{2,4}} => [3,4,5,2,1] => [1,3,5,2,4] => [1,5,3,2,4] => 2
{{1,3},{2,4,5}} => [3,4,1,5,2] => [1,3,2,4,5] => [1,3,2,4,5] => 1
{{1,3},{2,4},{5}} => [3,4,1,2,5] => [1,3,2,4,5] => [1,3,2,4,5] => 1
{{1,3,5},{2},{4}} => [3,2,5,4,1] => [1,3,5,2,4] => [1,5,3,2,4] => 2
{{1,3},{2,5},{4}} => [3,5,1,4,2] => [1,3,2,5,4] => [1,3,2,5,4] => 4
{{1,3},{2},{4,5}} => [3,2,1,5,4] => [1,3,2,4,5] => [1,3,2,4,5] => 1
{{1,3},{2},{4},{5}} => [3,2,1,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 1
{{1,4,5},{2,3}} => [4,3,2,5,1] => [1,4,5,2,3] => [1,5,2,4,3] => 3
{{1,4},{2,3,5}} => [4,3,5,1,2] => [1,4,2,3,5] => [1,4,2,3,5] => 1
{{1,4},{2,3},{5}} => [4,3,2,1,5] => [1,4,2,3,5] => [1,4,2,3,5] => 1
{{1,5},{2,3,4}} => [5,3,4,2,1] => [1,5,2,3,4] => [1,5,2,3,4] => 1
{{1},{2,3,4,5}} => [1,3,4,5,2] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1},{2,3,4},{5}} => [1,3,4,2,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,5},{2,3},{4}} => [5,3,2,4,1] => [1,5,2,3,4] => [1,5,2,3,4] => 1
{{1},{2,3,5},{4}} => [1,3,5,4,2] => [1,2,3,5,4] => [1,2,3,5,4] => 3
{{1},{2,3},{4,5}} => [1,3,2,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1},{2,3},{4},{5}} => [1,3,2,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,4,5},{2},{3}} => [4,2,3,5,1] => [1,4,5,2,3] => [1,5,2,4,3] => 3
{{1,4},{2,5},{3}} => [4,5,3,1,2] => [1,4,2,5,3] => [1,5,4,2,3] => 2
{{1,4},{2},{3,5}} => [4,2,5,1,3] => [1,4,2,3,5] => [1,4,2,3,5] => 1
{{1,4},{2},{3},{5}} => [4,2,3,1,5] => [1,4,2,3,5] => [1,4,2,3,5] => 1
{{1,5},{2,4},{3}} => [5,4,3,2,1] => [1,5,2,4,3] => [1,4,5,2,3] => 2
{{1},{2,4,5},{3}} => [1,4,3,5,2] => [1,2,4,5,3] => [1,2,5,4,3] => 4
{{1},{2,4},{3,5}} => [1,4,5,2,3] => [1,2,4,3,5] => [1,2,4,3,5] => 2
{{1},{2,4},{3},{5}} => [1,4,3,2,5] => [1,2,4,3,5] => [1,2,4,3,5] => 2
{{1,5},{2},{3,4}} => [5,2,4,3,1] => [1,5,2,3,4] => [1,5,2,3,4] => 1
{{1},{2,5},{3,4}} => [1,5,4,3,2] => [1,2,5,3,4] => [1,2,5,3,4] => 2
{{1},{2},{3,4,5}} => [1,2,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1},{2},{3,4},{5}} => [1,2,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,5},{2},{3},{4}} => [5,2,3,4,1] => [1,5,2,3,4] => [1,5,2,3,4] => 1
{{1},{2,5},{3},{4}} => [1,5,3,4,2] => [1,2,5,3,4] => [1,2,5,3,4] => 2
{{1},{2},{3,5},{4}} => [1,2,5,4,3] => [1,2,3,5,4] => [1,2,3,5,4] => 3
{{1},{2},{3},{4,5}} => [1,2,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1},{2},{3},{4},{5}} => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 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
cycle-as-one-line notation
Description
Return the permutation obtained by concatenating the cycles of a permutation, each written with minimal element first, sorted by minimal element.
Map
to permutation
Description
Sends the set partition to the permutation obtained by considering the blocks as increasing cycles.
Map
Clarke-Steingrimsson-Zeng inverse
Description
The inverse of the Clarke-Steingrimsson-Zeng map, sending excedances to descents.
This is the inverse of the map $\Phi$ in [1, sec.3].