Identifier
Values
[[1]] => [1] => [1,0] => [2,1] => 0
[[1,2]] => [2] => [1,1,0,0] => [2,3,1] => 0
[[1],[2]] => [1,1] => [1,0,1,0] => [3,1,2] => 0
[[1,2,3]] => [3] => [1,1,1,0,0,0] => [2,3,4,1] => 0
[[1,3],[2]] => [1,2] => [1,0,1,1,0,0] => [3,1,4,2] => 1
[[1,2],[3]] => [2,1] => [1,1,0,0,1,0] => [2,4,1,3] => 1
[[1],[2],[3]] => [1,1,1] => [1,0,1,0,1,0] => [4,1,2,3] => 0
[[1,2,3,4]] => [4] => [1,1,1,1,0,0,0,0] => [2,3,4,5,1] => 0
[[1,3,4],[2]] => [1,3] => [1,0,1,1,1,0,0,0] => [3,1,4,5,2] => 1
[[1,2,4],[3]] => [2,2] => [1,1,0,0,1,1,0,0] => [2,4,1,5,3] => 2
[[1,2,3],[4]] => [3,1] => [1,1,1,0,0,0,1,0] => [2,3,5,1,4] => 2
[[1,3],[2,4]] => [1,2,1] => [1,0,1,1,0,0,1,0] => [3,1,5,2,4] => 2
[[1,2],[3,4]] => [2,2] => [1,1,0,0,1,1,0,0] => [2,4,1,5,3] => 2
[[1,4],[2],[3]] => [1,1,2] => [1,0,1,0,1,1,0,0] => [4,1,2,5,3] => 2
[[1,3],[2],[4]] => [1,2,1] => [1,0,1,1,0,0,1,0] => [3,1,5,2,4] => 2
[[1,2],[3],[4]] => [2,1,1] => [1,1,0,0,1,0,1,0] => [2,5,1,3,4] => 1
[[1],[2],[3],[4]] => [1,1,1,1] => [1,0,1,0,1,0,1,0] => [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
bounce path
Description
The bounce path determined by an integer composition.
Map
horizontal strip sizes
Description
The composition of horizontal strip sizes.
We associate to a standard Young tableau $T$ the composition $(c_1,\dots,c_k)$, such that $k$ is minimal and the numbers $c_1+\dots+c_i + 1,\dots,c_1+\dots+c_{i+1}$ form a horizontal strip in $T$ for all $i$.
Map
Ringel
Description
The Ringel permutation of the LNakayama algebra corresponding to a Dyck path.