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Identifier

Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001811: Permutations ⟶ ℤ

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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$F_{1} = 1$

$F_{2} = 2$

$F_{3} = 2 + 2\ q$

$F_{4} = 2 + 2\ q + 6\ q^{2}$

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.