- FindStat

Identifier

Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St001174: Permutations ⟶ ℤ

Values

[1] => [1,0] => [2,1] => [1,2] => 0
[1,1] => [1,0,1,0] => [3,1,2] => [1,3,2] => 0
[2] => [1,1,0,0] => [2,3,1] => [1,2,3] => 0
[1,1,1] => [1,0,1,0,1,0] => [4,1,2,3] => [1,4,3,2] => 0
[1,2] => [1,0,1,1,0,0] => [3,1,4,2] => [1,3,4,2] => 1
[2,1] => [1,1,0,0,1,0] => [2,4,1,3] => [1,2,4,3] => 0
[3] => [1,1,1,0,0,0] => [2,3,4,1] => [1,2,3,4] => 0
[1,1,1,1] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => [1,5,4,3,2] => 0
[1,1,2] => [1,0,1,0,1,1,0,0] => [4,1,2,5,3] => [1,4,5,3,2] => 1
[1,2,1] => [1,0,1,1,0,0,1,0] => [3,1,5,2,4] => [1,3,5,4,2] => 1
[1,3] => [1,0,1,1,1,0,0,0] => [3,1,4,5,2] => [1,3,4,5,2] => 1
[2,1,1] => [1,1,0,0,1,0,1,0] => [2,5,1,3,4] => [1,2,5,4,3] => 0
[2,2] => [1,1,0,0,1,1,0,0] => [2,4,1,5,3] => [1,2,4,5,3] => 1
[3,1] => [1,1,1,0,0,0,1,0] => [2,3,5,1,4] => [1,2,3,5,4] => 0
[4] => [1,1,1,1,0,0,0,0] => [2,3,4,5,1] => [1,2,3,4,5] => 0
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0] => [6,1,2,3,4,5] => [1,6,5,4,3,2] => 0
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0] => [5,1,2,3,6,4] => [1,5,6,4,3,2] => 1
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => [4,1,2,6,3,5] => [1,4,6,5,3,2] => 1
[1,1,3] => [1,0,1,0,1,1,1,0,0,0] => [4,1,2,5,6,3] => [1,4,5,6,3,2] => 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => [3,1,6,2,4,5] => [1,3,6,5,4,2] => 1
[1,2,2] => [1,0,1,1,0,0,1,1,0,0] => [3,1,5,2,6,4] => [1,3,5,6,4,2] => 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0] => [3,1,4,6,2,5] => [1,3,4,6,5,2] => 1
[1,4] => [1,0,1,1,1,1,0,0,0,0] => [3,1,4,5,6,2] => [1,3,4,5,6,2] => 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => [2,6,1,3,4,5] => [1,2,6,5,4,3] => 0
[2,1,2] => [1,1,0,0,1,0,1,1,0,0] => [2,5,1,3,6,4] => [1,2,5,6,4,3] => 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0] => [2,4,1,6,3,5] => [1,2,4,6,5,3] => 1
[2,3] => [1,1,0,0,1,1,1,0,0,0] => [2,4,1,5,6,3] => [1,2,4,5,6,3] => 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0] => [2,3,6,1,4,5] => [1,2,3,6,5,4] => 0
[3,2] => [1,1,1,0,0,0,1,1,0,0] => [2,3,5,1,6,4] => [1,2,3,5,6,4] => 1
[4,1] => [1,1,1,1,0,0,0,0,1,0] => [2,3,4,6,1,5] => [1,2,3,4,6,5] => 0
[5] => [1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => [1,2,3,4,5,6] => 0

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$F_{1} = 1$

$F_{2} = 2$

$F_{3} = 3 + q$

$F_{4} = 4 + 4\ q$

$F_{5} = 5 + 11\ q$

Description

The Gorenstein dimension of the algebra

$A/I$ when

$I$ is the tilting module corresponding to the permutation in the Auslander algebra of

$K[x]/(x^n)$ .

Map

bounce path

Description

The bounce path determined by an integer composition.

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

Ringel

Description

The Ringel permutation of the LNakayama algebra corresponding to a Dyck path.