- FindStat

Identifier

Mp00201: Dyck paths —Ringel⟶ Permutations
St001005: Permutations ⟶ ℤ

Values

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

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$F_{1} = q^{2}$

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

$F_{3} = q^{3} + 4\ q^{4}$

$F_{4} = q^{3} + 5\ q^{4} + 8\ q^{5}$

$F_{5} = q^{3} + 8\ q^{4} + 16\ q^{5} + 17\ q^{6}$

Description

The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both.

Map

Ringel

Description

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