- FindStat

Identifier

Mp00201: Dyck paths —Ringel⟶ Permutations
St001074: Permutations ⟶ ℤ

Values

[1,0] => [2,1] => 2
[1,0,1,0] => [3,1,2] => 3
[1,1,0,0] => [2,3,1] => 5
[1,0,1,0,1,0] => [4,1,2,3] => 4
[1,0,1,1,0,0] => [3,1,4,2] => 6
[1,1,0,0,1,0] => [2,4,1,3] => 6
[1,1,0,1,0,0] => [4,3,1,2] => 4
[1,1,1,0,0,0] => [2,3,4,1] => 8
[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] => 7
[1,0,1,1,0,0,1,0] => [3,1,5,2,4] => 9
[1,0,1,1,0,1,0,0] => [5,1,4,2,3] => 9
[1,0,1,1,1,0,0,0] => [3,1,4,5,2] => 9
[1,1,0,0,1,0,1,0] => [2,5,1,3,4] => 9
[1,1,0,0,1,1,0,0] => [2,4,1,5,3] => 11
[1,1,0,1,0,0,1,0] => [5,3,1,2,4] => 5
[1,1,0,1,0,1,0,0] => [5,4,1,2,3] => 5
[1,1,0,1,1,0,0,0] => [4,3,1,5,2] => 9
[1,1,1,0,0,0,1,0] => [2,3,5,1,4] => 9
[1,1,1,0,0,1,0,0] => [2,5,4,1,3] => 9
[1,1,1,0,1,0,0,0] => [5,3,4,1,2] => 7
[1,1,1,1,0,0,0,0] => [2,3,4,5,1] => 11
[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] => 8
[1,0,1,0,1,1,0,0,1,0] => [4,1,2,6,3,5] => 10
[1,0,1,0,1,1,0,1,0,0] => [6,1,2,5,3,4] => 10
[1,0,1,0,1,1,1,0,0,0] => [4,1,2,5,6,3] => 10
[1,0,1,1,0,0,1,0,1,0] => [3,1,6,2,4,5] => 12
[1,0,1,1,0,0,1,1,0,0] => [3,1,5,2,6,4] => 12
[1,0,1,1,0,1,0,0,1,0] => [6,1,4,2,3,5] => 10
[1,0,1,1,0,1,0,1,0,0] => [6,1,5,2,3,4] => 12
[1,0,1,1,0,1,1,0,0,0] => [5,1,4,2,6,3] => 12
[1,0,1,1,1,0,0,0,1,0] => [3,1,4,6,2,5] => 10
[1,0,1,1,1,0,0,1,0,0] => [3,1,6,5,2,4] => 12
[1,0,1,1,1,0,1,0,0,0] => [6,1,4,5,2,3] => 12
[1,0,1,1,1,1,0,0,0,0] => [3,1,4,5,6,2] => 12
[1,1,0,0,1,0,1,0,1,0] => [2,6,1,3,4,5] => 12
[1,1,0,0,1,0,1,1,0,0] => [2,5,1,3,6,4] => 12
[1,1,0,0,1,1,0,0,1,0] => [2,4,1,6,3,5] => 14
[1,1,0,0,1,1,0,1,0,0] => [2,6,1,5,3,4] => 16
[1,1,0,0,1,1,1,0,0,0] => [2,4,1,5,6,3] => 14
[1,1,0,1,0,0,1,0,1,0] => [6,3,1,2,4,5] => 6
[1,1,0,1,0,0,1,1,0,0] => [5,3,1,2,6,4] => 8
[1,1,0,1,0,1,0,0,1,0] => [6,4,1,2,3,5] => 6
[1,1,0,1,0,1,0,1,0,0] => [5,6,1,2,3,4] => 8
[1,1,0,1,0,1,1,0,0,0] => [5,4,1,2,6,3] => 10
[1,1,0,1,1,0,0,0,1,0] => [4,3,1,6,2,5] => 12
[1,1,0,1,1,0,0,1,0,0] => [6,3,1,5,2,4] => 10
[1,1,0,1,1,0,1,0,0,0] => [6,4,1,5,2,3] => 10
[1,1,0,1,1,1,0,0,0,0] => [4,3,1,5,6,2] => 12
[1,1,1,0,0,0,1,0,1,0] => [2,3,6,1,4,5] => 12
[1,1,1,0,0,0,1,1,0,0] => [2,3,5,1,6,4] => 14
[1,1,1,0,0,1,0,0,1,0] => [2,6,4,1,3,5] => 12
[1,1,1,0,0,1,0,1,0,0] => [2,6,5,1,3,4] => 12
[1,1,1,0,0,1,1,0,0,0] => [2,5,4,1,6,3] => 16
[1,1,1,0,1,0,0,0,1,0] => [6,3,4,1,2,5] => 8
[1,1,1,0,1,0,0,1,0,0] => [6,3,5,1,2,4] => 8
[1,1,1,0,1,0,1,0,0,0] => [6,5,4,1,2,3] => 6
[1,1,1,0,1,1,0,0,0,0] => [5,3,4,1,6,2] => 14
[1,1,1,1,0,0,0,0,1,0] => [2,3,4,6,1,5] => 12
[1,1,1,1,0,0,0,1,0,0] => [2,3,6,5,1,4] => 12
[1,1,1,1,0,0,1,0,0,0] => [2,6,4,5,1,3] => 14
[1,1,1,1,0,1,0,0,0,0] => [6,3,4,5,1,2] => 10
[1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => 14

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

$F_{2} = q^{3} + q^{5}$

$F_{3} = 2\ q^{4} + 2\ q^{6} + q^{8}$

$F_{4} = 3\ q^{5} + 2\ q^{7} + 7\ q^{9} + 2\ q^{11}$

$F_{5} = 4\ q^{6} + 5\ q^{8} + 9\ q^{10} + 16\ q^{12} + 6\ q^{14} + 2\ q^{16}$

Description

The number of inversions of the cyclic embedding of a permutation.
The cyclic embedding of a permutation

$\pi$ of length

$n$ is given by the permutation of length

$n+1$ represented in cycle notation by

$(\pi_1,\ldots,\pi_n,n+1)$ .
This reflects in particular the fact that the number of long cycles of length

$n+1$ equals

$n!$ .
This statistic counts the number of inversions of this embedding, see [1]. As shown in [2], the sum of this statistic on all permutations of length

$n$ equals

$n!\cdot(3n-1)/12$ .

Map

Ringel

Description

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