- FindStat

Identifier

Mp00201: Dyck paths —Ringel⟶ Permutations
St000030: Permutations ⟶ ℤ

Values

[1,0] => [2,1] => 1
[1,0,1,0] => [3,1,2] => 2
[1,1,0,0] => [2,3,1] => 2
[1,0,1,0,1,0] => [4,1,2,3] => 3
[1,0,1,1,0,0] => [3,1,4,2] => 4
[1,1,0,0,1,0] => [2,4,1,3] => 3
[1,1,0,1,0,0] => [4,3,1,2] => 3
[1,1,1,0,0,0] => [2,3,4,1] => 3
[1,0,1,0,1,0,1,0] => [5,1,2,3,4] => 4
[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] => 6
[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] => 4
[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] => 6
[1,1,1,0,0,0,1,0] => [2,3,5,1,4] => 4
[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] => 5
[1,1,1,1,0,0,0,0] => [2,3,4,5,1] => 4
[1,0,1,0,1,0,1,0,1,0] => [6,1,2,3,4,5] => 5
[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] => 7
[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] => 7
[1,0,1,1,0,1,0,0,1,0] => [6,1,4,2,3,5] => 7
[1,0,1,1,0,1,0,1,0,0] => [6,1,5,2,3,4] => 8
[1,0,1,1,0,1,1,0,0,0] => [5,1,4,2,6,3] => 9
[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] => 6
[1,0,1,1,1,0,1,0,0,0] => [6,1,4,5,2,3] => 8
[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] => 5
[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] => 7
[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] => 6
[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] => 5
[1,1,0,1,0,1,1,0,0,0] => [5,4,1,2,6,3] => 7
[1,1,0,1,1,0,0,0,1,0] => [4,3,1,6,2,5] => 7
[1,1,0,1,1,0,0,1,0,0] => [6,3,1,5,2,4] => 8
[1,1,0,1,1,0,1,0,0,0] => [6,4,1,5,2,3] => 8
[1,1,0,1,1,1,0,0,0,0] => [4,3,1,5,6,2] => 7
[1,1,1,0,0,0,1,0,1,0] => [2,3,6,1,4,5] => 5
[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] => 7
[1,1,1,0,1,0,0,0,1,0] => [6,3,4,1,2,5] => 6
[1,1,1,0,1,0,0,1,0,0] => [6,3,5,1,2,4] => 7
[1,1,1,0,1,0,1,0,0,0] => [6,5,4,1,2,3] => 5
[1,1,1,0,1,1,0,0,0,0] => [5,3,4,1,6,2] => 9
[1,1,1,1,0,0,0,0,1,0] => [2,3,4,6,1,5] => 5
[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] => 6
[1,1,1,1,0,1,0,0,0,0] => [6,3,4,5,1,2] => 7
[1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => 5
[] => [1] => 0

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$F_{1} = q$

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

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

$F_{4} = 7\ q^{4} + 5\ q^{5} + 2\ q^{6}$

$F_{5} = 12\ q^{5} + 14\ q^{6} + 10\ q^{7} + 4\ q^{8} + 2\ q^{9}$

Description

The sum of the descent differences of a permutations.
This statistic is given by

$\pi \mapsto \sum_{i\in\operatorname{Des}(\pi)} (\pi_i-\pi_{i+1}).$
See St000111The sum of the descent tops (or Genocchi descents) of a permutation. and St000154The sum of the descent bottoms of a permutation. for the sum of the descent tops and the descent bottoms, respectively. This statistic was studied in [1] and [2] where is was called the drop of a permutation.

Map

Ringel

Description

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