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Matching statistic: St000471
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St000471: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => 2
[2,1] => 0
[1,2,3] => 5
[1,3,2] => 3
[2,1,3] => 3
[2,3,1] => 3
[3,1,2] => 2
[3,2,1] => 0
[1,2,3,4] => 9
[1,2,4,3] => 6
[1,3,2,4] => 7
[1,3,4,2] => 7
[1,4,2,3] => 7
[1,4,3,2] => 4
[2,1,3,4] => 7
[2,1,4,3] => 4
[2,3,1,4] => 7
[2,3,4,1] => 7
[2,4,1,3] => 7
[2,4,3,1] => 4
[3,1,2,4] => 6
[3,1,4,2] => 4
[3,2,1,4] => 4
[3,2,4,1] => 4
[3,4,1,2] => 6
[3,4,2,1] => 4
[4,1,2,3] => 5
[4,1,3,2] => 3
[4,2,1,3] => 3
[4,2,3,1] => 3
[4,3,1,2] => 2
[4,3,2,1] => 0
[1,2,3,4,5] => 14
[1,2,3,5,4] => 10
[1,2,4,3,5] => 11
[1,2,4,5,3] => 11
[1,2,5,3,4] => 11
[1,2,5,4,3] => 7
[1,3,2,4,5] => 12
[1,3,2,5,4] => 8
[1,3,4,2,5] => 12
[1,3,4,5,2] => 12
[1,3,5,2,4] => 12
[1,3,5,4,2] => 8
[1,4,2,3,5] => 12
[1,4,2,5,3] => 9
[1,4,3,2,5] => 9
[1,4,3,5,2] => 9
[1,4,5,2,3] => 12
[1,4,5,3,2] => 9
Description
The sum of the ascent tops of a permutation.
Matching statistic: St000111
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Mp00064: Permutations —reverse⟶ Permutations
St000111: Permutations ⟶ ℤResult quality: 73% ●values known / values provided: 73%●distinct values known / distinct values provided: 73%
St000111: Permutations ⟶ ℤResult quality: 73% ●values known / values provided: 73%●distinct values known / distinct values provided: 73%
Values
[1,2] => [2,1] => 2
[2,1] => [1,2] => 0
[1,2,3] => [3,2,1] => 5
[1,3,2] => [2,3,1] => 3
[2,1,3] => [3,1,2] => 3
[2,3,1] => [1,3,2] => 3
[3,1,2] => [2,1,3] => 2
[3,2,1] => [1,2,3] => 0
[1,2,3,4] => [4,3,2,1] => 9
[1,2,4,3] => [3,4,2,1] => 6
[1,3,2,4] => [4,2,3,1] => 7
[1,3,4,2] => [2,4,3,1] => 7
[1,4,2,3] => [3,2,4,1] => 7
[1,4,3,2] => [2,3,4,1] => 4
[2,1,3,4] => [4,3,1,2] => 7
[2,1,4,3] => [3,4,1,2] => 4
[2,3,1,4] => [4,1,3,2] => 7
[2,3,4,1] => [1,4,3,2] => 7
[2,4,1,3] => [3,1,4,2] => 7
[2,4,3,1] => [1,3,4,2] => 4
[3,1,2,4] => [4,2,1,3] => 6
[3,1,4,2] => [2,4,1,3] => 4
[3,2,1,4] => [4,1,2,3] => 4
[3,2,4,1] => [1,4,2,3] => 4
[3,4,1,2] => [2,1,4,3] => 6
[3,4,2,1] => [1,2,4,3] => 4
[4,1,2,3] => [3,2,1,4] => 5
[4,1,3,2] => [2,3,1,4] => 3
[4,2,1,3] => [3,1,2,4] => 3
[4,2,3,1] => [1,3,2,4] => 3
[4,3,1,2] => [2,1,3,4] => 2
[4,3,2,1] => [1,2,3,4] => 0
[1,2,3,4,5] => [5,4,3,2,1] => 14
[1,2,3,5,4] => [4,5,3,2,1] => 10
[1,2,4,3,5] => [5,3,4,2,1] => 11
[1,2,4,5,3] => [3,5,4,2,1] => 11
[1,2,5,3,4] => [4,3,5,2,1] => 11
[1,2,5,4,3] => [3,4,5,2,1] => 7
[1,3,2,4,5] => [5,4,2,3,1] => 12
[1,3,2,5,4] => [4,5,2,3,1] => 8
[1,3,4,2,5] => [5,2,4,3,1] => 12
[1,3,4,5,2] => [2,5,4,3,1] => 12
[1,3,5,2,4] => [4,2,5,3,1] => 12
[1,3,5,4,2] => [2,4,5,3,1] => 8
[1,4,2,3,5] => [5,3,2,4,1] => 12
[1,4,2,5,3] => [3,5,2,4,1] => 9
[1,4,3,2,5] => [5,2,3,4,1] => 9
[1,4,3,5,2] => [2,5,3,4,1] => 9
[1,4,5,2,3] => [3,2,5,4,1] => 12
[1,4,5,3,2] => [2,3,5,4,1] => 9
[1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 27
[1,2,3,4,5,7,6] => [6,7,5,4,3,2,1] => ? = 21
[1,2,3,4,6,5,7] => [7,5,6,4,3,2,1] => ? = 22
[1,2,3,4,6,7,5] => [5,7,6,4,3,2,1] => ? = 22
[1,2,3,4,7,5,6] => [6,5,7,4,3,2,1] => ? = 22
[1,2,3,4,7,6,5] => [5,6,7,4,3,2,1] => ? = 16
[1,2,3,5,4,6,7] => [7,6,4,5,3,2,1] => ? = 23
[1,2,3,5,4,7,6] => [6,7,4,5,3,2,1] => ? = 17
[1,2,3,5,6,4,7] => [7,4,6,5,3,2,1] => ? = 23
[1,2,3,5,6,7,4] => [4,7,6,5,3,2,1] => ? = 23
[1,2,3,5,7,4,6] => [6,4,7,5,3,2,1] => ? = 23
[1,2,3,5,7,6,4] => [4,6,7,5,3,2,1] => ? = 17
[1,2,3,6,4,5,7] => [7,5,4,6,3,2,1] => ? = 23
[1,2,3,6,4,7,5] => [5,7,4,6,3,2,1] => ? = 18
[1,2,3,6,5,4,7] => [7,4,5,6,3,2,1] => ? = 18
[1,2,3,6,5,7,4] => [4,7,5,6,3,2,1] => ? = 18
[1,2,3,6,7,4,5] => [5,4,7,6,3,2,1] => ? = 23
[1,2,3,6,7,5,4] => [4,5,7,6,3,2,1] => ? = 18
[1,2,3,7,4,5,6] => [6,5,4,7,3,2,1] => ? = 23
[1,2,3,7,4,6,5] => [5,6,4,7,3,2,1] => ? = 18
[1,2,3,7,5,4,6] => [6,4,5,7,3,2,1] => ? = 18
[1,2,3,7,5,6,4] => [4,6,5,7,3,2,1] => ? = 18
[1,2,3,7,6,4,5] => [5,4,6,7,3,2,1] => ? = 17
[1,2,3,7,6,5,4] => [4,5,6,7,3,2,1] => ? = 12
[1,2,4,3,5,6,7] => [7,6,5,3,4,2,1] => ? = 24
[1,2,4,3,5,7,6] => [6,7,5,3,4,2,1] => ? = 18
[1,2,4,3,6,5,7] => [7,5,6,3,4,2,1] => ? = 19
[1,2,4,3,6,7,5] => [5,7,6,3,4,2,1] => ? = 19
[1,2,4,3,7,5,6] => [6,5,7,3,4,2,1] => ? = 19
[1,2,4,3,7,6,5] => [5,6,7,3,4,2,1] => ? = 13
[1,2,4,5,3,6,7] => [7,6,3,5,4,2,1] => ? = 24
[1,2,4,5,3,7,6] => [6,7,3,5,4,2,1] => ? = 18
[1,2,4,5,6,3,7] => [7,3,6,5,4,2,1] => ? = 24
[1,2,4,5,6,7,3] => [3,7,6,5,4,2,1] => ? = 24
[1,2,4,5,7,3,6] => [6,3,7,5,4,2,1] => ? = 24
[1,2,4,5,7,6,3] => [3,6,7,5,4,2,1] => ? = 18
[1,2,4,6,3,5,7] => [7,5,3,6,4,2,1] => ? = 24
[1,2,4,6,3,7,5] => [5,7,3,6,4,2,1] => ? = 19
[1,2,4,6,5,3,7] => [7,3,5,6,4,2,1] => ? = 19
[1,2,4,6,5,7,3] => [3,7,5,6,4,2,1] => ? = 19
[1,2,4,6,7,3,5] => [5,3,7,6,4,2,1] => ? = 24
[1,2,4,6,7,5,3] => [3,5,7,6,4,2,1] => ? = 19
[1,2,4,7,3,5,6] => [6,5,3,7,4,2,1] => ? = 24
[1,2,4,7,3,6,5] => [5,6,3,7,4,2,1] => ? = 19
[1,2,4,7,5,3,6] => [6,3,5,7,4,2,1] => ? = 19
[1,2,4,7,5,6,3] => [3,6,5,7,4,2,1] => ? = 19
[1,2,4,7,6,3,5] => [5,3,6,7,4,2,1] => ? = 18
[1,2,4,7,6,5,3] => [3,5,6,7,4,2,1] => ? = 13
[1,2,5,3,4,6,7] => [7,6,4,3,5,2,1] => ? = 24
[1,2,5,3,4,7,6] => [6,7,4,3,5,2,1] => ? = 18
Description
The sum of the descent tops (or Genocchi descents) of a permutation.
This statistic is given by
$$\pi \mapsto \sum_{i\in\operatorname{Des}(\pi)} \pi_i.$$
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