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Matching statistic: St000471
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Mp00223: Permutations —runsort⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000471: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00069: Permutations —complement⟶ Permutations
St000471: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [2,1] => 0
[2,1] => [1,2] => [2,1] => 0
[1,2,3] => [1,2,3] => [3,2,1] => 0
[1,3,2] => [1,3,2] => [3,1,2] => 2
[2,1,3] => [1,3,2] => [3,1,2] => 2
[2,3,1] => [1,2,3] => [3,2,1] => 0
[3,1,2] => [1,2,3] => [3,2,1] => 0
[3,2,1] => [1,2,3] => [3,2,1] => 0
[1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 0
[1,2,4,3] => [1,2,4,3] => [4,3,1,2] => 2
[1,3,2,4] => [1,3,2,4] => [4,2,3,1] => 3
[1,3,4,2] => [1,3,4,2] => [4,2,1,3] => 3
[1,4,2,3] => [1,4,2,3] => [4,1,3,2] => 3
[1,4,3,2] => [1,4,2,3] => [4,1,3,2] => 3
[2,1,3,4] => [1,3,4,2] => [4,2,1,3] => 3
[2,1,4,3] => [1,4,2,3] => [4,1,3,2] => 3
[2,3,1,4] => [1,4,2,3] => [4,1,3,2] => 3
[2,3,4,1] => [1,2,3,4] => [4,3,2,1] => 0
[2,4,1,3] => [1,3,2,4] => [4,2,3,1] => 3
[2,4,3,1] => [1,2,4,3] => [4,3,1,2] => 2
[3,1,2,4] => [1,2,4,3] => [4,3,1,2] => 2
[3,1,4,2] => [1,4,2,3] => [4,1,3,2] => 3
[3,2,1,4] => [1,4,2,3] => [4,1,3,2] => 3
[3,2,4,1] => [1,2,4,3] => [4,3,1,2] => 2
[3,4,1,2] => [1,2,3,4] => [4,3,2,1] => 0
[3,4,2,1] => [1,2,3,4] => [4,3,2,1] => 0
[4,1,2,3] => [1,2,3,4] => [4,3,2,1] => 0
[4,1,3,2] => [1,3,2,4] => [4,2,3,1] => 3
[4,2,1,3] => [1,3,2,4] => [4,2,3,1] => 3
[4,2,3,1] => [1,2,3,4] => [4,3,2,1] => 0
[4,3,1,2] => [1,2,3,4] => [4,3,2,1] => 0
[4,3,2,1] => [1,2,3,4] => [4,3,2,1] => 0
[1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 0
[1,2,3,5,4] => [1,2,3,5,4] => [5,4,3,1,2] => 2
[1,2,4,3,5] => [1,2,4,3,5] => [5,4,2,3,1] => 3
[1,2,4,5,3] => [1,2,4,5,3] => [5,4,2,1,3] => 3
[1,2,5,3,4] => [1,2,5,3,4] => [5,4,1,3,2] => 3
[1,2,5,4,3] => [1,2,5,3,4] => [5,4,1,3,2] => 3
[1,3,2,4,5] => [1,3,2,4,5] => [5,3,4,2,1] => 4
[1,3,2,5,4] => [1,3,2,5,4] => [5,3,4,1,2] => 6
[1,3,4,2,5] => [1,3,4,2,5] => [5,3,2,4,1] => 4
[1,3,4,5,2] => [1,3,4,5,2] => [5,3,2,1,4] => 4
[1,3,5,2,4] => [1,3,5,2,4] => [5,3,1,4,2] => 4
[1,3,5,4,2] => [1,3,5,2,4] => [5,3,1,4,2] => 4
[1,4,2,3,5] => [1,4,2,3,5] => [5,2,4,3,1] => 4
[1,4,2,5,3] => [1,4,2,5,3] => [5,2,4,1,3] => 7
[1,4,3,2,5] => [1,4,2,5,3] => [5,2,4,1,3] => 7
[1,4,3,5,2] => [1,4,2,3,5] => [5,2,4,3,1] => 4
[1,4,5,2,3] => [1,4,5,2,3] => [5,2,1,4,3] => 4
[1,4,5,3,2] => [1,4,5,2,3] => [5,2,1,4,3] => 4
Description
The sum of the ascent tops of a permutation.
Matching statistic: St000726
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(load all 3 compositions to match this statistic)
Mp00223: Permutations —runsort⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000726: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00069: Permutations —complement⟶ Permutations
St000726: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [2,1] => 0
[2,1] => [1,2] => [2,1] => 0
[1,2,3] => [1,2,3] => [3,2,1] => 0
[1,3,2] => [1,3,2] => [3,1,2] => 2
[2,1,3] => [1,3,2] => [3,1,2] => 2
[2,3,1] => [1,2,3] => [3,2,1] => 0
[3,1,2] => [1,2,3] => [3,2,1] => 0
[3,2,1] => [1,2,3] => [3,2,1] => 0
[1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 0
[1,2,4,3] => [1,2,4,3] => [4,3,1,2] => 2
[1,3,2,4] => [1,3,2,4] => [4,2,3,1] => 3
[1,3,4,2] => [1,3,4,2] => [4,2,1,3] => 3
[1,4,2,3] => [1,4,2,3] => [4,1,3,2] => 3
[1,4,3,2] => [1,4,2,3] => [4,1,3,2] => 3
[2,1,3,4] => [1,3,4,2] => [4,2,1,3] => 3
[2,1,4,3] => [1,4,2,3] => [4,1,3,2] => 3
[2,3,1,4] => [1,4,2,3] => [4,1,3,2] => 3
[2,3,4,1] => [1,2,3,4] => [4,3,2,1] => 0
[2,4,1,3] => [1,3,2,4] => [4,2,3,1] => 3
[2,4,3,1] => [1,2,4,3] => [4,3,1,2] => 2
[3,1,2,4] => [1,2,4,3] => [4,3,1,2] => 2
[3,1,4,2] => [1,4,2,3] => [4,1,3,2] => 3
[3,2,1,4] => [1,4,2,3] => [4,1,3,2] => 3
[3,2,4,1] => [1,2,4,3] => [4,3,1,2] => 2
[3,4,1,2] => [1,2,3,4] => [4,3,2,1] => 0
[3,4,2,1] => [1,2,3,4] => [4,3,2,1] => 0
[4,1,2,3] => [1,2,3,4] => [4,3,2,1] => 0
[4,1,3,2] => [1,3,2,4] => [4,2,3,1] => 3
[4,2,1,3] => [1,3,2,4] => [4,2,3,1] => 3
[4,2,3,1] => [1,2,3,4] => [4,3,2,1] => 0
[4,3,1,2] => [1,2,3,4] => [4,3,2,1] => 0
[4,3,2,1] => [1,2,3,4] => [4,3,2,1] => 0
[1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 0
[1,2,3,5,4] => [1,2,3,5,4] => [5,4,3,1,2] => 2
[1,2,4,3,5] => [1,2,4,3,5] => [5,4,2,3,1] => 3
[1,2,4,5,3] => [1,2,4,5,3] => [5,4,2,1,3] => 3
[1,2,5,3,4] => [1,2,5,3,4] => [5,4,1,3,2] => 3
[1,2,5,4,3] => [1,2,5,3,4] => [5,4,1,3,2] => 3
[1,3,2,4,5] => [1,3,2,4,5] => [5,3,4,2,1] => 4
[1,3,2,5,4] => [1,3,2,5,4] => [5,3,4,1,2] => 6
[1,3,4,2,5] => [1,3,4,2,5] => [5,3,2,4,1] => 4
[1,3,4,5,2] => [1,3,4,5,2] => [5,3,2,1,4] => 4
[1,3,5,2,4] => [1,3,5,2,4] => [5,3,1,4,2] => 4
[1,3,5,4,2] => [1,3,5,2,4] => [5,3,1,4,2] => 4
[1,4,2,3,5] => [1,4,2,3,5] => [5,2,4,3,1] => 4
[1,4,2,5,3] => [1,4,2,5,3] => [5,2,4,1,3] => 7
[1,4,3,2,5] => [1,4,2,5,3] => [5,2,4,1,3] => 7
[1,4,3,5,2] => [1,4,2,3,5] => [5,2,4,3,1] => 4
[1,4,5,2,3] => [1,4,5,2,3] => [5,2,1,4,3] => 4
[1,4,5,3,2] => [1,4,5,2,3] => [5,2,1,4,3] => 4
Description
The normalized sum of the leaf labels of the increasing binary tree associated to a permutation.
The sum of the leaf labels is at least the size of the permutation, equality is attained for the binary trees that have only one leaf. This statistic is the sum of the leaf labels minus the size of the permutation.
Matching statistic: St000111
Mp00223: Permutations —runsort⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000111: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00064: Permutations —reverse⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000111: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [2,1] => [1,2] => 0
[2,1] => [1,2] => [2,1] => [1,2] => 0
[1,2,3] => [1,2,3] => [3,2,1] => [1,2,3] => 0
[1,3,2] => [1,3,2] => [2,3,1] => [2,1,3] => 2
[2,1,3] => [1,3,2] => [2,3,1] => [2,1,3] => 2
[2,3,1] => [1,2,3] => [3,2,1] => [1,2,3] => 0
[3,1,2] => [1,2,3] => [3,2,1] => [1,2,3] => 0
[3,2,1] => [1,2,3] => [3,2,1] => [1,2,3] => 0
[1,2,3,4] => [1,2,3,4] => [4,3,2,1] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,4,3] => [3,4,2,1] => [2,1,3,4] => 2
[1,3,2,4] => [1,3,2,4] => [4,2,3,1] => [1,3,2,4] => 3
[1,3,4,2] => [1,3,4,2] => [2,4,3,1] => [3,1,2,4] => 3
[1,4,2,3] => [1,4,2,3] => [3,2,4,1] => [2,3,1,4] => 3
[1,4,3,2] => [1,4,2,3] => [3,2,4,1] => [2,3,1,4] => 3
[2,1,3,4] => [1,3,4,2] => [2,4,3,1] => [3,1,2,4] => 3
[2,1,4,3] => [1,4,2,3] => [3,2,4,1] => [2,3,1,4] => 3
[2,3,1,4] => [1,4,2,3] => [3,2,4,1] => [2,3,1,4] => 3
[2,3,4,1] => [1,2,3,4] => [4,3,2,1] => [1,2,3,4] => 0
[2,4,1,3] => [1,3,2,4] => [4,2,3,1] => [1,3,2,4] => 3
[2,4,3,1] => [1,2,4,3] => [3,4,2,1] => [2,1,3,4] => 2
[3,1,2,4] => [1,2,4,3] => [3,4,2,1] => [2,1,3,4] => 2
[3,1,4,2] => [1,4,2,3] => [3,2,4,1] => [2,3,1,4] => 3
[3,2,1,4] => [1,4,2,3] => [3,2,4,1] => [2,3,1,4] => 3
[3,2,4,1] => [1,2,4,3] => [3,4,2,1] => [2,1,3,4] => 2
[3,4,1,2] => [1,2,3,4] => [4,3,2,1] => [1,2,3,4] => 0
[3,4,2,1] => [1,2,3,4] => [4,3,2,1] => [1,2,3,4] => 0
[4,1,2,3] => [1,2,3,4] => [4,3,2,1] => [1,2,3,4] => 0
[4,1,3,2] => [1,3,2,4] => [4,2,3,1] => [1,3,2,4] => 3
[4,2,1,3] => [1,3,2,4] => [4,2,3,1] => [1,3,2,4] => 3
[4,2,3,1] => [1,2,3,4] => [4,3,2,1] => [1,2,3,4] => 0
[4,3,1,2] => [1,2,3,4] => [4,3,2,1] => [1,2,3,4] => 0
[4,3,2,1] => [1,2,3,4] => [4,3,2,1] => [1,2,3,4] => 0
[1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,2,3,5,4] => [4,5,3,2,1] => [2,1,3,4,5] => 2
[1,2,4,3,5] => [1,2,4,3,5] => [5,3,4,2,1] => [1,3,2,4,5] => 3
[1,2,4,5,3] => [1,2,4,5,3] => [3,5,4,2,1] => [3,1,2,4,5] => 3
[1,2,5,3,4] => [1,2,5,3,4] => [4,3,5,2,1] => [2,3,1,4,5] => 3
[1,2,5,4,3] => [1,2,5,3,4] => [4,3,5,2,1] => [2,3,1,4,5] => 3
[1,3,2,4,5] => [1,3,2,4,5] => [5,4,2,3,1] => [1,2,4,3,5] => 4
[1,3,2,5,4] => [1,3,2,5,4] => [4,5,2,3,1] => [2,1,4,3,5] => 6
[1,3,4,2,5] => [1,3,4,2,5] => [5,2,4,3,1] => [1,4,2,3,5] => 4
[1,3,4,5,2] => [1,3,4,5,2] => [2,5,4,3,1] => [4,1,2,3,5] => 4
[1,3,5,2,4] => [1,3,5,2,4] => [4,2,5,3,1] => [2,4,1,3,5] => 4
[1,3,5,4,2] => [1,3,5,2,4] => [4,2,5,3,1] => [2,4,1,3,5] => 4
[1,4,2,3,5] => [1,4,2,3,5] => [5,3,2,4,1] => [1,3,4,2,5] => 4
[1,4,2,5,3] => [1,4,2,5,3] => [3,5,2,4,1] => [3,1,4,2,5] => 7
[1,4,3,2,5] => [1,4,2,5,3] => [3,5,2,4,1] => [3,1,4,2,5] => 7
[1,4,3,5,2] => [1,4,2,3,5] => [5,3,2,4,1] => [1,3,4,2,5] => 4
[1,4,5,2,3] => [1,4,5,2,3] => [3,2,5,4,1] => [3,4,1,2,5] => 4
[1,4,5,3,2] => [1,4,5,2,3] => [3,2,5,4,1] => [3,4,1,2,5] => 4
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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