Processing math: 30%

Your data matches 22 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000740
(load all 28 compositions to match this statistic)
Mapping
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000740: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => 1
[1,2] => [1,2] => 2
[2,1] => [1,2] => 2
[1,2,3] => [1,2,3] => 3
[1,3,2] => [1,2,3] => 3
[2,1,3] => [1,2,3] => 3
[2,3,1] => [1,2,3] => 3
[3,1,2] => [1,3,2] => 2
[3,2,1] => [1,3,2] => 2
[1,2,3,4] => [1,2,3,4] => 4
[1,2,4,3] => [1,2,3,4] => 4
[1,3,2,4] => [1,2,3,4] => 4
[1,3,4,2] => [1,2,3,4] => 4
[1,4,2,3] => [1,2,4,3] => 3
[1,4,3,2] => [1,2,4,3] => 3
[2,1,3,4] => [1,2,3,4] => 4
[2,1,4,3] => [1,2,3,4] => 4
[2,3,1,4] => [1,2,3,4] => 4
[2,3,4,1] => [1,2,3,4] => 4
[2,4,1,3] => [1,2,4,3] => 3
[2,4,3,1] => [1,2,4,3] => 3
[3,1,2,4] => [1,3,2,4] => 4
[3,1,4,2] => [1,3,4,2] => 2
[3,2,1,4] => [1,3,2,4] => 4
[3,2,4,1] => [1,3,4,2] => 2
[3,4,1,2] => [1,3,2,4] => 4
[3,4,2,1] => [1,3,2,4] => 4
[4,1,2,3] => [1,4,3,2] => 2
[4,1,3,2] => [1,4,2,3] => 3
[4,2,1,3] => [1,4,3,2] => 2
[4,2,3,1] => [1,4,2,3] => 3
[4,3,1,2] => [1,4,2,3] => 3
[4,3,2,1] => [1,4,2,3] => 3
[1,2,3,4,5] => [1,2,3,4,5] => 5
[1,2,3,5,4] => [1,2,3,4,5] => 5
[1,2,4,3,5] => [1,2,3,4,5] => 5
[1,2,4,5,3] => [1,2,3,4,5] => 5
[1,2,5,3,4] => [1,2,3,5,4] => 4
[1,2,5,4,3] => [1,2,3,5,4] => 4
[1,3,2,4,5] => [1,2,3,4,5] => 5
[1,3,2,5,4] => [1,2,3,4,5] => 5
[1,3,4,2,5] => [1,2,3,4,5] => 5
[1,3,4,5,2] => [1,2,3,4,5] => 5
[1,3,5,2,4] => [1,2,3,5,4] => 4
[1,3,5,4,2] => [1,2,3,5,4] => 4
[1,4,2,3,5] => [1,2,4,3,5] => 5
[1,4,2,5,3] => [1,2,4,5,3] => 3
[1,4,3,2,5] => [1,2,4,3,5] => 5
[1,4,3,5,2] => [1,2,4,5,3] => 3
[1,4,5,2,3] => [1,2,4,3,5] => 5
Description
The last entry of a permutation. This statistic is undefined for the empty permutation.
Matching statistic: St000025
Mapping
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000025: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1,0]
 => 1
[1,2] => [1,2] => [2,1] => [1,1,0,0]
 => 2
[2,1] => [1,2] => [2,1] => [1,1,0,0]
 => 2
[1,2,3] => [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
 => 3
[1,3,2] => [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
 => 3
[2,1,3] => [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
 => 3
[2,3,1] => [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
 => 3
[3,1,2] => [1,3,2] => [2,3,1] => [1,1,0,1,0,0]
 => 2
[3,2,1] => [1,3,2] => [2,3,1] => [1,1,0,1,0,0]
 => 2
[1,2,3,4] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => 4
[1,2,4,3] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => 4
[1,3,2,4] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => 4
[1,3,4,2] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => 4
[1,4,2,3] => [1,2,4,3] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
 => 3
[1,4,3,2] => [1,2,4,3] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
 => 3
[2,1,3,4] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => 4
[2,1,4,3] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => 4
[2,3,1,4] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => 4
[2,3,4,1] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => 4
[2,4,1,3] => [1,2,4,3] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
 => 3
[2,4,3,1] => [1,2,4,3] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
 => 3
[3,1,2,4] => [1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => 4
[3,1,4,2] => [1,3,4,2] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
 => 2
[3,2,1,4] => [1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => 4
[3,2,4,1] => [1,3,4,2] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
 => 2
[3,4,1,2] => [1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => 4
[3,4,2,1] => [1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => 4
[4,1,2,3] => [1,4,3,2] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
 => 2
[4,1,3,2] => [1,4,2,3] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
 => 3
[4,2,1,3] => [1,4,3,2] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
 => 2
[4,2,3,1] => [1,4,2,3] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
 => 3
[4,3,1,2] => [1,4,2,3] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
 => 3
[4,3,2,1] => [1,4,2,3] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
 => 3
[1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => 5
[1,2,3,5,4] => [1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => 5
[1,2,4,3,5] => [1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => 5
[1,2,4,5,3] => [1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => 5
[1,2,5,3,4] => [1,2,3,5,4] => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
 => 4
[1,2,5,4,3] => [1,2,3,5,4] => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
 => 4
[1,3,2,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => 5
[1,3,2,5,4] => [1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => 5
[1,3,4,2,5] => [1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => 5
[1,3,4,5,2] => [1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => 5
[1,3,5,2,4] => [1,2,3,5,4] => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
 => 4
[1,3,5,4,2] => [1,2,3,5,4] => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
 => 4
[1,4,2,3,5] => [1,2,4,3,5] => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => 5
[1,4,2,5,3] => [1,2,4,5,3] => [3,5,4,2,1] => [1,1,1,0,1,1,0,0,0,0]
 => 3
[1,4,3,2,5] => [1,2,4,3,5] => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => 5
[1,4,3,5,2] => [1,2,4,5,3] => [3,5,4,2,1] => [1,1,1,0,1,1,0,0,0,0]
 => 3
[1,4,5,2,3] => [1,2,4,3,5] => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => 5
Description
The number of initial rises of a Dyck path. In other words, this is the height of the first peak of D.
Matching statistic: St000439
Mapping
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000439: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1,0]
 => 2 = 1 + 1
[1,2] => [1,2] => [2,1] => [1,1,0,0]
 => 3 = 2 + 1
[2,1] => [1,2] => [2,1] => [1,1,0,0]
 => 3 = 2 + 1
[1,2,3] => [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
 => 4 = 3 + 1
[1,3,2] => [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
 => 4 = 3 + 1
[2,1,3] => [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
 => 4 = 3 + 1
[2,3,1] => [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
 => 4 = 3 + 1
[3,1,2] => [1,3,2] => [2,3,1] => [1,1,0,1,0,0]
 => 3 = 2 + 1
[3,2,1] => [1,3,2] => [2,3,1] => [1,1,0,1,0,0]
 => 3 = 2 + 1
[1,2,3,4] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => 5 = 4 + 1
[1,2,4,3] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => 5 = 4 + 1
[1,3,2,4] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => 5 = 4 + 1
[1,3,4,2] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => 5 = 4 + 1
[1,4,2,3] => [1,2,4,3] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
 => 4 = 3 + 1
[1,4,3,2] => [1,2,4,3] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
 => 4 = 3 + 1
[2,1,3,4] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => 5 = 4 + 1
[2,1,4,3] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => 5 = 4 + 1
[2,3,1,4] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => 5 = 4 + 1
[2,3,4,1] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => 5 = 4 + 1
[2,4,1,3] => [1,2,4,3] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
 => 4 = 3 + 1
[2,4,3,1] => [1,2,4,3] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
 => 4 = 3 + 1
[3,1,2,4] => [1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => 5 = 4 + 1
[3,1,4,2] => [1,3,4,2] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
 => 3 = 2 + 1
[3,2,1,4] => [1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => 5 = 4 + 1
[3,2,4,1] => [1,3,4,2] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
 => 3 = 2 + 1
[3,4,1,2] => [1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => 5 = 4 + 1
[3,4,2,1] => [1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => 5 = 4 + 1
[4,1,2,3] => [1,4,3,2] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
 => 3 = 2 + 1
[4,1,3,2] => [1,4,2,3] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
 => 4 = 3 + 1
[4,2,1,3] => [1,4,3,2] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
 => 3 = 2 + 1
[4,2,3,1] => [1,4,2,3] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
 => 4 = 3 + 1
[4,3,1,2] => [1,4,2,3] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
 => 4 = 3 + 1
[4,3,2,1] => [1,4,2,3] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
 => 4 = 3 + 1
[1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => 6 = 5 + 1
[1,2,3,5,4] => [1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => 6 = 5 + 1
[1,2,4,3,5] => [1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => 6 = 5 + 1
[1,2,4,5,3] => [1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => 6 = 5 + 1
[1,2,5,3,4] => [1,2,3,5,4] => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
 => 5 = 4 + 1
[1,2,5,4,3] => [1,2,3,5,4] => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
 => 5 = 4 + 1
[1,3,2,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => 6 = 5 + 1
[1,3,2,5,4] => [1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => 6 = 5 + 1
[1,3,4,2,5] => [1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => 6 = 5 + 1
[1,3,4,5,2] => [1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => 6 = 5 + 1
[1,3,5,2,4] => [1,2,3,5,4] => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
 => 5 = 4 + 1
[1,3,5,4,2] => [1,2,3,5,4] => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
 => 5 = 4 + 1
[1,4,2,3,5] => [1,2,4,3,5] => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => 6 = 5 + 1
[1,4,2,5,3] => [1,2,4,5,3] => [3,5,4,2,1] => [1,1,1,0,1,1,0,0,0,0]
 => 4 = 3 + 1
[1,4,3,2,5] => [1,2,4,3,5] => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => 6 = 5 + 1
[1,4,3,5,2] => [1,2,4,5,3] => [3,5,4,2,1] => [1,1,1,0,1,1,0,0,0,0]
 => 4 = 3 + 1
[1,4,5,2,3] => [1,2,4,3,5] => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => 6 = 5 + 1
Description
The position of the first down step of a Dyck path.
Matching statistic: St000054
(load all 9 compositions to match this statistic)
Mapping
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00088: Permutations —Kreweras complement⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000054: Permutations ⟶ ℤResult quality: 33% ●values known / values provided: 33%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1] => 1
[1,2] => [1,2] => [2,1] => [2,1] => 2
[2,1] => [1,2] => [2,1] => [2,1] => 2
[1,2,3] => [1,2,3] => [2,3,1] => [3,1,2] => 3
[1,3,2] => [1,2,3] => [2,3,1] => [3,1,2] => 3
[2,1,3] => [1,2,3] => [2,3,1] => [3,1,2] => 3
[2,3,1] => [1,2,3] => [2,3,1] => [3,1,2] => 3
[3,1,2] => [1,3,2] => [2,1,3] => [2,1,3] => 2
[3,2,1] => [1,3,2] => [2,1,3] => [2,1,3] => 2
[1,2,3,4] => [1,2,3,4] => [2,3,4,1] => [4,1,2,3] => 4
[1,2,4,3] => [1,2,3,4] => [2,3,4,1] => [4,1,2,3] => 4
[1,3,2,4] => [1,2,3,4] => [2,3,4,1] => [4,1,2,3] => 4
[1,3,4,2] => [1,2,3,4] => [2,3,4,1] => [4,1,2,3] => 4
[1,4,2,3] => [1,2,4,3] => [2,3,1,4] => [3,1,2,4] => 3
[1,4,3,2] => [1,2,4,3] => [2,3,1,4] => [3,1,2,4] => 3
[2,1,3,4] => [1,2,3,4] => [2,3,4,1] => [4,1,2,3] => 4
[2,1,4,3] => [1,2,3,4] => [2,3,4,1] => [4,1,2,3] => 4
[2,3,1,4] => [1,2,3,4] => [2,3,4,1] => [4,1,2,3] => 4
[2,3,4,1] => [1,2,3,4] => [2,3,4,1] => [4,1,2,3] => 4
[2,4,1,3] => [1,2,4,3] => [2,3,1,4] => [3,1,2,4] => 3
[2,4,3,1] => [1,2,4,3] => [2,3,1,4] => [3,1,2,4] => 3
[3,1,2,4] => [1,3,2,4] => [2,4,3,1] => [4,1,3,2] => 4
[3,1,4,2] => [1,3,4,2] => [2,1,3,4] => [2,1,3,4] => 2
[3,2,1,4] => [1,3,2,4] => [2,4,3,1] => [4,1,3,2] => 4
[3,2,4,1] => [1,3,4,2] => [2,1,3,4] => [2,1,3,4] => 2
[3,4,1,2] => [1,3,2,4] => [2,4,3,1] => [4,1,3,2] => 4
[3,4,2,1] => [1,3,2,4] => [2,4,3,1] => [4,1,3,2] => 4
[4,1,2,3] => [1,4,3,2] => [2,1,4,3] => [2,1,4,3] => 2
[4,1,3,2] => [1,4,2,3] => [2,4,1,3] => [3,1,4,2] => 3
[4,2,1,3] => [1,4,3,2] => [2,1,4,3] => [2,1,4,3] => 2
[4,2,3,1] => [1,4,2,3] => [2,4,1,3] => [3,1,4,2] => 3
[4,3,1,2] => [1,4,2,3] => [2,4,1,3] => [3,1,4,2] => 3
[4,3,2,1] => [1,4,2,3] => [2,4,1,3] => [3,1,4,2] => 3
[1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => [5,1,2,3,4] => 5
[1,2,3,5,4] => [1,2,3,4,5] => [2,3,4,5,1] => [5,1,2,3,4] => 5
[1,2,4,3,5] => [1,2,3,4,5] => [2,3,4,5,1] => [5,1,2,3,4] => 5
[1,2,4,5,3] => [1,2,3,4,5] => [2,3,4,5,1] => [5,1,2,3,4] => 5
[1,2,5,3,4] => [1,2,3,5,4] => [2,3,4,1,5] => [4,1,2,3,5] => 4
[1,2,5,4,3] => [1,2,3,5,4] => [2,3,4,1,5] => [4,1,2,3,5] => 4
[1,3,2,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => [5,1,2,3,4] => 5
[1,3,2,5,4] => [1,2,3,4,5] => [2,3,4,5,1] => [5,1,2,3,4] => 5
[1,3,4,2,5] => [1,2,3,4,5] => [2,3,4,5,1] => [5,1,2,3,4] => 5
[1,3,4,5,2] => [1,2,3,4,5] => [2,3,4,5,1] => [5,1,2,3,4] => 5
[1,3,5,2,4] => [1,2,3,5,4] => [2,3,4,1,5] => [4,1,2,3,5] => 4
[1,3,5,4,2] => [1,2,3,5,4] => [2,3,4,1,5] => [4,1,2,3,5] => 4
[1,4,2,3,5] => [1,2,4,3,5] => [2,3,5,4,1] => [5,1,2,4,3] => 5
[1,4,2,5,3] => [1,2,4,5,3] => [2,3,1,4,5] => [3,1,2,4,5] => 3
[1,4,3,2,5] => [1,2,4,3,5] => [2,3,5,4,1] => [5,1,2,4,3] => 5
[1,4,3,5,2] => [1,2,4,5,3] => [2,3,1,4,5] => [3,1,2,4,5] => 3
[1,4,5,2,3] => [1,2,4,3,5] => [2,3,5,4,1] => [5,1,2,4,3] => 5
[1,2,3,6,4,5,7] => [1,2,3,4,6,5,7] => [2,3,4,5,7,6,1] => [7,1,2,3,4,6,5] => ? = 7
[1,2,3,6,5,4,7] => [1,2,3,4,6,5,7] => [2,3,4,5,7,6,1] => [7,1,2,3,4,6,5] => ? = 7
[1,2,3,6,7,4,5] => [1,2,3,4,6,5,7] => [2,3,4,5,7,6,1] => [7,1,2,3,4,6,5] => ? = 7
[1,2,3,6,7,5,4] => [1,2,3,4,6,5,7] => [2,3,4,5,7,6,1] => [7,1,2,3,4,6,5] => ? = 7
[1,2,4,6,3,5,7] => [1,2,3,4,6,5,7] => [2,3,4,5,7,6,1] => [7,1,2,3,4,6,5] => ? = 7
[1,2,4,6,5,3,7] => [1,2,3,4,6,5,7] => [2,3,4,5,7,6,1] => [7,1,2,3,4,6,5] => ? = 7
[1,2,4,6,7,3,5] => [1,2,3,4,6,5,7] => [2,3,4,5,7,6,1] => [7,1,2,3,4,6,5] => ? = 7
[1,2,4,6,7,5,3] => [1,2,3,4,6,5,7] => [2,3,4,5,7,6,1] => [7,1,2,3,4,6,5] => ? = 7
[1,2,5,3,4,6,7] => [1,2,3,5,4,6,7] => [2,3,4,6,5,7,1] => [7,1,2,3,5,4,6] => ? = 7
[1,2,5,3,4,7,6] => [1,2,3,5,4,6,7] => [2,3,4,6,5,7,1] => [7,1,2,3,5,4,6] => ? = 7
[1,2,5,3,6,4,7] => [1,2,3,5,6,4,7] => [2,3,4,7,5,6,1] => [7,1,2,3,5,6,4] => ? = 7
[1,2,5,3,7,6,4] => [1,2,3,5,7,4,6] => [2,3,4,7,5,1,6] => [6,1,2,3,5,7,4] => ? = 6
[1,2,5,4,3,6,7] => [1,2,3,5,4,6,7] => [2,3,4,6,5,7,1] => [7,1,2,3,5,4,6] => ? = 7
[1,2,5,4,3,7,6] => [1,2,3,5,4,6,7] => [2,3,4,6,5,7,1] => [7,1,2,3,5,4,6] => ? = 7
[1,2,5,4,6,3,7] => [1,2,3,5,6,4,7] => [2,3,4,7,5,6,1] => [7,1,2,3,5,6,4] => ? = 7
[1,2,5,4,7,6,3] => [1,2,3,5,7,4,6] => [2,3,4,7,5,1,6] => [6,1,2,3,5,7,4] => ? = 6
[1,2,5,6,3,4,7] => [1,2,3,5,4,6,7] => [2,3,4,6,5,7,1] => [7,1,2,3,5,4,6] => ? = 7
[1,2,5,6,3,7,4] => [1,2,3,5,4,6,7] => [2,3,4,6,5,7,1] => [7,1,2,3,5,4,6] => ? = 7
[1,2,5,6,4,3,7] => [1,2,3,5,4,6,7] => [2,3,4,6,5,7,1] => [7,1,2,3,5,4,6] => ? = 7
[1,2,5,6,4,7,3] => [1,2,3,5,4,6,7] => [2,3,4,6,5,7,1] => [7,1,2,3,5,4,6] => ? = 7
[1,2,5,6,7,3,4] => [1,2,3,5,7,4,6] => [2,3,4,7,5,1,6] => [6,1,2,3,5,7,4] => ? = 6
[1,2,5,6,7,4,3] => [1,2,3,5,7,4,6] => [2,3,4,7,5,1,6] => [6,1,2,3,5,7,4] => ? = 6
[1,2,5,7,3,4,6] => [1,2,3,5,4,7,6] => [2,3,4,6,5,1,7] => [6,1,2,3,5,4,7] => ? = 6
[1,2,5,7,3,6,4] => [1,2,3,5,4,7,6] => [2,3,4,6,5,1,7] => [6,1,2,3,5,4,7] => ? = 6
[1,2,5,7,4,3,6] => [1,2,3,5,4,7,6] => [2,3,4,6,5,1,7] => [6,1,2,3,5,4,7] => ? = 6
[1,2,5,7,4,6,3] => [1,2,3,5,4,7,6] => [2,3,4,6,5,1,7] => [6,1,2,3,5,4,7] => ? = 6
[1,2,5,7,6,3,4] => [1,2,3,5,6,4,7] => [2,3,4,7,5,6,1] => [7,1,2,3,5,6,4] => ? = 7
[1,2,5,7,6,4,3] => [1,2,3,5,6,4,7] => [2,3,4,7,5,6,1] => [7,1,2,3,5,6,4] => ? = 7
[1,2,6,3,4,5,7] => [1,2,3,6,5,4,7] => [2,3,4,7,6,5,1] => [7,1,2,3,6,5,4] => ? = 7
[1,2,6,3,4,7,5] => [1,2,3,6,7,5,4] => [2,3,4,1,7,5,6] => [4,1,2,3,6,7,5] => ? = 4
[1,2,6,3,5,4,7] => [1,2,3,6,4,5,7] => [2,3,4,6,7,5,1] => [7,1,2,3,6,4,5] => ? = 7
[1,2,6,3,7,4,5] => [1,2,3,6,4,5,7] => [2,3,4,6,7,5,1] => [7,1,2,3,6,4,5] => ? = 7
[1,2,6,3,7,5,4] => [1,2,3,6,5,7,4] => [2,3,4,1,6,5,7] => [4,1,2,3,6,5,7] => ? = 4
[1,2,6,4,3,5,7] => [1,2,3,6,5,4,7] => [2,3,4,7,6,5,1] => [7,1,2,3,6,5,4] => ? = 7
[1,2,6,4,3,7,5] => [1,2,3,6,7,5,4] => [2,3,4,1,7,5,6] => [4,1,2,3,6,7,5] => ? = 4
[1,2,6,4,5,3,7] => [1,2,3,6,4,5,7] => [2,3,4,6,7,5,1] => [7,1,2,3,6,4,5] => ? = 7
[1,2,6,4,7,3,5] => [1,2,3,6,4,5,7] => [2,3,4,6,7,5,1] => [7,1,2,3,6,4,5] => ? = 7
[1,2,6,4,7,5,3] => [1,2,3,6,5,7,4] => [2,3,4,1,6,5,7] => [4,1,2,3,6,5,7] => ? = 4
[1,2,6,5,3,4,7] => [1,2,3,6,4,5,7] => [2,3,4,6,7,5,1] => [7,1,2,3,6,4,5] => ? = 7
[1,2,6,5,4,3,7] => [1,2,3,6,4,5,7] => [2,3,4,6,7,5,1] => [7,1,2,3,6,4,5] => ? = 7
[1,2,6,5,7,3,4] => [1,2,3,6,4,5,7] => [2,3,4,6,7,5,1] => [7,1,2,3,6,4,5] => ? = 7
[1,2,6,5,7,4,3] => [1,2,3,6,4,5,7] => [2,3,4,6,7,5,1] => [7,1,2,3,6,4,5] => ? = 7
[1,2,6,7,3,5,4] => [1,2,3,6,5,4,7] => [2,3,4,7,6,5,1] => [7,1,2,3,6,5,4] => ? = 7
[1,2,6,7,4,5,3] => [1,2,3,6,5,4,7] => [2,3,4,7,6,5,1] => [7,1,2,3,6,5,4] => ? = 7
[1,2,7,3,4,5,6] => [1,2,3,7,6,5,4] => [2,3,4,1,7,6,5] => [4,1,2,3,7,6,5] => ? = 4
[1,2,7,3,4,6,5] => [1,2,3,7,5,4,6] => [2,3,4,7,6,1,5] => [6,1,2,3,7,5,4] => ? = 6
[1,2,7,3,5,4,6] => [1,2,3,7,6,4,5] => [2,3,4,7,1,6,5] => [5,1,2,3,7,6,4] => ? = 5
[1,2,7,3,6,4,5] => [1,2,3,7,5,6,4] => [2,3,4,1,6,7,5] => [4,1,2,3,7,5,6] => ? = 4
[1,2,7,4,3,5,6] => [1,2,3,7,6,5,4] => [2,3,4,1,7,6,5] => [4,1,2,3,7,6,5] => ? = 4
[1,2,7,4,3,6,5] => [1,2,3,7,5,4,6] => [2,3,4,7,6,1,5] => [6,1,2,3,7,5,4] => ? = 6
Description
The first entry of the permutation. This can be described as 1 plus the number of occurrences of the vincular pattern ([2,1], {(0,0),(0,1),(0,2)}), i.e., the first column is shaded, see [1]. This statistic is related to the number of deficiencies [[St000703]] as follows: consider the arc diagram of a permutation π of n, together with its rotations, obtained by conjugating with the long cycle (1,…,n). Drawing the labels 1 to n in this order on a circle, and the arcs (i,π(i)) as straight lines, the rotation of π is obtained by replacing each number i by (imod. Then, \pi(1)-1 is the number of rotations of \pi where the arc (1, \pi(1)) is a deficiency. In particular, if O(\pi) is the orbit of rotations of \pi, then the number of deficiencies of \pi equals \frac{1}{|O(\pi)|}\sum_{\sigma\in O(\pi)} (\sigma(1)-1).
Matching statistic: St001291
(load all 3 compositions to match this statistic)
Mapping
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001291: Dyck paths ⟶ ℤResult quality: 24% ●values known / values provided: 24%●distinct values known / distinct values provided: 86%
Values
[1] => [1] => [1] => [1,0]
 => 1
[1,2] => [1,2] => [1,2] => [1,0,1,0]
 => 2
[2,1] => [1,2] => [1,2] => [1,0,1,0]
 => 2
[1,2,3] => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
 => 3
[1,3,2] => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
 => 3
[2,1,3] => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
 => 3
[2,3,1] => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
 => 3
[3,1,2] => [1,3,2] => [1,3,2] => [1,0,1,1,0,0]
 => 2
[3,2,1] => [1,3,2] => [1,3,2] => [1,0,1,1,0,0]
 => 2
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => 4
[1,2,4,3] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => 4
[1,3,2,4] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => 4
[1,3,4,2] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => 4
[1,4,2,3] => [1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
 => 3
[1,4,3,2] => [1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
 => 3
[2,1,3,4] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => 4
[2,1,4,3] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => 4
[2,3,1,4] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => 4
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => 4
[2,4,1,3] => [1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
 => 3
[2,4,3,1] => [1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
 => 3
[3,1,2,4] => [1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
 => 4
[3,1,4,2] => [1,3,4,2] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
 => 2
[3,2,1,4] => [1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
 => 4
[3,2,4,1] => [1,3,4,2] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
 => 2
[3,4,1,2] => [1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
 => 4
[3,4,2,1] => [1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
 => 4
[4,1,2,3] => [1,4,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => 2
[4,1,3,2] => [1,4,2,3] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
 => 3
[4,2,1,3] => [1,4,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => 2
[4,2,3,1] => [1,4,2,3] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
 => 3
[4,3,1,2] => [1,4,2,3] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
 => 3
[4,3,2,1] => [1,4,2,3] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
 => 3
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => 5
[1,2,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => 5
[1,2,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => 5
[1,2,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => 5
[1,2,5,3,4] => [1,2,3,5,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
 => 4
[1,2,5,4,3] => [1,2,3,5,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
 => 4
[1,3,2,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => 5
[1,3,2,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => 5
[1,3,4,2,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => 5
[1,3,4,5,2] => [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => 5
[1,3,5,2,4] => [1,2,3,5,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
 => 4
[1,3,5,4,2] => [1,2,3,5,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
 => 4
[1,4,2,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
 => 5
[1,4,2,5,3] => [1,2,4,5,3] => [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
 => 3
[1,4,3,2,5] => [1,2,4,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
 => 5
[1,4,3,5,2] => [1,2,4,5,3] => [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
 => 3
[1,4,5,2,3] => [1,2,4,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
 => 5
[1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 7
[1,2,3,4,5,7,6] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 7
[1,2,3,4,6,5,7] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 7
[1,2,3,4,6,7,5] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 7
[1,2,3,4,7,5,6] => [1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 6
[1,2,3,4,7,6,5] => [1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 6
[1,2,3,5,4,6,7] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 7
[1,2,3,5,4,7,6] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 7
[1,2,3,5,6,4,7] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 7
[1,2,3,5,6,7,4] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 7
[1,2,3,5,7,4,6] => [1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 6
[1,2,3,5,7,6,4] => [1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 6
[1,2,3,6,4,5,7] => [1,2,3,4,6,5,7] => [1,2,3,4,6,5,7] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 7
[1,2,3,6,4,7,5] => [1,2,3,4,6,7,5] => [1,2,3,4,7,5,6] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 5
[1,2,3,6,5,4,7] => [1,2,3,4,6,5,7] => [1,2,3,4,6,5,7] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 7
[1,2,3,6,5,7,4] => [1,2,3,4,6,7,5] => [1,2,3,4,7,5,6] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 5
[1,2,3,6,7,4,5] => [1,2,3,4,6,5,7] => [1,2,3,4,6,5,7] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 7
[1,2,3,6,7,5,4] => [1,2,3,4,6,5,7] => [1,2,3,4,6,5,7] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 7
[1,2,3,7,4,5,6] => [1,2,3,4,7,6,5] => [1,2,3,4,7,6,5] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 5
[1,2,3,7,4,6,5] => [1,2,3,4,7,5,6] => [1,2,3,4,6,7,5] => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 6
[1,2,3,7,5,4,6] => [1,2,3,4,7,6,5] => [1,2,3,4,7,6,5] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 5
[1,2,3,7,5,6,4] => [1,2,3,4,7,5,6] => [1,2,3,4,6,7,5] => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 6
[1,2,3,7,6,4,5] => [1,2,3,4,7,5,6] => [1,2,3,4,6,7,5] => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 6
[1,2,3,7,6,5,4] => [1,2,3,4,7,5,6] => [1,2,3,4,6,7,5] => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 6
[1,2,4,3,5,6,7] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 7
[1,2,4,3,5,7,6] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 7
[1,2,4,3,6,5,7] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 7
[1,2,4,3,6,7,5] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 7
[1,2,4,3,7,5,6] => [1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 6
[1,2,4,3,7,6,5] => [1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 6
[1,2,4,5,3,6,7] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 7
[1,2,4,5,3,7,6] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 7
[1,2,4,5,6,3,7] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 7
[1,2,4,5,6,7,3] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 7
[1,2,4,5,7,3,6] => [1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 6
[1,2,4,5,7,6,3] => [1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 6
[1,2,4,6,3,5,7] => [1,2,3,4,6,5,7] => [1,2,3,4,6,5,7] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 7
[1,2,4,6,3,7,5] => [1,2,3,4,6,7,5] => [1,2,3,4,7,5,6] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 5
[1,2,4,6,5,3,7] => [1,2,3,4,6,5,7] => [1,2,3,4,6,5,7] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 7
[1,2,4,6,5,7,3] => [1,2,3,4,6,7,5] => [1,2,3,4,7,5,6] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 5
[1,2,4,6,7,3,5] => [1,2,3,4,6,5,7] => [1,2,3,4,6,5,7] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 7
[1,2,4,6,7,5,3] => [1,2,3,4,6,5,7] => [1,2,3,4,6,5,7] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 7
[1,2,4,7,3,5,6] => [1,2,3,4,7,6,5] => [1,2,3,4,7,6,5] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 5
[1,2,4,7,3,6,5] => [1,2,3,4,7,5,6] => [1,2,3,4,6,7,5] => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 6
[1,2,4,7,5,3,6] => [1,2,3,4,7,6,5] => [1,2,3,4,7,6,5] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 5
[1,2,4,7,5,6,3] => [1,2,3,4,7,5,6] => [1,2,3,4,6,7,5] => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 6
[1,2,4,7,6,3,5] => [1,2,3,4,7,5,6] => [1,2,3,4,6,7,5] => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 6
[1,2,4,7,6,5,3] => [1,2,3,4,7,5,6] => [1,2,3,4,6,7,5] => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 6
[1,2,5,3,4,6,7] => [1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => ? = 7
[1,2,5,3,4,7,6] => [1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => ? = 7
Description
The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. Let A be the Nakayama algebra associated to a Dyck path as given in [[DyckPaths/NakayamaAlgebras]]. This statistics is the number of indecomposable summands of D(A) \otimes D(A), where D(A) is the natural dual of A.
Matching statistic: St000051
(load all 2 compositions to match this statistic)
Mapping
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00088: Permutations —Kreweras complement⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
St000051: Binary trees ⟶ ℤResult quality: 24% ●values known / values provided: 24%●distinct values known / distinct values provided: 86%
Values
[1] => [1] => [1] => [.,.]
 => 0 = 1 - 1
[1,2] => [1,2] => [2,1] => [[.,.],.]
 => 1 = 2 - 1
[2,1] => [1,2] => [2,1] => [[.,.],.]
 => 1 = 2 - 1
[1,2,3] => [1,2,3] => [2,3,1] => [[.,[.,.]],.]
 => 2 = 3 - 1
[1,3,2] => [1,2,3] => [2,3,1] => [[.,[.,.]],.]
 => 2 = 3 - 1
[2,1,3] => [1,2,3] => [2,3,1] => [[.,[.,.]],.]
 => 2 = 3 - 1
[2,3,1] => [1,2,3] => [2,3,1] => [[.,[.,.]],.]
 => 2 = 3 - 1
[3,1,2] => [1,3,2] => [2,1,3] => [[.,.],[.,.]]
 => 1 = 2 - 1
[3,2,1] => [1,3,2] => [2,1,3] => [[.,.],[.,.]]
 => 1 = 2 - 1
[1,2,3,4] => [1,2,3,4] => [2,3,4,1] => [[.,[.,[.,.]]],.]
 => 3 = 4 - 1
[1,2,4,3] => [1,2,3,4] => [2,3,4,1] => [[.,[.,[.,.]]],.]
 => 3 = 4 - 1
[1,3,2,4] => [1,2,3,4] => [2,3,4,1] => [[.,[.,[.,.]]],.]
 => 3 = 4 - 1
[1,3,4,2] => [1,2,3,4] => [2,3,4,1] => [[.,[.,[.,.]]],.]
 => 3 = 4 - 1
[1,4,2,3] => [1,2,4,3] => [2,3,1,4] => [[.,[.,.]],[.,.]]
 => 2 = 3 - 1
[1,4,3,2] => [1,2,4,3] => [2,3,1,4] => [[.,[.,.]],[.,.]]
 => 2 = 3 - 1
[2,1,3,4] => [1,2,3,4] => [2,3,4,1] => [[.,[.,[.,.]]],.]
 => 3 = 4 - 1
[2,1,4,3] => [1,2,3,4] => [2,3,4,1] => [[.,[.,[.,.]]],.]
 => 3 = 4 - 1
[2,3,1,4] => [1,2,3,4] => [2,3,4,1] => [[.,[.,[.,.]]],.]
 => 3 = 4 - 1
[2,3,4,1] => [1,2,3,4] => [2,3,4,1] => [[.,[.,[.,.]]],.]
 => 3 = 4 - 1
[2,4,1,3] => [1,2,4,3] => [2,3,1,4] => [[.,[.,.]],[.,.]]
 => 2 = 3 - 1
[2,4,3,1] => [1,2,4,3] => [2,3,1,4] => [[.,[.,.]],[.,.]]
 => 2 = 3 - 1
[3,1,2,4] => [1,3,2,4] => [2,4,3,1] => [[.,[[.,.],.]],.]
 => 3 = 4 - 1
[3,1,4,2] => [1,3,4,2] => [2,1,3,4] => [[.,.],[.,[.,.]]]
 => 1 = 2 - 1
[3,2,1,4] => [1,3,2,4] => [2,4,3,1] => [[.,[[.,.],.]],.]
 => 3 = 4 - 1
[3,2,4,1] => [1,3,4,2] => [2,1,3,4] => [[.,.],[.,[.,.]]]
 => 1 = 2 - 1
[3,4,1,2] => [1,3,2,4] => [2,4,3,1] => [[.,[[.,.],.]],.]
 => 3 = 4 - 1
[3,4,2,1] => [1,3,2,4] => [2,4,3,1] => [[.,[[.,.],.]],.]
 => 3 = 4 - 1
[4,1,2,3] => [1,4,3,2] => [2,1,4,3] => [[.,.],[[.,.],.]]
 => 1 = 2 - 1
[4,1,3,2] => [1,4,2,3] => [2,4,1,3] => [[.,[.,.]],[.,.]]
 => 2 = 3 - 1
[4,2,1,3] => [1,4,3,2] => [2,1,4,3] => [[.,.],[[.,.],.]]
 => 1 = 2 - 1
[4,2,3,1] => [1,4,2,3] => [2,4,1,3] => [[.,[.,.]],[.,.]]
 => 2 = 3 - 1
[4,3,1,2] => [1,4,2,3] => [2,4,1,3] => [[.,[.,.]],[.,.]]
 => 2 = 3 - 1
[4,3,2,1] => [1,4,2,3] => [2,4,1,3] => [[.,[.,.]],[.,.]]
 => 2 = 3 - 1
[1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => [[.,[.,[.,[.,.]]]],.]
 => 4 = 5 - 1
[1,2,3,5,4] => [1,2,3,4,5] => [2,3,4,5,1] => [[.,[.,[.,[.,.]]]],.]
 => 4 = 5 - 1
[1,2,4,3,5] => [1,2,3,4,5] => [2,3,4,5,1] => [[.,[.,[.,[.,.]]]],.]
 => 4 = 5 - 1
[1,2,4,5,3] => [1,2,3,4,5] => [2,3,4,5,1] => [[.,[.,[.,[.,.]]]],.]
 => 4 = 5 - 1
[1,2,5,3,4] => [1,2,3,5,4] => [2,3,4,1,5] => [[.,[.,[.,.]]],[.,.]]
 => 3 = 4 - 1
[1,2,5,4,3] => [1,2,3,5,4] => [2,3,4,1,5] => [[.,[.,[.,.]]],[.,.]]
 => 3 = 4 - 1
[1,3,2,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => [[.,[.,[.,[.,.]]]],.]
 => 4 = 5 - 1
[1,3,2,5,4] => [1,2,3,4,5] => [2,3,4,5,1] => [[.,[.,[.,[.,.]]]],.]
 => 4 = 5 - 1
[1,3,4,2,5] => [1,2,3,4,5] => [2,3,4,5,1] => [[.,[.,[.,[.,.]]]],.]
 => 4 = 5 - 1
[1,3,4,5,2] => [1,2,3,4,5] => [2,3,4,5,1] => [[.,[.,[.,[.,.]]]],.]
 => 4 = 5 - 1
[1,3,5,2,4] => [1,2,3,5,4] => [2,3,4,1,5] => [[.,[.,[.,.]]],[.,.]]
 => 3 = 4 - 1
[1,3,5,4,2] => [1,2,3,5,4] => [2,3,4,1,5] => [[.,[.,[.,.]]],[.,.]]
 => 3 = 4 - 1
[1,4,2,3,5] => [1,2,4,3,5] => [2,3,5,4,1] => [[.,[.,[[.,.],.]]],.]
 => 4 = 5 - 1
[1,4,2,5,3] => [1,2,4,5,3] => [2,3,1,4,5] => [[.,[.,.]],[.,[.,.]]]
 => 2 = 3 - 1
[1,4,3,2,5] => [1,2,4,3,5] => [2,3,5,4,1] => [[.,[.,[[.,.],.]]],.]
 => 4 = 5 - 1
[1,4,3,5,2] => [1,2,4,5,3] => [2,3,1,4,5] => [[.,[.,.]],[.,[.,.]]]
 => 2 = 3 - 1
[1,4,5,2,3] => [1,2,4,3,5] => [2,3,5,4,1] => [[.,[.,[[.,.],.]]],.]
 => 4 = 5 - 1
[1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => [2,3,4,5,6,7,1] => [[.,[.,[.,[.,[.,[.,.]]]]]],.]
 => ? = 7 - 1
[1,2,3,4,5,7,6] => [1,2,3,4,5,6,7] => [2,3,4,5,6,7,1] => [[.,[.,[.,[.,[.,[.,.]]]]]],.]
 => ? = 7 - 1
[1,2,3,4,6,5,7] => [1,2,3,4,5,6,7] => [2,3,4,5,6,7,1] => [[.,[.,[.,[.,[.,[.,.]]]]]],.]
 => ? = 7 - 1
[1,2,3,4,6,7,5] => [1,2,3,4,5,6,7] => [2,3,4,5,6,7,1] => [[.,[.,[.,[.,[.,[.,.]]]]]],.]
 => ? = 7 - 1
[1,2,3,4,7,5,6] => [1,2,3,4,5,7,6] => [2,3,4,5,6,1,7] => [[.,[.,[.,[.,[.,.]]]]],[.,.]]
 => ? = 6 - 1
[1,2,3,4,7,6,5] => [1,2,3,4,5,7,6] => [2,3,4,5,6,1,7] => [[.,[.,[.,[.,[.,.]]]]],[.,.]]
 => ? = 6 - 1
[1,2,3,5,4,6,7] => [1,2,3,4,5,6,7] => [2,3,4,5,6,7,1] => [[.,[.,[.,[.,[.,[.,.]]]]]],.]
 => ? = 7 - 1
[1,2,3,5,4,7,6] => [1,2,3,4,5,6,7] => [2,3,4,5,6,7,1] => [[.,[.,[.,[.,[.,[.,.]]]]]],.]
 => ? = 7 - 1
[1,2,3,5,6,4,7] => [1,2,3,4,5,6,7] => [2,3,4,5,6,7,1] => [[.,[.,[.,[.,[.,[.,.]]]]]],.]
 => ? = 7 - 1
[1,2,3,5,6,7,4] => [1,2,3,4,5,6,7] => [2,3,4,5,6,7,1] => [[.,[.,[.,[.,[.,[.,.]]]]]],.]
 => ? = 7 - 1
[1,2,3,5,7,4,6] => [1,2,3,4,5,7,6] => [2,3,4,5,6,1,7] => [[.,[.,[.,[.,[.,.]]]]],[.,.]]
 => ? = 6 - 1
[1,2,3,5,7,6,4] => [1,2,3,4,5,7,6] => [2,3,4,5,6,1,7] => [[.,[.,[.,[.,[.,.]]]]],[.,.]]
 => ? = 6 - 1
[1,2,3,6,4,5,7] => [1,2,3,4,6,5,7] => [2,3,4,5,7,6,1] => [[.,[.,[.,[.,[[.,.],.]]]]],.]
 => ? = 7 - 1
[1,2,3,6,4,7,5] => [1,2,3,4,6,7,5] => [2,3,4,5,1,6,7] => [[.,[.,[.,[.,.]]]],[.,[.,.]]]
 => ? = 5 - 1
[1,2,3,6,5,4,7] => [1,2,3,4,6,5,7] => [2,3,4,5,7,6,1] => [[.,[.,[.,[.,[[.,.],.]]]]],.]
 => ? = 7 - 1
[1,2,3,6,5,7,4] => [1,2,3,4,6,7,5] => [2,3,4,5,1,6,7] => [[.,[.,[.,[.,.]]]],[.,[.,.]]]
 => ? = 5 - 1
[1,2,3,6,7,4,5] => [1,2,3,4,6,5,7] => [2,3,4,5,7,6,1] => [[.,[.,[.,[.,[[.,.],.]]]]],.]
 => ? = 7 - 1
[1,2,3,6,7,5,4] => [1,2,3,4,6,5,7] => [2,3,4,5,7,6,1] => [[.,[.,[.,[.,[[.,.],.]]]]],.]
 => ? = 7 - 1
[1,2,3,7,4,5,6] => [1,2,3,4,7,6,5] => [2,3,4,5,1,7,6] => [[.,[.,[.,[.,.]]]],[[.,.],.]]
 => ? = 5 - 1
[1,2,3,7,4,6,5] => [1,2,3,4,7,5,6] => [2,3,4,5,7,1,6] => [[.,[.,[.,[.,[.,.]]]]],[.,.]]
 => ? = 6 - 1
[1,2,3,7,5,4,6] => [1,2,3,4,7,6,5] => [2,3,4,5,1,7,6] => [[.,[.,[.,[.,.]]]],[[.,.],.]]
 => ? = 5 - 1
[1,2,3,7,5,6,4] => [1,2,3,4,7,5,6] => [2,3,4,5,7,1,6] => [[.,[.,[.,[.,[.,.]]]]],[.,.]]
 => ? = 6 - 1
[1,2,3,7,6,4,5] => [1,2,3,4,7,5,6] => [2,3,4,5,7,1,6] => [[.,[.,[.,[.,[.,.]]]]],[.,.]]
 => ? = 6 - 1
[1,2,3,7,6,5,4] => [1,2,3,4,7,5,6] => [2,3,4,5,7,1,6] => [[.,[.,[.,[.,[.,.]]]]],[.,.]]
 => ? = 6 - 1
[1,2,4,3,5,6,7] => [1,2,3,4,5,6,7] => [2,3,4,5,6,7,1] => [[.,[.,[.,[.,[.,[.,.]]]]]],.]
 => ? = 7 - 1
[1,2,4,3,5,7,6] => [1,2,3,4,5,6,7] => [2,3,4,5,6,7,1] => [[.,[.,[.,[.,[.,[.,.]]]]]],.]
 => ? = 7 - 1
[1,2,4,3,6,5,7] => [1,2,3,4,5,6,7] => [2,3,4,5,6,7,1] => [[.,[.,[.,[.,[.,[.,.]]]]]],.]
 => ? = 7 - 1
[1,2,4,3,6,7,5] => [1,2,3,4,5,6,7] => [2,3,4,5,6,7,1] => [[.,[.,[.,[.,[.,[.,.]]]]]],.]
 => ? = 7 - 1
[1,2,4,3,7,5,6] => [1,2,3,4,5,7,6] => [2,3,4,5,6,1,7] => [[.,[.,[.,[.,[.,.]]]]],[.,.]]
 => ? = 6 - 1
[1,2,4,3,7,6,5] => [1,2,3,4,5,7,6] => [2,3,4,5,6,1,7] => [[.,[.,[.,[.,[.,.]]]]],[.,.]]
 => ? = 6 - 1
[1,2,4,5,3,6,7] => [1,2,3,4,5,6,7] => [2,3,4,5,6,7,1] => [[.,[.,[.,[.,[.,[.,.]]]]]],.]
 => ? = 7 - 1
[1,2,4,5,3,7,6] => [1,2,3,4,5,6,7] => [2,3,4,5,6,7,1] => [[.,[.,[.,[.,[.,[.,.]]]]]],.]
 => ? = 7 - 1
[1,2,4,5,6,3,7] => [1,2,3,4,5,6,7] => [2,3,4,5,6,7,1] => [[.,[.,[.,[.,[.,[.,.]]]]]],.]
 => ? = 7 - 1
[1,2,4,5,6,7,3] => [1,2,3,4,5,6,7] => [2,3,4,5,6,7,1] => [[.,[.,[.,[.,[.,[.,.]]]]]],.]
 => ? = 7 - 1
[1,2,4,5,7,3,6] => [1,2,3,4,5,7,6] => [2,3,4,5,6,1,7] => [[.,[.,[.,[.,[.,.]]]]],[.,.]]
 => ? = 6 - 1
[1,2,4,5,7,6,3] => [1,2,3,4,5,7,6] => [2,3,4,5,6,1,7] => [[.,[.,[.,[.,[.,.]]]]],[.,.]]
 => ? = 6 - 1
[1,2,4,6,3,5,7] => [1,2,3,4,6,5,7] => [2,3,4,5,7,6,1] => [[.,[.,[.,[.,[[.,.],.]]]]],.]
 => ? = 7 - 1
[1,2,4,6,3,7,5] => [1,2,3,4,6,7,5] => [2,3,4,5,1,6,7] => [[.,[.,[.,[.,.]]]],[.,[.,.]]]
 => ? = 5 - 1
[1,2,4,6,5,3,7] => [1,2,3,4,6,5,7] => [2,3,4,5,7,6,1] => [[.,[.,[.,[.,[[.,.],.]]]]],.]
 => ? = 7 - 1
[1,2,4,6,5,7,3] => [1,2,3,4,6,7,5] => [2,3,4,5,1,6,7] => [[.,[.,[.,[.,.]]]],[.,[.,.]]]
 => ? = 5 - 1
[1,2,4,6,7,3,5] => [1,2,3,4,6,5,7] => [2,3,4,5,7,6,1] => [[.,[.,[.,[.,[[.,.],.]]]]],.]
 => ? = 7 - 1
[1,2,4,6,7,5,3] => [1,2,3,4,6,5,7] => [2,3,4,5,7,6,1] => [[.,[.,[.,[.,[[.,.],.]]]]],.]
 => ? = 7 - 1
[1,2,4,7,3,5,6] => [1,2,3,4,7,6,5] => [2,3,4,5,1,7,6] => [[.,[.,[.,[.,.]]]],[[.,.],.]]
 => ? = 5 - 1
[1,2,4,7,3,6,5] => [1,2,3,4,7,5,6] => [2,3,4,5,7,1,6] => [[.,[.,[.,[.,[.,.]]]]],[.,.]]
 => ? = 6 - 1
[1,2,4,7,5,3,6] => [1,2,3,4,7,6,5] => [2,3,4,5,1,7,6] => [[.,[.,[.,[.,.]]]],[[.,.],.]]
 => ? = 5 - 1
[1,2,4,7,5,6,3] => [1,2,3,4,7,5,6] => [2,3,4,5,7,1,6] => [[.,[.,[.,[.,[.,.]]]]],[.,.]]
 => ? = 6 - 1
[1,2,4,7,6,3,5] => [1,2,3,4,7,5,6] => [2,3,4,5,7,1,6] => [[.,[.,[.,[.,[.,.]]]]],[.,.]]
 => ? = 6 - 1
[1,2,4,7,6,5,3] => [1,2,3,4,7,5,6] => [2,3,4,5,7,1,6] => [[.,[.,[.,[.,[.,.]]]]],[.,.]]
 => ? = 6 - 1
[1,2,5,3,4,6,7] => [1,2,3,5,4,6,7] => [2,3,4,6,5,7,1] => [[.,[.,[.,[[.,.],[.,.]]]]],.]
 => ? = 7 - 1
[1,2,5,3,4,7,6] => [1,2,3,5,4,6,7] => [2,3,4,6,5,7,1] => [[.,[.,[.,[[.,.],[.,.]]]]],.]
 => ? = 7 - 1
Description
The size of the left subtree of a binary tree.
Matching statistic: St000133
Mapping
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
Mp00088: Permutations —Kreweras complement⟶ Permutations
St000133: Permutations ⟶ ℤResult quality: 24% ●values known / values provided: 24%●distinct values known / distinct values provided: 86%
Values
[1] => [1] => [1] => [1] => 0 = 1 - 1
[1,2] => [1,2] => [2,1] => [1,2] => 1 = 2 - 1
[2,1] => [1,2] => [2,1] => [1,2] => 1 = 2 - 1
[1,2,3] => [1,2,3] => [3,2,1] => [1,3,2] => 2 = 3 - 1
[1,3,2] => [1,2,3] => [3,2,1] => [1,3,2] => 2 = 3 - 1
[2,1,3] => [1,2,3] => [3,2,1] => [1,3,2] => 2 = 3 - 1
[2,3,1] => [1,2,3] => [3,2,1] => [1,3,2] => 2 = 3 - 1
[3,1,2] => [1,3,2] => [3,1,2] => [3,1,2] => 1 = 2 - 1
[3,2,1] => [1,3,2] => [3,1,2] => [3,1,2] => 1 = 2 - 1
[1,2,3,4] => [1,2,3,4] => [4,3,2,1] => [1,4,3,2] => 3 = 4 - 1
[1,2,4,3] => [1,2,3,4] => [4,3,2,1] => [1,4,3,2] => 3 = 4 - 1
[1,3,2,4] => [1,2,3,4] => [4,3,2,1] => [1,4,3,2] => 3 = 4 - 1
[1,3,4,2] => [1,2,3,4] => [4,3,2,1] => [1,4,3,2] => 3 = 4 - 1
[1,4,2,3] => [1,2,4,3] => [4,3,1,2] => [4,1,3,2] => 2 = 3 - 1
[1,4,3,2] => [1,2,4,3] => [4,3,1,2] => [4,1,3,2] => 2 = 3 - 1
[2,1,3,4] => [1,2,3,4] => [4,3,2,1] => [1,4,3,2] => 3 = 4 - 1
[2,1,4,3] => [1,2,3,4] => [4,3,2,1] => [1,4,3,2] => 3 = 4 - 1
[2,3,1,4] => [1,2,3,4] => [4,3,2,1] => [1,4,3,2] => 3 = 4 - 1
[2,3,4,1] => [1,2,3,4] => [4,3,2,1] => [1,4,3,2] => 3 = 4 - 1
[2,4,1,3] => [1,2,4,3] => [4,3,1,2] => [4,1,3,2] => 2 = 3 - 1
[2,4,3,1] => [1,2,4,3] => [4,3,1,2] => [4,1,3,2] => 2 = 3 - 1
[3,1,2,4] => [1,3,2,4] => [4,2,3,1] => [1,3,4,2] => 3 = 4 - 1
[3,1,4,2] => [1,3,4,2] => [4,2,1,3] => [4,3,1,2] => 1 = 2 - 1
[3,2,1,4] => [1,3,2,4] => [4,2,3,1] => [1,3,4,2] => 3 = 4 - 1
[3,2,4,1] => [1,3,4,2] => [4,2,1,3] => [4,3,1,2] => 1 = 2 - 1
[3,4,1,2] => [1,3,2,4] => [4,2,3,1] => [1,3,4,2] => 3 = 4 - 1
[3,4,2,1] => [1,3,2,4] => [4,2,3,1] => [1,3,4,2] => 3 = 4 - 1
[4,1,2,3] => [1,4,3,2] => [4,1,2,3] => [3,4,1,2] => 1 = 2 - 1
[4,1,3,2] => [1,4,2,3] => [4,1,3,2] => [3,1,4,2] => 2 = 3 - 1
[4,2,1,3] => [1,4,3,2] => [4,1,2,3] => [3,4,1,2] => 1 = 2 - 1
[4,2,3,1] => [1,4,2,3] => [4,1,3,2] => [3,1,4,2] => 2 = 3 - 1
[4,3,1,2] => [1,4,2,3] => [4,1,3,2] => [3,1,4,2] => 2 = 3 - 1
[4,3,2,1] => [1,4,2,3] => [4,1,3,2] => [3,1,4,2] => 2 = 3 - 1
[1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => [1,5,4,3,2] => 4 = 5 - 1
[1,2,3,5,4] => [1,2,3,4,5] => [5,4,3,2,1] => [1,5,4,3,2] => 4 = 5 - 1
[1,2,4,3,5] => [1,2,3,4,5] => [5,4,3,2,1] => [1,5,4,3,2] => 4 = 5 - 1
[1,2,4,5,3] => [1,2,3,4,5] => [5,4,3,2,1] => [1,5,4,3,2] => 4 = 5 - 1
[1,2,5,3,4] => [1,2,3,5,4] => [5,4,3,1,2] => [5,1,4,3,2] => 3 = 4 - 1
[1,2,5,4,3] => [1,2,3,5,4] => [5,4,3,1,2] => [5,1,4,3,2] => 3 = 4 - 1
[1,3,2,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => [1,5,4,3,2] => 4 = 5 - 1
[1,3,2,5,4] => [1,2,3,4,5] => [5,4,3,2,1] => [1,5,4,3,2] => 4 = 5 - 1
[1,3,4,2,5] => [1,2,3,4,5] => [5,4,3,2,1] => [1,5,4,3,2] => 4 = 5 - 1
[1,3,4,5,2] => [1,2,3,4,5] => [5,4,3,2,1] => [1,5,4,3,2] => 4 = 5 - 1
[1,3,5,2,4] => [1,2,3,5,4] => [5,4,3,1,2] => [5,1,4,3,2] => 3 = 4 - 1
[1,3,5,4,2] => [1,2,3,5,4] => [5,4,3,1,2] => [5,1,4,3,2] => 3 = 4 - 1
[1,4,2,3,5] => [1,2,4,3,5] => [5,4,2,3,1] => [1,4,5,3,2] => 4 = 5 - 1
[1,4,2,5,3] => [1,2,4,5,3] => [5,4,2,1,3] => [5,4,1,3,2] => 2 = 3 - 1
[1,4,3,2,5] => [1,2,4,3,5] => [5,4,2,3,1] => [1,4,5,3,2] => 4 = 5 - 1
[1,4,3,5,2] => [1,2,4,5,3] => [5,4,2,1,3] => [5,4,1,3,2] => 2 = 3 - 1
[1,4,5,2,3] => [1,2,4,3,5] => [5,4,2,3,1] => [1,4,5,3,2] => 4 = 5 - 1
[1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => [1,7,6,5,4,3,2] => ? = 7 - 1
[1,2,3,4,5,7,6] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => [1,7,6,5,4,3,2] => ? = 7 - 1
[1,2,3,4,6,5,7] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => [1,7,6,5,4,3,2] => ? = 7 - 1
[1,2,3,4,6,7,5] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => [1,7,6,5,4,3,2] => ? = 7 - 1
[1,2,3,4,7,5,6] => [1,2,3,4,5,7,6] => [7,6,5,4,3,1,2] => [7,1,6,5,4,3,2] => ? = 6 - 1
[1,2,3,4,7,6,5] => [1,2,3,4,5,7,6] => [7,6,5,4,3,1,2] => [7,1,6,5,4,3,2] => ? = 6 - 1
[1,2,3,5,4,6,7] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => [1,7,6,5,4,3,2] => ? = 7 - 1
[1,2,3,5,4,7,6] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => [1,7,6,5,4,3,2] => ? = 7 - 1
[1,2,3,5,6,4,7] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => [1,7,6,5,4,3,2] => ? = 7 - 1
[1,2,3,5,6,7,4] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => [1,7,6,5,4,3,2] => ? = 7 - 1
[1,2,3,5,7,4,6] => [1,2,3,4,5,7,6] => [7,6,5,4,3,1,2] => [7,1,6,5,4,3,2] => ? = 6 - 1
[1,2,3,5,7,6,4] => [1,2,3,4,5,7,6] => [7,6,5,4,3,1,2] => [7,1,6,5,4,3,2] => ? = 6 - 1
[1,2,3,6,4,5,7] => [1,2,3,4,6,5,7] => [7,6,5,4,2,3,1] => [1,6,7,5,4,3,2] => ? = 7 - 1
[1,2,3,6,4,7,5] => [1,2,3,4,6,7,5] => [7,6,5,4,2,1,3] => [7,6,1,5,4,3,2] => ? = 5 - 1
[1,2,3,6,5,4,7] => [1,2,3,4,6,5,7] => [7,6,5,4,2,3,1] => [1,6,7,5,4,3,2] => ? = 7 - 1
[1,2,3,6,5,7,4] => [1,2,3,4,6,7,5] => [7,6,5,4,2,1,3] => [7,6,1,5,4,3,2] => ? = 5 - 1
[1,2,3,6,7,4,5] => [1,2,3,4,6,5,7] => [7,6,5,4,2,3,1] => [1,6,7,5,4,3,2] => ? = 7 - 1
[1,2,3,6,7,5,4] => [1,2,3,4,6,5,7] => [7,6,5,4,2,3,1] => [1,6,7,5,4,3,2] => ? = 7 - 1
[1,2,3,7,4,5,6] => [1,2,3,4,7,6,5] => [7,6,5,4,1,2,3] => [6,7,1,5,4,3,2] => ? = 5 - 1
[1,2,3,7,4,6,5] => [1,2,3,4,7,5,6] => [7,6,5,4,1,3,2] => [6,1,7,5,4,3,2] => ? = 6 - 1
[1,2,3,7,5,4,6] => [1,2,3,4,7,6,5] => [7,6,5,4,1,2,3] => [6,7,1,5,4,3,2] => ? = 5 - 1
[1,2,3,7,5,6,4] => [1,2,3,4,7,5,6] => [7,6,5,4,1,3,2] => [6,1,7,5,4,3,2] => ? = 6 - 1
[1,2,3,7,6,4,5] => [1,2,3,4,7,5,6] => [7,6,5,4,1,3,2] => [6,1,7,5,4,3,2] => ? = 6 - 1
[1,2,3,7,6,5,4] => [1,2,3,4,7,5,6] => [7,6,5,4,1,3,2] => [6,1,7,5,4,3,2] => ? = 6 - 1
[1,2,4,3,5,6,7] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => [1,7,6,5,4,3,2] => ? = 7 - 1
[1,2,4,3,5,7,6] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => [1,7,6,5,4,3,2] => ? = 7 - 1
[1,2,4,3,6,5,7] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => [1,7,6,5,4,3,2] => ? = 7 - 1
[1,2,4,3,6,7,5] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => [1,7,6,5,4,3,2] => ? = 7 - 1
[1,2,4,3,7,5,6] => [1,2,3,4,5,7,6] => [7,6,5,4,3,1,2] => [7,1,6,5,4,3,2] => ? = 6 - 1
[1,2,4,3,7,6,5] => [1,2,3,4,5,7,6] => [7,6,5,4,3,1,2] => [7,1,6,5,4,3,2] => ? = 6 - 1
[1,2,4,5,3,6,7] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => [1,7,6,5,4,3,2] => ? = 7 - 1
[1,2,4,5,3,7,6] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => [1,7,6,5,4,3,2] => ? = 7 - 1
[1,2,4,5,6,3,7] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => [1,7,6,5,4,3,2] => ? = 7 - 1
[1,2,4,5,6,7,3] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => [1,7,6,5,4,3,2] => ? = 7 - 1
[1,2,4,5,7,3,6] => [1,2,3,4,5,7,6] => [7,6,5,4,3,1,2] => [7,1,6,5,4,3,2] => ? = 6 - 1
[1,2,4,5,7,6,3] => [1,2,3,4,5,7,6] => [7,6,5,4,3,1,2] => [7,1,6,5,4,3,2] => ? = 6 - 1
[1,2,4,6,3,5,7] => [1,2,3,4,6,5,7] => [7,6,5,4,2,3,1] => [1,6,7,5,4,3,2] => ? = 7 - 1
[1,2,4,6,3,7,5] => [1,2,3,4,6,7,5] => [7,6,5,4,2,1,3] => [7,6,1,5,4,3,2] => ? = 5 - 1
[1,2,4,6,5,3,7] => [1,2,3,4,6,5,7] => [7,6,5,4,2,3,1] => [1,6,7,5,4,3,2] => ? = 7 - 1
[1,2,4,6,5,7,3] => [1,2,3,4,6,7,5] => [7,6,5,4,2,1,3] => [7,6,1,5,4,3,2] => ? = 5 - 1
[1,2,4,6,7,3,5] => [1,2,3,4,6,5,7] => [7,6,5,4,2,3,1] => [1,6,7,5,4,3,2] => ? = 7 - 1
[1,2,4,6,7,5,3] => [1,2,3,4,6,5,7] => [7,6,5,4,2,3,1] => [1,6,7,5,4,3,2] => ? = 7 - 1
[1,2,4,7,3,5,6] => [1,2,3,4,7,6,5] => [7,6,5,4,1,2,3] => [6,7,1,5,4,3,2] => ? = 5 - 1
[1,2,4,7,3,6,5] => [1,2,3,4,7,5,6] => [7,6,5,4,1,3,2] => [6,1,7,5,4,3,2] => ? = 6 - 1
[1,2,4,7,5,3,6] => [1,2,3,4,7,6,5] => [7,6,5,4,1,2,3] => [6,7,1,5,4,3,2] => ? = 5 - 1
[1,2,4,7,5,6,3] => [1,2,3,4,7,5,6] => [7,6,5,4,1,3,2] => [6,1,7,5,4,3,2] => ? = 6 - 1
[1,2,4,7,6,3,5] => [1,2,3,4,7,5,6] => [7,6,5,4,1,3,2] => [6,1,7,5,4,3,2] => ? = 6 - 1
[1,2,4,7,6,5,3] => [1,2,3,4,7,5,6] => [7,6,5,4,1,3,2] => [6,1,7,5,4,3,2] => ? = 6 - 1
[1,2,5,3,4,6,7] => [1,2,3,5,4,6,7] => [7,6,5,3,4,2,1] => [1,7,5,6,4,3,2] => ? = 7 - 1
[1,2,5,3,4,7,6] => [1,2,3,5,4,6,7] => [7,6,5,3,4,2,1] => [1,7,5,6,4,3,2] => ? = 7 - 1
Description
The "bounce" of a permutation.
Matching statistic: St000800
(load all 6 compositions to match this statistic)
Mapping
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00088: Permutations —Kreweras complement⟶ Permutations
St000800: Permutations ⟶ ℤResult quality: 24% ●values known / values provided: 24%●distinct values known / distinct values provided: 71%
Values
[1] => [1] => [1] => ? = 1 - 2
[1,2] => [1,2] => [2,1] => 0 = 2 - 2
[2,1] => [1,2] => [2,1] => 0 = 2 - 2
[1,2,3] => [1,2,3] => [2,3,1] => 1 = 3 - 2
[1,3,2] => [1,2,3] => [2,3,1] => 1 = 3 - 2
[2,1,3] => [1,2,3] => [2,3,1] => 1 = 3 - 2
[2,3,1] => [1,2,3] => [2,3,1] => 1 = 3 - 2
[3,1,2] => [1,3,2] => [2,1,3] => 0 = 2 - 2
[3,2,1] => [1,3,2] => [2,1,3] => 0 = 2 - 2
[1,2,3,4] => [1,2,3,4] => [2,3,4,1] => 2 = 4 - 2
[1,2,4,3] => [1,2,3,4] => [2,3,4,1] => 2 = 4 - 2
[1,3,2,4] => [1,2,3,4] => [2,3,4,1] => 2 = 4 - 2
[1,3,4,2] => [1,2,3,4] => [2,3,4,1] => 2 = 4 - 2
[1,4,2,3] => [1,2,4,3] => [2,3,1,4] => 1 = 3 - 2
[1,4,3,2] => [1,2,4,3] => [2,3,1,4] => 1 = 3 - 2
[2,1,3,4] => [1,2,3,4] => [2,3,4,1] => 2 = 4 - 2
[2,1,4,3] => [1,2,3,4] => [2,3,4,1] => 2 = 4 - 2
[2,3,1,4] => [1,2,3,4] => [2,3,4,1] => 2 = 4 - 2
[2,3,4,1] => [1,2,3,4] => [2,3,4,1] => 2 = 4 - 2
[2,4,1,3] => [1,2,4,3] => [2,3,1,4] => 1 = 3 - 2
[2,4,3,1] => [1,2,4,3] => [2,3,1,4] => 1 = 3 - 2
[3,1,2,4] => [1,3,2,4] => [2,4,3,1] => 2 = 4 - 2
[3,1,4,2] => [1,3,4,2] => [2,1,3,4] => 0 = 2 - 2
[3,2,1,4] => [1,3,2,4] => [2,4,3,1] => 2 = 4 - 2
[3,2,4,1] => [1,3,4,2] => [2,1,3,4] => 0 = 2 - 2
[3,4,1,2] => [1,3,2,4] => [2,4,3,1] => 2 = 4 - 2
[3,4,2,1] => [1,3,2,4] => [2,4,3,1] => 2 = 4 - 2
[4,1,2,3] => [1,4,3,2] => [2,1,4,3] => 0 = 2 - 2
[4,1,3,2] => [1,4,2,3] => [2,4,1,3] => 1 = 3 - 2
[4,2,1,3] => [1,4,3,2] => [2,1,4,3] => 0 = 2 - 2
[4,2,3,1] => [1,4,2,3] => [2,4,1,3] => 1 = 3 - 2
[4,3,1,2] => [1,4,2,3] => [2,4,1,3] => 1 = 3 - 2
[4,3,2,1] => [1,4,2,3] => [2,4,1,3] => 1 = 3 - 2
[1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => 3 = 5 - 2
[1,2,3,5,4] => [1,2,3,4,5] => [2,3,4,5,1] => 3 = 5 - 2
[1,2,4,3,5] => [1,2,3,4,5] => [2,3,4,5,1] => 3 = 5 - 2
[1,2,4,5,3] => [1,2,3,4,5] => [2,3,4,5,1] => 3 = 5 - 2
[1,2,5,3,4] => [1,2,3,5,4] => [2,3,4,1,5] => 2 = 4 - 2
[1,2,5,4,3] => [1,2,3,5,4] => [2,3,4,1,5] => 2 = 4 - 2
[1,3,2,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => 3 = 5 - 2
[1,3,2,5,4] => [1,2,3,4,5] => [2,3,4,5,1] => 3 = 5 - 2
[1,3,4,2,5] => [1,2,3,4,5] => [2,3,4,5,1] => 3 = 5 - 2
[1,3,4,5,2] => [1,2,3,4,5] => [2,3,4,5,1] => 3 = 5 - 2
[1,3,5,2,4] => [1,2,3,5,4] => [2,3,4,1,5] => 2 = 4 - 2
[1,3,5,4,2] => [1,2,3,5,4] => [2,3,4,1,5] => 2 = 4 - 2
[1,4,2,3,5] => [1,2,4,3,5] => [2,3,5,4,1] => 3 = 5 - 2
[1,4,2,5,3] => [1,2,4,5,3] => [2,3,1,4,5] => 1 = 3 - 2
[1,4,3,2,5] => [1,2,4,3,5] => [2,3,5,4,1] => 3 = 5 - 2
[1,4,3,5,2] => [1,2,4,5,3] => [2,3,1,4,5] => 1 = 3 - 2
[1,4,5,2,3] => [1,2,4,3,5] => [2,3,5,4,1] => 3 = 5 - 2
[1,4,5,3,2] => [1,2,4,3,5] => [2,3,5,4,1] => 3 = 5 - 2
[1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => [2,3,4,5,6,7,1] => ? = 7 - 2
[1,2,3,4,5,7,6] => [1,2,3,4,5,6,7] => [2,3,4,5,6,7,1] => ? = 7 - 2
[1,2,3,4,6,5,7] => [1,2,3,4,5,6,7] => [2,3,4,5,6,7,1] => ? = 7 - 2
[1,2,3,4,6,7,5] => [1,2,3,4,5,6,7] => [2,3,4,5,6,7,1] => ? = 7 - 2
[1,2,3,4,7,5,6] => [1,2,3,4,5,7,6] => [2,3,4,5,6,1,7] => ? = 6 - 2
[1,2,3,4,7,6,5] => [1,2,3,4,5,7,6] => [2,3,4,5,6,1,7] => ? = 6 - 2
[1,2,3,5,4,6,7] => [1,2,3,4,5,6,7] => [2,3,4,5,6,7,1] => ? = 7 - 2
[1,2,3,5,4,7,6] => [1,2,3,4,5,6,7] => [2,3,4,5,6,7,1] => ? = 7 - 2
[1,2,3,5,6,4,7] => [1,2,3,4,5,6,7] => [2,3,4,5,6,7,1] => ? = 7 - 2
[1,2,3,5,6,7,4] => [1,2,3,4,5,6,7] => [2,3,4,5,6,7,1] => ? = 7 - 2
[1,2,3,5,7,4,6] => [1,2,3,4,5,7,6] => [2,3,4,5,6,1,7] => ? = 6 - 2
[1,2,3,5,7,6,4] => [1,2,3,4,5,7,6] => [2,3,4,5,6,1,7] => ? = 6 - 2
[1,2,3,6,4,5,7] => [1,2,3,4,6,5,7] => [2,3,4,5,7,6,1] => ? = 7 - 2
[1,2,3,6,4,7,5] => [1,2,3,4,6,7,5] => [2,3,4,5,1,6,7] => ? = 5 - 2
[1,2,3,6,5,4,7] => [1,2,3,4,6,5,7] => [2,3,4,5,7,6,1] => ? = 7 - 2
[1,2,3,6,5,7,4] => [1,2,3,4,6,7,5] => [2,3,4,5,1,6,7] => ? = 5 - 2
[1,2,3,6,7,4,5] => [1,2,3,4,6,5,7] => [2,3,4,5,7,6,1] => ? = 7 - 2
[1,2,3,6,7,5,4] => [1,2,3,4,6,5,7] => [2,3,4,5,7,6,1] => ? = 7 - 2
[1,2,3,7,4,5,6] => [1,2,3,4,7,6,5] => [2,3,4,5,1,7,6] => ? = 5 - 2
[1,2,3,7,4,6,5] => [1,2,3,4,7,5,6] => [2,3,4,5,7,1,6] => ? = 6 - 2
[1,2,3,7,5,4,6] => [1,2,3,4,7,6,5] => [2,3,4,5,1,7,6] => ? = 5 - 2
[1,2,3,7,5,6,4] => [1,2,3,4,7,5,6] => [2,3,4,5,7,1,6] => ? = 6 - 2
[1,2,3,7,6,4,5] => [1,2,3,4,7,5,6] => [2,3,4,5,7,1,6] => ? = 6 - 2
[1,2,3,7,6,5,4] => [1,2,3,4,7,5,6] => [2,3,4,5,7,1,6] => ? = 6 - 2
[1,2,4,3,5,6,7] => [1,2,3,4,5,6,7] => [2,3,4,5,6,7,1] => ? = 7 - 2
[1,2,4,3,5,7,6] => [1,2,3,4,5,6,7] => [2,3,4,5,6,7,1] => ? = 7 - 2
[1,2,4,3,6,5,7] => [1,2,3,4,5,6,7] => [2,3,4,5,6,7,1] => ? = 7 - 2
[1,2,4,3,6,7,5] => [1,2,3,4,5,6,7] => [2,3,4,5,6,7,1] => ? = 7 - 2
[1,2,4,3,7,5,6] => [1,2,3,4,5,7,6] => [2,3,4,5,6,1,7] => ? = 6 - 2
[1,2,4,3,7,6,5] => [1,2,3,4,5,7,6] => [2,3,4,5,6,1,7] => ? = 6 - 2
[1,2,4,5,3,6,7] => [1,2,3,4,5,6,7] => [2,3,4,5,6,7,1] => ? = 7 - 2
[1,2,4,5,3,7,6] => [1,2,3,4,5,6,7] => [2,3,4,5,6,7,1] => ? = 7 - 2
[1,2,4,5,6,3,7] => [1,2,3,4,5,6,7] => [2,3,4,5,6,7,1] => ? = 7 - 2
[1,2,4,5,6,7,3] => [1,2,3,4,5,6,7] => [2,3,4,5,6,7,1] => ? = 7 - 2
[1,2,4,5,7,3,6] => [1,2,3,4,5,7,6] => [2,3,4,5,6,1,7] => ? = 6 - 2
[1,2,4,5,7,6,3] => [1,2,3,4,5,7,6] => [2,3,4,5,6,1,7] => ? = 6 - 2
[1,2,4,6,3,5,7] => [1,2,3,4,6,5,7] => [2,3,4,5,7,6,1] => ? = 7 - 2
[1,2,4,6,3,7,5] => [1,2,3,4,6,7,5] => [2,3,4,5,1,6,7] => ? = 5 - 2
[1,2,4,6,5,3,7] => [1,2,3,4,6,5,7] => [2,3,4,5,7,6,1] => ? = 7 - 2
[1,2,4,6,5,7,3] => [1,2,3,4,6,7,5] => [2,3,4,5,1,6,7] => ? = 5 - 2
[1,2,4,6,7,3,5] => [1,2,3,4,6,5,7] => [2,3,4,5,7,6,1] => ? = 7 - 2
[1,2,4,6,7,5,3] => [1,2,3,4,6,5,7] => [2,3,4,5,7,6,1] => ? = 7 - 2
[1,2,4,7,3,5,6] => [1,2,3,4,7,6,5] => [2,3,4,5,1,7,6] => ? = 5 - 2
[1,2,4,7,3,6,5] => [1,2,3,4,7,5,6] => [2,3,4,5,7,1,6] => ? = 6 - 2
[1,2,4,7,5,3,6] => [1,2,3,4,7,6,5] => [2,3,4,5,1,7,6] => ? = 5 - 2
[1,2,4,7,5,6,3] => [1,2,3,4,7,5,6] => [2,3,4,5,7,1,6] => ? = 6 - 2
[1,2,4,7,6,3,5] => [1,2,3,4,7,5,6] => [2,3,4,5,7,1,6] => ? = 6 - 2
[1,2,4,7,6,5,3] => [1,2,3,4,7,5,6] => [2,3,4,5,7,1,6] => ? = 6 - 2
[1,2,5,3,4,6,7] => [1,2,3,5,4,6,7] => [2,3,4,6,5,7,1] => ? = 7 - 2
Description
The number of occurrences of the vincular pattern |231 in a permutation. This is the number of occurrences of the pattern (2,3,1), such that the letter matched by 2 is the first entry of the permutation.
Matching statistic: St000060
Mapping
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
St000060: Permutations ⟶ ℤResult quality: 24% ●values known / values provided: 24%●distinct values known / distinct values provided: 71%
Values
[1] => [1] => [1] => [1] => ? = 1 - 1
[1,2] => [1,2] => [2,1] => [1,2] => 1 = 2 - 1
[2,1] => [1,2] => [2,1] => [1,2] => 1 = 2 - 1
[1,2,3] => [1,2,3] => [2,3,1] => [1,2,3] => 2 = 3 - 1
[1,3,2] => [1,2,3] => [2,3,1] => [1,2,3] => 2 = 3 - 1
[2,1,3] => [1,2,3] => [2,3,1] => [1,2,3] => 2 = 3 - 1
[2,3,1] => [1,2,3] => [2,3,1] => [1,2,3] => 2 = 3 - 1
[3,1,2] => [1,3,2] => [3,2,1] => [2,1,3] => 1 = 2 - 1
[3,2,1] => [1,3,2] => [3,2,1] => [2,1,3] => 1 = 2 - 1
[1,2,3,4] => [1,2,3,4] => [2,3,4,1] => [1,2,3,4] => 3 = 4 - 1
[1,2,4,3] => [1,2,3,4] => [2,3,4,1] => [1,2,3,4] => 3 = 4 - 1
[1,3,2,4] => [1,2,3,4] => [2,3,4,1] => [1,2,3,4] => 3 = 4 - 1
[1,3,4,2] => [1,2,3,4] => [2,3,4,1] => [1,2,3,4] => 3 = 4 - 1
[1,4,2,3] => [1,2,4,3] => [2,4,3,1] => [1,3,2,4] => 2 = 3 - 1
[1,4,3,2] => [1,2,4,3] => [2,4,3,1] => [1,3,2,4] => 2 = 3 - 1
[2,1,3,4] => [1,2,3,4] => [2,3,4,1] => [1,2,3,4] => 3 = 4 - 1
[2,1,4,3] => [1,2,3,4] => [2,3,4,1] => [1,2,3,4] => 3 = 4 - 1
[2,3,1,4] => [1,2,3,4] => [2,3,4,1] => [1,2,3,4] => 3 = 4 - 1
[2,3,4,1] => [1,2,3,4] => [2,3,4,1] => [1,2,3,4] => 3 = 4 - 1
[2,4,1,3] => [1,2,4,3] => [2,4,3,1] => [1,3,2,4] => 2 = 3 - 1
[2,4,3,1] => [1,2,4,3] => [2,4,3,1] => [1,3,2,4] => 2 = 3 - 1
[3,1,2,4] => [1,3,2,4] => [3,2,4,1] => [2,1,3,4] => 3 = 4 - 1
[3,1,4,2] => [1,3,4,2] => [4,2,3,1] => [2,3,1,4] => 1 = 2 - 1
[3,2,1,4] => [1,3,2,4] => [3,2,4,1] => [2,1,3,4] => 3 = 4 - 1
[3,2,4,1] => [1,3,4,2] => [4,2,3,1] => [2,3,1,4] => 1 = 2 - 1
[3,4,1,2] => [1,3,2,4] => [3,2,4,1] => [2,1,3,4] => 3 = 4 - 1
[3,4,2,1] => [1,3,2,4] => [3,2,4,1] => [2,1,3,4] => 3 = 4 - 1
[4,1,2,3] => [1,4,3,2] => [4,3,2,1] => [3,2,1,4] => 1 = 2 - 1
[4,1,3,2] => [1,4,2,3] => [3,4,2,1] => [3,1,2,4] => 2 = 3 - 1
[4,2,1,3] => [1,4,3,2] => [4,3,2,1] => [3,2,1,4] => 1 = 2 - 1
[4,2,3,1] => [1,4,2,3] => [3,4,2,1] => [3,1,2,4] => 2 = 3 - 1
[4,3,1,2] => [1,4,2,3] => [3,4,2,1] => [3,1,2,4] => 2 = 3 - 1
[4,3,2,1] => [1,4,2,3] => [3,4,2,1] => [3,1,2,4] => 2 = 3 - 1
[1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => [1,2,3,4,5] => 4 = 5 - 1
[1,2,3,5,4] => [1,2,3,4,5] => [2,3,4,5,1] => [1,2,3,4,5] => 4 = 5 - 1
[1,2,4,3,5] => [1,2,3,4,5] => [2,3,4,5,1] => [1,2,3,4,5] => 4 = 5 - 1
[1,2,4,5,3] => [1,2,3,4,5] => [2,3,4,5,1] => [1,2,3,4,5] => 4 = 5 - 1
[1,2,5,3,4] => [1,2,3,5,4] => [2,3,5,4,1] => [1,2,4,3,5] => 3 = 4 - 1
[1,2,5,4,3] => [1,2,3,5,4] => [2,3,5,4,1] => [1,2,4,3,5] => 3 = 4 - 1
[1,3,2,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => [1,2,3,4,5] => 4 = 5 - 1
[1,3,2,5,4] => [1,2,3,4,5] => [2,3,4,5,1] => [1,2,3,4,5] => 4 = 5 - 1
[1,3,4,2,5] => [1,2,3,4,5] => [2,3,4,5,1] => [1,2,3,4,5] => 4 = 5 - 1
[1,3,4,5,2] => [1,2,3,4,5] => [2,3,4,5,1] => [1,2,3,4,5] => 4 = 5 - 1
[1,3,5,2,4] => [1,2,3,5,4] => [2,3,5,4,1] => [1,2,4,3,5] => 3 = 4 - 1
[1,3,5,4,2] => [1,2,3,5,4] => [2,3,5,4,1] => [1,2,4,3,5] => 3 = 4 - 1
[1,4,2,3,5] => [1,2,4,3,5] => [2,4,3,5,1] => [1,3,2,4,5] => 4 = 5 - 1
[1,4,2,5,3] => [1,2,4,5,3] => [2,5,3,4,1] => [1,3,4,2,5] => 2 = 3 - 1
[1,4,3,2,5] => [1,2,4,3,5] => [2,4,3,5,1] => [1,3,2,4,5] => 4 = 5 - 1
[1,4,3,5,2] => [1,2,4,5,3] => [2,5,3,4,1] => [1,3,4,2,5] => 2 = 3 - 1
[1,4,5,2,3] => [1,2,4,3,5] => [2,4,3,5,1] => [1,3,2,4,5] => 4 = 5 - 1
[1,4,5,3,2] => [1,2,4,3,5] => [2,4,3,5,1] => [1,3,2,4,5] => 4 = 5 - 1
[1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => [2,3,4,5,6,7,1] => [1,2,3,4,5,6,7] => ? = 7 - 1
[1,2,3,4,5,7,6] => [1,2,3,4,5,6,7] => [2,3,4,5,6,7,1] => [1,2,3,4,5,6,7] => ? = 7 - 1
[1,2,3,4,6,5,7] => [1,2,3,4,5,6,7] => [2,3,4,5,6,7,1] => [1,2,3,4,5,6,7] => ? = 7 - 1
[1,2,3,4,6,7,5] => [1,2,3,4,5,6,7] => [2,3,4,5,6,7,1] => [1,2,3,4,5,6,7] => ? = 7 - 1
[1,2,3,4,7,5,6] => [1,2,3,4,5,7,6] => [2,3,4,5,7,6,1] => [1,2,3,4,6,5,7] => ? = 6 - 1
[1,2,3,4,7,6,5] => [1,2,3,4,5,7,6] => [2,3,4,5,7,6,1] => [1,2,3,4,6,5,7] => ? = 6 - 1
[1,2,3,5,4,6,7] => [1,2,3,4,5,6,7] => [2,3,4,5,6,7,1] => [1,2,3,4,5,6,7] => ? = 7 - 1
[1,2,3,5,4,7,6] => [1,2,3,4,5,6,7] => [2,3,4,5,6,7,1] => [1,2,3,4,5,6,7] => ? = 7 - 1
[1,2,3,5,6,4,7] => [1,2,3,4,5,6,7] => [2,3,4,5,6,7,1] => [1,2,3,4,5,6,7] => ? = 7 - 1
[1,2,3,5,6,7,4] => [1,2,3,4,5,6,7] => [2,3,4,5,6,7,1] => [1,2,3,4,5,6,7] => ? = 7 - 1
[1,2,3,5,7,4,6] => [1,2,3,4,5,7,6] => [2,3,4,5,7,6,1] => [1,2,3,4,6,5,7] => ? = 6 - 1
[1,2,3,5,7,6,4] => [1,2,3,4,5,7,6] => [2,3,4,5,7,6,1] => [1,2,3,4,6,5,7] => ? = 6 - 1
[1,2,3,6,4,5,7] => [1,2,3,4,6,5,7] => [2,3,4,6,5,7,1] => [1,2,3,5,4,6,7] => ? = 7 - 1
[1,2,3,6,4,7,5] => [1,2,3,4,6,7,5] => [2,3,4,7,5,6,1] => [1,2,3,5,6,4,7] => ? = 5 - 1
[1,2,3,6,5,4,7] => [1,2,3,4,6,5,7] => [2,3,4,6,5,7,1] => [1,2,3,5,4,6,7] => ? = 7 - 1
[1,2,3,6,5,7,4] => [1,2,3,4,6,7,5] => [2,3,4,7,5,6,1] => [1,2,3,5,6,4,7] => ? = 5 - 1
[1,2,3,6,7,4,5] => [1,2,3,4,6,5,7] => [2,3,4,6,5,7,1] => [1,2,3,5,4,6,7] => ? = 7 - 1
[1,2,3,6,7,5,4] => [1,2,3,4,6,5,7] => [2,3,4,6,5,7,1] => [1,2,3,5,4,6,7] => ? = 7 - 1
[1,2,3,7,4,5,6] => [1,2,3,4,7,6,5] => [2,3,4,7,6,5,1] => [1,2,3,6,5,4,7] => ? = 5 - 1
[1,2,3,7,4,6,5] => [1,2,3,4,7,5,6] => [2,3,4,6,7,5,1] => [1,2,3,6,4,5,7] => ? = 6 - 1
[1,2,3,7,5,4,6] => [1,2,3,4,7,6,5] => [2,3,4,7,6,5,1] => [1,2,3,6,5,4,7] => ? = 5 - 1
[1,2,3,7,5,6,4] => [1,2,3,4,7,5,6] => [2,3,4,6,7,5,1] => [1,2,3,6,4,5,7] => ? = 6 - 1
[1,2,3,7,6,4,5] => [1,2,3,4,7,5,6] => [2,3,4,6,7,5,1] => [1,2,3,6,4,5,7] => ? = 6 - 1
[1,2,3,7,6,5,4] => [1,2,3,4,7,5,6] => [2,3,4,6,7,5,1] => [1,2,3,6,4,5,7] => ? = 6 - 1
[1,2,4,3,5,6,7] => [1,2,3,4,5,6,7] => [2,3,4,5,6,7,1] => [1,2,3,4,5,6,7] => ? = 7 - 1
[1,2,4,3,5,7,6] => [1,2,3,4,5,6,7] => [2,3,4,5,6,7,1] => [1,2,3,4,5,6,7] => ? = 7 - 1
[1,2,4,3,6,5,7] => [1,2,3,4,5,6,7] => [2,3,4,5,6,7,1] => [1,2,3,4,5,6,7] => ? = 7 - 1
[1,2,4,3,6,7,5] => [1,2,3,4,5,6,7] => [2,3,4,5,6,7,1] => [1,2,3,4,5,6,7] => ? = 7 - 1
[1,2,4,3,7,5,6] => [1,2,3,4,5,7,6] => [2,3,4,5,7,6,1] => [1,2,3,4,6,5,7] => ? = 6 - 1
[1,2,4,3,7,6,5] => [1,2,3,4,5,7,6] => [2,3,4,5,7,6,1] => [1,2,3,4,6,5,7] => ? = 6 - 1
[1,2,4,5,3,6,7] => [1,2,3,4,5,6,7] => [2,3,4,5,6,7,1] => [1,2,3,4,5,6,7] => ? = 7 - 1
[1,2,4,5,3,7,6] => [1,2,3,4,5,6,7] => [2,3,4,5,6,7,1] => [1,2,3,4,5,6,7] => ? = 7 - 1
[1,2,4,5,6,3,7] => [1,2,3,4,5,6,7] => [2,3,4,5,6,7,1] => [1,2,3,4,5,6,7] => ? = 7 - 1
[1,2,4,5,6,7,3] => [1,2,3,4,5,6,7] => [2,3,4,5,6,7,1] => [1,2,3,4,5,6,7] => ? = 7 - 1
[1,2,4,5,7,3,6] => [1,2,3,4,5,7,6] => [2,3,4,5,7,6,1] => [1,2,3,4,6,5,7] => ? = 6 - 1
[1,2,4,5,7,6,3] => [1,2,3,4,5,7,6] => [2,3,4,5,7,6,1] => [1,2,3,4,6,5,7] => ? = 6 - 1
[1,2,4,6,3,5,7] => [1,2,3,4,6,5,7] => [2,3,4,6,5,7,1] => [1,2,3,5,4,6,7] => ? = 7 - 1
[1,2,4,6,3,7,5] => [1,2,3,4,6,7,5] => [2,3,4,7,5,6,1] => [1,2,3,5,6,4,7] => ? = 5 - 1
[1,2,4,6,5,3,7] => [1,2,3,4,6,5,7] => [2,3,4,6,5,7,1] => [1,2,3,5,4,6,7] => ? = 7 - 1
[1,2,4,6,5,7,3] => [1,2,3,4,6,7,5] => [2,3,4,7,5,6,1] => [1,2,3,5,6,4,7] => ? = 5 - 1
[1,2,4,6,7,3,5] => [1,2,3,4,6,5,7] => [2,3,4,6,5,7,1] => [1,2,3,5,4,6,7] => ? = 7 - 1
[1,2,4,6,7,5,3] => [1,2,3,4,6,5,7] => [2,3,4,6,5,7,1] => [1,2,3,5,4,6,7] => ? = 7 - 1
[1,2,4,7,3,5,6] => [1,2,3,4,7,6,5] => [2,3,4,7,6,5,1] => [1,2,3,6,5,4,7] => ? = 5 - 1
[1,2,4,7,3,6,5] => [1,2,3,4,7,5,6] => [2,3,4,6,7,5,1] => [1,2,3,6,4,5,7] => ? = 6 - 1
[1,2,4,7,5,3,6] => [1,2,3,4,7,6,5] => [2,3,4,7,6,5,1] => [1,2,3,6,5,4,7] => ? = 5 - 1
[1,2,4,7,5,6,3] => [1,2,3,4,7,5,6] => [2,3,4,6,7,5,1] => [1,2,3,6,4,5,7] => ? = 6 - 1
[1,2,4,7,6,3,5] => [1,2,3,4,7,5,6] => [2,3,4,6,7,5,1] => [1,2,3,6,4,5,7] => ? = 6 - 1
[1,2,4,7,6,5,3] => [1,2,3,4,7,5,6] => [2,3,4,6,7,5,1] => [1,2,3,6,4,5,7] => ? = 6 - 1
[1,2,5,3,4,6,7] => [1,2,3,5,4,6,7] => [2,3,5,4,6,7,1] => [1,2,4,3,5,6,7] => ? = 7 - 1
Description
The greater neighbor of the maximum. Han [2] showed that this statistic is (up to a shift) equidistributed on zigzag permutations (permutations \pi such that \pi(1) < \pi(2) > \pi(3) \cdots) with the smallest path leaf label of the binary tree associated to a permutation ([[St000724]]), see also [3].
Matching statistic: St000799
Mapping
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00088: Permutations —Kreweras complement⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000799: Permutations ⟶ ℤResult quality: 24% ●values known / values provided: 24%●distinct values known / distinct values provided: 71%
Values
[1] => [1] => [1] => [1] => ? = 1 - 2
[1,2] => [1,2] => [2,1] => [1,2] => 0 = 2 - 2
[2,1] => [1,2] => [2,1] => [1,2] => 0 = 2 - 2
[1,2,3] => [1,2,3] => [2,3,1] => [2,1,3] => 1 = 3 - 2
[1,3,2] => [1,2,3] => [2,3,1] => [2,1,3] => 1 = 3 - 2
[2,1,3] => [1,2,3] => [2,3,1] => [2,1,3] => 1 = 3 - 2
[2,3,1] => [1,2,3] => [2,3,1] => [2,1,3] => 1 = 3 - 2
[3,1,2] => [1,3,2] => [2,1,3] => [2,3,1] => 0 = 2 - 2
[3,2,1] => [1,3,2] => [2,1,3] => [2,3,1] => 0 = 2 - 2
[1,2,3,4] => [1,2,3,4] => [2,3,4,1] => [3,2,1,4] => 2 = 4 - 2
[1,2,4,3] => [1,2,3,4] => [2,3,4,1] => [3,2,1,4] => 2 = 4 - 2
[1,3,2,4] => [1,2,3,4] => [2,3,4,1] => [3,2,1,4] => 2 = 4 - 2
[1,3,4,2] => [1,2,3,4] => [2,3,4,1] => [3,2,1,4] => 2 = 4 - 2
[1,4,2,3] => [1,2,4,3] => [2,3,1,4] => [3,2,4,1] => 1 = 3 - 2
[1,4,3,2] => [1,2,4,3] => [2,3,1,4] => [3,2,4,1] => 1 = 3 - 2
[2,1,3,4] => [1,2,3,4] => [2,3,4,1] => [3,2,1,4] => 2 = 4 - 2
[2,1,4,3] => [1,2,3,4] => [2,3,4,1] => [3,2,1,4] => 2 = 4 - 2
[2,3,1,4] => [1,2,3,4] => [2,3,4,1] => [3,2,1,4] => 2 = 4 - 2
[2,3,4,1] => [1,2,3,4] => [2,3,4,1] => [3,2,1,4] => 2 = 4 - 2
[2,4,1,3] => [1,2,4,3] => [2,3,1,4] => [3,2,4,1] => 1 = 3 - 2
[2,4,3,1] => [1,2,4,3] => [2,3,1,4] => [3,2,4,1] => 1 = 3 - 2
[3,1,2,4] => [1,3,2,4] => [2,4,3,1] => [3,1,2,4] => 2 = 4 - 2
[3,1,4,2] => [1,3,4,2] => [2,1,3,4] => [3,4,2,1] => 0 = 2 - 2
[3,2,1,4] => [1,3,2,4] => [2,4,3,1] => [3,1,2,4] => 2 = 4 - 2
[3,2,4,1] => [1,3,4,2] => [2,1,3,4] => [3,4,2,1] => 0 = 2 - 2
[3,4,1,2] => [1,3,2,4] => [2,4,3,1] => [3,1,2,4] => 2 = 4 - 2
[3,4,2,1] => [1,3,2,4] => [2,4,3,1] => [3,1,2,4] => 2 = 4 - 2
[4,1,2,3] => [1,4,3,2] => [2,1,4,3] => [3,4,1,2] => 0 = 2 - 2
[4,1,3,2] => [1,4,2,3] => [2,4,1,3] => [3,1,4,2] => 1 = 3 - 2
[4,2,1,3] => [1,4,3,2] => [2,1,4,3] => [3,4,1,2] => 0 = 2 - 2
[4,2,3,1] => [1,4,2,3] => [2,4,1,3] => [3,1,4,2] => 1 = 3 - 2
[4,3,1,2] => [1,4,2,3] => [2,4,1,3] => [3,1,4,2] => 1 = 3 - 2
[4,3,2,1] => [1,4,2,3] => [2,4,1,3] => [3,1,4,2] => 1 = 3 - 2
[1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => [4,3,2,1,5] => 3 = 5 - 2
[1,2,3,5,4] => [1,2,3,4,5] => [2,3,4,5,1] => [4,3,2,1,5] => 3 = 5 - 2
[1,2,4,3,5] => [1,2,3,4,5] => [2,3,4,5,1] => [4,3,2,1,5] => 3 = 5 - 2
[1,2,4,5,3] => [1,2,3,4,5] => [2,3,4,5,1] => [4,3,2,1,5] => 3 = 5 - 2
[1,2,5,3,4] => [1,2,3,5,4] => [2,3,4,1,5] => [4,3,2,5,1] => 2 = 4 - 2
[1,2,5,4,3] => [1,2,3,5,4] => [2,3,4,1,5] => [4,3,2,5,1] => 2 = 4 - 2
[1,3,2,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => [4,3,2,1,5] => 3 = 5 - 2
[1,3,2,5,4] => [1,2,3,4,5] => [2,3,4,5,1] => [4,3,2,1,5] => 3 = 5 - 2
[1,3,4,2,5] => [1,2,3,4,5] => [2,3,4,5,1] => [4,3,2,1,5] => 3 = 5 - 2
[1,3,4,5,2] => [1,2,3,4,5] => [2,3,4,5,1] => [4,3,2,1,5] => 3 = 5 - 2
[1,3,5,2,4] => [1,2,3,5,4] => [2,3,4,1,5] => [4,3,2,5,1] => 2 = 4 - 2
[1,3,5,4,2] => [1,2,3,5,4] => [2,3,4,1,5] => [4,3,2,5,1] => 2 = 4 - 2
[1,4,2,3,5] => [1,2,4,3,5] => [2,3,5,4,1] => [4,3,1,2,5] => 3 = 5 - 2
[1,4,2,5,3] => [1,2,4,5,3] => [2,3,1,4,5] => [4,3,5,2,1] => 1 = 3 - 2
[1,4,3,2,5] => [1,2,4,3,5] => [2,3,5,4,1] => [4,3,1,2,5] => 3 = 5 - 2
[1,4,3,5,2] => [1,2,4,5,3] => [2,3,1,4,5] => [4,3,5,2,1] => 1 = 3 - 2
[1,4,5,2,3] => [1,2,4,3,5] => [2,3,5,4,1] => [4,3,1,2,5] => 3 = 5 - 2
[1,4,5,3,2] => [1,2,4,3,5] => [2,3,5,4,1] => [4,3,1,2,5] => 3 = 5 - 2
[1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => [2,3,4,5,6,7,1] => [6,5,4,3,2,1,7] => ? = 7 - 2
[1,2,3,4,5,7,6] => [1,2,3,4,5,6,7] => [2,3,4,5,6,7,1] => [6,5,4,3,2,1,7] => ? = 7 - 2
[1,2,3,4,6,5,7] => [1,2,3,4,5,6,7] => [2,3,4,5,6,7,1] => [6,5,4,3,2,1,7] => ? = 7 - 2
[1,2,3,4,6,7,5] => [1,2,3,4,5,6,7] => [2,3,4,5,6,7,1] => [6,5,4,3,2,1,7] => ? = 7 - 2
[1,2,3,4,7,5,6] => [1,2,3,4,5,7,6] => [2,3,4,5,6,1,7] => [6,5,4,3,2,7,1] => ? = 6 - 2
[1,2,3,4,7,6,5] => [1,2,3,4,5,7,6] => [2,3,4,5,6,1,7] => [6,5,4,3,2,7,1] => ? = 6 - 2
[1,2,3,5,4,6,7] => [1,2,3,4,5,6,7] => [2,3,4,5,6,7,1] => [6,5,4,3,2,1,7] => ? = 7 - 2
[1,2,3,5,4,7,6] => [1,2,3,4,5,6,7] => [2,3,4,5,6,7,1] => [6,5,4,3,2,1,7] => ? = 7 - 2
[1,2,3,5,6,4,7] => [1,2,3,4,5,6,7] => [2,3,4,5,6,7,1] => [6,5,4,3,2,1,7] => ? = 7 - 2
[1,2,3,5,6,7,4] => [1,2,3,4,5,6,7] => [2,3,4,5,6,7,1] => [6,5,4,3,2,1,7] => ? = 7 - 2
[1,2,3,5,7,4,6] => [1,2,3,4,5,7,6] => [2,3,4,5,6,1,7] => [6,5,4,3,2,7,1] => ? = 6 - 2
[1,2,3,5,7,6,4] => [1,2,3,4,5,7,6] => [2,3,4,5,6,1,7] => [6,5,4,3,2,7,1] => ? = 6 - 2
[1,2,3,6,4,5,7] => [1,2,3,4,6,5,7] => [2,3,4,5,7,6,1] => [6,5,4,3,1,2,7] => ? = 7 - 2
[1,2,3,6,4,7,5] => [1,2,3,4,6,7,5] => [2,3,4,5,1,6,7] => [6,5,4,3,7,2,1] => ? = 5 - 2
[1,2,3,6,5,4,7] => [1,2,3,4,6,5,7] => [2,3,4,5,7,6,1] => [6,5,4,3,1,2,7] => ? = 7 - 2
[1,2,3,6,5,7,4] => [1,2,3,4,6,7,5] => [2,3,4,5,1,6,7] => [6,5,4,3,7,2,1] => ? = 5 - 2
[1,2,3,6,7,4,5] => [1,2,3,4,6,5,7] => [2,3,4,5,7,6,1] => [6,5,4,3,1,2,7] => ? = 7 - 2
[1,2,3,6,7,5,4] => [1,2,3,4,6,5,7] => [2,3,4,5,7,6,1] => [6,5,4,3,1,2,7] => ? = 7 - 2
[1,2,3,7,4,5,6] => [1,2,3,4,7,6,5] => [2,3,4,5,1,7,6] => [6,5,4,3,7,1,2] => ? = 5 - 2
[1,2,3,7,4,6,5] => [1,2,3,4,7,5,6] => [2,3,4,5,7,1,6] => [6,5,4,3,1,7,2] => ? = 6 - 2
[1,2,3,7,5,4,6] => [1,2,3,4,7,6,5] => [2,3,4,5,1,7,6] => [6,5,4,3,7,1,2] => ? = 5 - 2
[1,2,3,7,5,6,4] => [1,2,3,4,7,5,6] => [2,3,4,5,7,1,6] => [6,5,4,3,1,7,2] => ? = 6 - 2
[1,2,3,7,6,4,5] => [1,2,3,4,7,5,6] => [2,3,4,5,7,1,6] => [6,5,4,3,1,7,2] => ? = 6 - 2
[1,2,3,7,6,5,4] => [1,2,3,4,7,5,6] => [2,3,4,5,7,1,6] => [6,5,4,3,1,7,2] => ? = 6 - 2
[1,2,4,3,5,6,7] => [1,2,3,4,5,6,7] => [2,3,4,5,6,7,1] => [6,5,4,3,2,1,7] => ? = 7 - 2
[1,2,4,3,5,7,6] => [1,2,3,4,5,6,7] => [2,3,4,5,6,7,1] => [6,5,4,3,2,1,7] => ? = 7 - 2
[1,2,4,3,6,5,7] => [1,2,3,4,5,6,7] => [2,3,4,5,6,7,1] => [6,5,4,3,2,1,7] => ? = 7 - 2
[1,2,4,3,6,7,5] => [1,2,3,4,5,6,7] => [2,3,4,5,6,7,1] => [6,5,4,3,2,1,7] => ? = 7 - 2
[1,2,4,3,7,5,6] => [1,2,3,4,5,7,6] => [2,3,4,5,6,1,7] => [6,5,4,3,2,7,1] => ? = 6 - 2
[1,2,4,3,7,6,5] => [1,2,3,4,5,7,6] => [2,3,4,5,6,1,7] => [6,5,4,3,2,7,1] => ? = 6 - 2
[1,2,4,5,3,6,7] => [1,2,3,4,5,6,7] => [2,3,4,5,6,7,1] => [6,5,4,3,2,1,7] => ? = 7 - 2
[1,2,4,5,3,7,6] => [1,2,3,4,5,6,7] => [2,3,4,5,6,7,1] => [6,5,4,3,2,1,7] => ? = 7 - 2
[1,2,4,5,6,3,7] => [1,2,3,4,5,6,7] => [2,3,4,5,6,7,1] => [6,5,4,3,2,1,7] => ? = 7 - 2
[1,2,4,5,6,7,3] => [1,2,3,4,5,6,7] => [2,3,4,5,6,7,1] => [6,5,4,3,2,1,7] => ? = 7 - 2
[1,2,4,5,7,3,6] => [1,2,3,4,5,7,6] => [2,3,4,5,6,1,7] => [6,5,4,3,2,7,1] => ? = 6 - 2
[1,2,4,5,7,6,3] => [1,2,3,4,5,7,6] => [2,3,4,5,6,1,7] => [6,5,4,3,2,7,1] => ? = 6 - 2
[1,2,4,6,3,5,7] => [1,2,3,4,6,5,7] => [2,3,4,5,7,6,1] => [6,5,4,3,1,2,7] => ? = 7 - 2
[1,2,4,6,3,7,5] => [1,2,3,4,6,7,5] => [2,3,4,5,1,6,7] => [6,5,4,3,7,2,1] => ? = 5 - 2
[1,2,4,6,5,3,7] => [1,2,3,4,6,5,7] => [2,3,4,5,7,6,1] => [6,5,4,3,1,2,7] => ? = 7 - 2
[1,2,4,6,5,7,3] => [1,2,3,4,6,7,5] => [2,3,4,5,1,6,7] => [6,5,4,3,7,2,1] => ? = 5 - 2
[1,2,4,6,7,3,5] => [1,2,3,4,6,5,7] => [2,3,4,5,7,6,1] => [6,5,4,3,1,2,7] => ? = 7 - 2
[1,2,4,6,7,5,3] => [1,2,3,4,6,5,7] => [2,3,4,5,7,6,1] => [6,5,4,3,1,2,7] => ? = 7 - 2
[1,2,4,7,3,5,6] => [1,2,3,4,7,6,5] => [2,3,4,5,1,7,6] => [6,5,4,3,7,1,2] => ? = 5 - 2
[1,2,4,7,3,6,5] => [1,2,3,4,7,5,6] => [2,3,4,5,7,1,6] => [6,5,4,3,1,7,2] => ? = 6 - 2
[1,2,4,7,5,3,6] => [1,2,3,4,7,6,5] => [2,3,4,5,1,7,6] => [6,5,4,3,7,1,2] => ? = 5 - 2
[1,2,4,7,5,6,3] => [1,2,3,4,7,5,6] => [2,3,4,5,7,1,6] => [6,5,4,3,1,7,2] => ? = 6 - 2
[1,2,4,7,6,3,5] => [1,2,3,4,7,5,6] => [2,3,4,5,7,1,6] => [6,5,4,3,1,7,2] => ? = 6 - 2
[1,2,4,7,6,5,3] => [1,2,3,4,7,5,6] => [2,3,4,5,7,1,6] => [6,5,4,3,1,7,2] => ? = 6 - 2
[1,2,5,3,4,6,7] => [1,2,3,5,4,6,7] => [2,3,4,6,5,7,1] => [6,5,4,2,3,1,7] => ? = 7 - 2
Description
The number of occurrences of the vincular pattern |213 in a permutation. This is the number of occurrences of the pattern (2,1,3), such that the letter matched by 2 is the first entry of the permutation.
The following 12 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000219The number of occurrences of the pattern 231 in a permutation. St000066The column of the unique '1' in the first row of the alternating sign matrix. St000200The row of the unique '1' in the last column of the alternating sign matrix. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000199The column of the unique '1' in the last row of the alternating sign matrix. St001557The number of inversions of the second entry of a permutation. St001330The hat guessing number of a graph. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001207The Lowey length of the algebra A/T when T is the 1-tilting module corresponding to the permutation in the Auslander algebra of K[x]/(x^n). St001927Sparre Andersen's number of positives of a signed permutation.