Your data matches 24 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000996
Mapping
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
St000996: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => [1] => [1] => 0
[1,0,1,0]
 => [2,1] => [1,2] => [2,1] => 1
[1,1,0,0]
 => [1,2] => [1,2] => [2,1] => 1
[1,0,1,0,1,0]
 => [3,2,1] => [1,2,3] => [2,3,1] => 2
[1,0,1,1,0,0]
 => [2,3,1] => [1,2,3] => [2,3,1] => 2
[1,1,0,0,1,0]
 => [3,1,2] => [1,2,3] => [2,3,1] => 2
[1,1,0,1,0,0]
 => [2,1,3] => [1,3,2] => [3,2,1] => 1
[1,1,1,0,0,0]
 => [1,2,3] => [1,2,3] => [2,3,1] => 2
[1,0,1,0,1,0,1,0]
 => [4,3,2,1] => [1,2,3,4] => [2,3,4,1] => 3
[1,0,1,0,1,1,0,0]
 => [3,4,2,1] => [1,2,3,4] => [2,3,4,1] => 3
[1,0,1,1,0,0,1,0]
 => [4,2,3,1] => [1,2,3,4] => [2,3,4,1] => 3
[1,0,1,1,0,1,0,0]
 => [3,2,4,1] => [1,2,4,3] => [2,4,3,1] => 2
[1,0,1,1,1,0,0,0]
 => [2,3,4,1] => [1,2,3,4] => [2,3,4,1] => 3
[1,1,0,0,1,0,1,0]
 => [4,3,1,2] => [1,2,3,4] => [2,3,4,1] => 3
[1,1,0,0,1,1,0,0]
 => [3,4,1,2] => [1,2,3,4] => [2,3,4,1] => 3
[1,1,0,1,0,0,1,0]
 => [4,2,1,3] => [1,3,2,4] => [3,2,4,1] => 2
[1,1,0,1,0,1,0,0]
 => [3,2,1,4] => [1,4,2,3] => [3,4,2,1] => 2
[1,1,0,1,1,0,0,0]
 => [2,3,1,4] => [1,4,2,3] => [3,4,2,1] => 2
[1,1,1,0,0,0,1,0]
 => [4,1,2,3] => [1,2,3,4] => [2,3,4,1] => 3
[1,1,1,0,0,1,0,0]
 => [3,1,2,4] => [1,2,4,3] => [2,4,3,1] => 2
[1,1,1,0,1,0,0,0]
 => [2,1,3,4] => [1,3,4,2] => [4,2,3,1] => 1
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [1,2,3,4] => [2,3,4,1] => 3
[1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => [1,2,3,4,5] => [2,3,4,5,1] => 4
[1,0,1,0,1,0,1,1,0,0]
 => [4,5,3,2,1] => [1,2,3,4,5] => [2,3,4,5,1] => 4
[1,0,1,0,1,1,0,0,1,0]
 => [5,3,4,2,1] => [1,2,3,4,5] => [2,3,4,5,1] => 4
[1,0,1,0,1,1,0,1,0,0]
 => [4,3,5,2,1] => [1,2,3,5,4] => [2,3,5,4,1] => 3
[1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => [1,2,3,4,5] => [2,3,4,5,1] => 4
[1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,3,1] => [1,2,3,4,5] => [2,3,4,5,1] => 4
[1,0,1,1,0,0,1,1,0,0]
 => [4,5,2,3,1] => [1,2,3,4,5] => [2,3,4,5,1] => 4
[1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,4,1] => [1,2,4,3,5] => [2,4,3,5,1] => 3
[1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,5,1] => [1,2,5,3,4] => [2,4,5,3,1] => 3
[1,0,1,1,0,1,1,0,0,0]
 => [3,4,2,5,1] => [1,2,5,3,4] => [2,4,5,3,1] => 3
[1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => [1,2,3,4,5] => [2,3,4,5,1] => 4
[1,0,1,1,1,0,0,1,0,0]
 => [4,2,3,5,1] => [1,2,3,5,4] => [2,3,5,4,1] => 3
[1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => [1,2,4,5,3] => [2,5,3,4,1] => 2
[1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [1,2,3,4,5] => [2,3,4,5,1] => 4
[1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,2] => [1,2,3,4,5] => [2,3,4,5,1] => 4
[1,1,0,0,1,0,1,1,0,0]
 => [4,5,3,1,2] => [1,2,3,4,5] => [2,3,4,5,1] => 4
[1,1,0,0,1,1,0,0,1,0]
 => [5,3,4,1,2] => [1,2,3,4,5] => [2,3,4,5,1] => 4
[1,1,0,0,1,1,0,1,0,0]
 => [4,3,5,1,2] => [1,2,3,5,4] => [2,3,5,4,1] => 3
[1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => [1,2,3,4,5] => [2,3,4,5,1] => 4
[1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,3] => [1,3,2,4,5] => [3,2,4,5,1] => 3
[1,1,0,1,0,0,1,1,0,0]
 => [4,5,2,1,3] => [1,3,2,4,5] => [3,2,4,5,1] => 3
[1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,4] => [1,4,2,3,5] => [3,4,2,5,1] => 3
[1,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1,5] => [1,5,2,3,4] => [3,4,5,2,1] => 3
[1,1,0,1,0,1,1,0,0,0]
 => [3,4,2,1,5] => [1,5,2,3,4] => [3,4,5,2,1] => 3
[1,1,0,1,1,0,0,0,1,0]
 => [5,2,3,1,4] => [1,4,2,3,5] => [3,4,2,5,1] => 3
[1,1,0,1,1,0,0,1,0,0]
 => [4,2,3,1,5] => [1,5,2,3,4] => [3,4,5,2,1] => 3
[1,1,0,1,1,0,1,0,0,0]
 => [3,2,4,1,5] => [1,5,2,4,3] => [3,5,4,2,1] => 2
[1,1,0,1,1,1,0,0,0,0]
 => [2,3,4,1,5] => [1,5,2,3,4] => [3,4,5,2,1] => 3
Description
The number of exclusive left-to-right maxima of a permutation. This is the number of left-to-right maxima that are not right-to-left minima.
Matching statistic: St001004
Mapping
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
St001004: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => [1] => [1] => 1 = 0 + 1
[1,0,1,0]
 => [2,1] => [1,2] => [2,1] => 2 = 1 + 1
[1,1,0,0]
 => [1,2] => [1,2] => [2,1] => 2 = 1 + 1
[1,0,1,0,1,0]
 => [3,2,1] => [1,2,3] => [2,3,1] => 3 = 2 + 1
[1,0,1,1,0,0]
 => [2,3,1] => [1,2,3] => [2,3,1] => 3 = 2 + 1
[1,1,0,0,1,0]
 => [3,1,2] => [1,2,3] => [2,3,1] => 3 = 2 + 1
[1,1,0,1,0,0]
 => [2,1,3] => [1,3,2] => [3,2,1] => 2 = 1 + 1
[1,1,1,0,0,0]
 => [1,2,3] => [1,2,3] => [2,3,1] => 3 = 2 + 1
[1,0,1,0,1,0,1,0]
 => [4,3,2,1] => [1,2,3,4] => [2,3,4,1] => 4 = 3 + 1
[1,0,1,0,1,1,0,0]
 => [3,4,2,1] => [1,2,3,4] => [2,3,4,1] => 4 = 3 + 1
[1,0,1,1,0,0,1,0]
 => [4,2,3,1] => [1,2,3,4] => [2,3,4,1] => 4 = 3 + 1
[1,0,1,1,0,1,0,0]
 => [3,2,4,1] => [1,2,4,3] => [2,4,3,1] => 3 = 2 + 1
[1,0,1,1,1,0,0,0]
 => [2,3,4,1] => [1,2,3,4] => [2,3,4,1] => 4 = 3 + 1
[1,1,0,0,1,0,1,0]
 => [4,3,1,2] => [1,2,3,4] => [2,3,4,1] => 4 = 3 + 1
[1,1,0,0,1,1,0,0]
 => [3,4,1,2] => [1,2,3,4] => [2,3,4,1] => 4 = 3 + 1
[1,1,0,1,0,0,1,0]
 => [4,2,1,3] => [1,3,2,4] => [3,2,4,1] => 3 = 2 + 1
[1,1,0,1,0,1,0,0]
 => [3,2,1,4] => [1,4,2,3] => [3,4,2,1] => 3 = 2 + 1
[1,1,0,1,1,0,0,0]
 => [2,3,1,4] => [1,4,2,3] => [3,4,2,1] => 3 = 2 + 1
[1,1,1,0,0,0,1,0]
 => [4,1,2,3] => [1,2,3,4] => [2,3,4,1] => 4 = 3 + 1
[1,1,1,0,0,1,0,0]
 => [3,1,2,4] => [1,2,4,3] => [2,4,3,1] => 3 = 2 + 1
[1,1,1,0,1,0,0,0]
 => [2,1,3,4] => [1,3,4,2] => [4,2,3,1] => 2 = 1 + 1
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [1,2,3,4] => [2,3,4,1] => 4 = 3 + 1
[1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => [1,2,3,4,5] => [2,3,4,5,1] => 5 = 4 + 1
[1,0,1,0,1,0,1,1,0,0]
 => [4,5,3,2,1] => [1,2,3,4,5] => [2,3,4,5,1] => 5 = 4 + 1
[1,0,1,0,1,1,0,0,1,0]
 => [5,3,4,2,1] => [1,2,3,4,5] => [2,3,4,5,1] => 5 = 4 + 1
[1,0,1,0,1,1,0,1,0,0]
 => [4,3,5,2,1] => [1,2,3,5,4] => [2,3,5,4,1] => 4 = 3 + 1
[1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => [1,2,3,4,5] => [2,3,4,5,1] => 5 = 4 + 1
[1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,3,1] => [1,2,3,4,5] => [2,3,4,5,1] => 5 = 4 + 1
[1,0,1,1,0,0,1,1,0,0]
 => [4,5,2,3,1] => [1,2,3,4,5] => [2,3,4,5,1] => 5 = 4 + 1
[1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,4,1] => [1,2,4,3,5] => [2,4,3,5,1] => 4 = 3 + 1
[1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,5,1] => [1,2,5,3,4] => [2,4,5,3,1] => 4 = 3 + 1
[1,0,1,1,0,1,1,0,0,0]
 => [3,4,2,5,1] => [1,2,5,3,4] => [2,4,5,3,1] => 4 = 3 + 1
[1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => [1,2,3,4,5] => [2,3,4,5,1] => 5 = 4 + 1
[1,0,1,1,1,0,0,1,0,0]
 => [4,2,3,5,1] => [1,2,3,5,4] => [2,3,5,4,1] => 4 = 3 + 1
[1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => [1,2,4,5,3] => [2,5,3,4,1] => 3 = 2 + 1
[1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [1,2,3,4,5] => [2,3,4,5,1] => 5 = 4 + 1
[1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,2] => [1,2,3,4,5] => [2,3,4,5,1] => 5 = 4 + 1
[1,1,0,0,1,0,1,1,0,0]
 => [4,5,3,1,2] => [1,2,3,4,5] => [2,3,4,5,1] => 5 = 4 + 1
[1,1,0,0,1,1,0,0,1,0]
 => [5,3,4,1,2] => [1,2,3,4,5] => [2,3,4,5,1] => 5 = 4 + 1
[1,1,0,0,1,1,0,1,0,0]
 => [4,3,5,1,2] => [1,2,3,5,4] => [2,3,5,4,1] => 4 = 3 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => [1,2,3,4,5] => [2,3,4,5,1] => 5 = 4 + 1
[1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,3] => [1,3,2,4,5] => [3,2,4,5,1] => 4 = 3 + 1
[1,1,0,1,0,0,1,1,0,0]
 => [4,5,2,1,3] => [1,3,2,4,5] => [3,2,4,5,1] => 4 = 3 + 1
[1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,4] => [1,4,2,3,5] => [3,4,2,5,1] => 4 = 3 + 1
[1,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1,5] => [1,5,2,3,4] => [3,4,5,2,1] => 4 = 3 + 1
[1,1,0,1,0,1,1,0,0,0]
 => [3,4,2,1,5] => [1,5,2,3,4] => [3,4,5,2,1] => 4 = 3 + 1
[1,1,0,1,1,0,0,0,1,0]
 => [5,2,3,1,4] => [1,4,2,3,5] => [3,4,2,5,1] => 4 = 3 + 1
[1,1,0,1,1,0,0,1,0,0]
 => [4,2,3,1,5] => [1,5,2,3,4] => [3,4,5,2,1] => 4 = 3 + 1
[1,1,0,1,1,0,1,0,0,0]
 => [3,2,4,1,5] => [1,5,2,4,3] => [3,5,4,2,1] => 3 = 2 + 1
[1,1,0,1,1,1,0,0,0,0]
 => [2,3,4,1,5] => [1,5,2,3,4] => [3,4,5,2,1] => 4 = 3 + 1
Description
The number of indices that are either left-to-right maxima or right-to-left minima. The (bivariate) generating function for this statistic is (essentially) given in [1], the mid points of a $321$ pattern in the permutation are those elements which are neither left-to-right maxima nor a right-to-left minima, see [[St000371]] and [[St000372]].
Matching statistic: St000007
Mapping
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000007: Permutations ⟶ ℤResult quality: 83% ●values known / values provided: 83%●distinct values known / distinct values provided: 91%
Values
[1,0]
 => [1] => [1] => [1] => 1 = 0 + 1
[1,0,1,0]
 => [2,1] => [1,2] => [2,1] => 2 = 1 + 1
[1,1,0,0]
 => [1,2] => [1,2] => [2,1] => 2 = 1 + 1
[1,0,1,0,1,0]
 => [3,2,1] => [1,2,3] => [3,2,1] => 3 = 2 + 1
[1,0,1,1,0,0]
 => [2,3,1] => [1,2,3] => [3,2,1] => 3 = 2 + 1
[1,1,0,0,1,0]
 => [3,1,2] => [1,2,3] => [3,2,1] => 3 = 2 + 1
[1,1,0,1,0,0]
 => [2,1,3] => [1,3,2] => [3,1,2] => 2 = 1 + 1
[1,1,1,0,0,0]
 => [1,2,3] => [1,2,3] => [3,2,1] => 3 = 2 + 1
[1,0,1,0,1,0,1,0]
 => [4,3,2,1] => [1,2,3,4] => [4,3,2,1] => 4 = 3 + 1
[1,0,1,0,1,1,0,0]
 => [3,4,2,1] => [1,2,3,4] => [4,3,2,1] => 4 = 3 + 1
[1,0,1,1,0,0,1,0]
 => [4,2,3,1] => [1,2,3,4] => [4,3,2,1] => 4 = 3 + 1
[1,0,1,1,0,1,0,0]
 => [3,2,4,1] => [1,2,4,3] => [4,3,1,2] => 3 = 2 + 1
[1,0,1,1,1,0,0,0]
 => [2,3,4,1] => [1,2,3,4] => [4,3,2,1] => 4 = 3 + 1
[1,1,0,0,1,0,1,0]
 => [4,3,1,2] => [1,2,3,4] => [4,3,2,1] => 4 = 3 + 1
[1,1,0,0,1,1,0,0]
 => [3,4,1,2] => [1,2,3,4] => [4,3,2,1] => 4 = 3 + 1
[1,1,0,1,0,0,1,0]
 => [4,2,1,3] => [1,3,2,4] => [4,2,3,1] => 3 = 2 + 1
[1,1,0,1,0,1,0,0]
 => [3,2,1,4] => [1,4,2,3] => [4,1,3,2] => 3 = 2 + 1
[1,1,0,1,1,0,0,0]
 => [2,3,1,4] => [1,4,2,3] => [4,1,3,2] => 3 = 2 + 1
[1,1,1,0,0,0,1,0]
 => [4,1,2,3] => [1,2,3,4] => [4,3,2,1] => 4 = 3 + 1
[1,1,1,0,0,1,0,0]
 => [3,1,2,4] => [1,2,4,3] => [4,3,1,2] => 3 = 2 + 1
[1,1,1,0,1,0,0,0]
 => [2,1,3,4] => [1,3,4,2] => [4,2,1,3] => 2 = 1 + 1
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 4 = 3 + 1
[1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => [1,2,3,4,5] => [5,4,3,2,1] => 5 = 4 + 1
[1,0,1,0,1,0,1,1,0,0]
 => [4,5,3,2,1] => [1,2,3,4,5] => [5,4,3,2,1] => 5 = 4 + 1
[1,0,1,0,1,1,0,0,1,0]
 => [5,3,4,2,1] => [1,2,3,4,5] => [5,4,3,2,1] => 5 = 4 + 1
[1,0,1,0,1,1,0,1,0,0]
 => [4,3,5,2,1] => [1,2,3,5,4] => [5,4,3,1,2] => 4 = 3 + 1
[1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => [1,2,3,4,5] => [5,4,3,2,1] => 5 = 4 + 1
[1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,3,1] => [1,2,3,4,5] => [5,4,3,2,1] => 5 = 4 + 1
[1,0,1,1,0,0,1,1,0,0]
 => [4,5,2,3,1] => [1,2,3,4,5] => [5,4,3,2,1] => 5 = 4 + 1
[1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,4,1] => [1,2,4,3,5] => [5,4,2,3,1] => 4 = 3 + 1
[1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,5,1] => [1,2,5,3,4] => [5,4,1,3,2] => 4 = 3 + 1
[1,0,1,1,0,1,1,0,0,0]
 => [3,4,2,5,1] => [1,2,5,3,4] => [5,4,1,3,2] => 4 = 3 + 1
[1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => [1,2,3,4,5] => [5,4,3,2,1] => 5 = 4 + 1
[1,0,1,1,1,0,0,1,0,0]
 => [4,2,3,5,1] => [1,2,3,5,4] => [5,4,3,1,2] => 4 = 3 + 1
[1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => [1,2,4,5,3] => [5,4,2,1,3] => 3 = 2 + 1
[1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [1,2,3,4,5] => [5,4,3,2,1] => 5 = 4 + 1
[1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,2] => [1,2,3,4,5] => [5,4,3,2,1] => 5 = 4 + 1
[1,1,0,0,1,0,1,1,0,0]
 => [4,5,3,1,2] => [1,2,3,4,5] => [5,4,3,2,1] => 5 = 4 + 1
[1,1,0,0,1,1,0,0,1,0]
 => [5,3,4,1,2] => [1,2,3,4,5] => [5,4,3,2,1] => 5 = 4 + 1
[1,1,0,0,1,1,0,1,0,0]
 => [4,3,5,1,2] => [1,2,3,5,4] => [5,4,3,1,2] => 4 = 3 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => [1,2,3,4,5] => [5,4,3,2,1] => 5 = 4 + 1
[1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,3] => [1,3,2,4,5] => [5,3,4,2,1] => 4 = 3 + 1
[1,1,0,1,0,0,1,1,0,0]
 => [4,5,2,1,3] => [1,3,2,4,5] => [5,3,4,2,1] => 4 = 3 + 1
[1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,4] => [1,4,2,3,5] => [5,2,4,3,1] => 4 = 3 + 1
[1,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1,5] => [1,5,2,3,4] => [5,1,4,3,2] => 4 = 3 + 1
[1,1,0,1,0,1,1,0,0,0]
 => [3,4,2,1,5] => [1,5,2,3,4] => [5,1,4,3,2] => 4 = 3 + 1
[1,1,0,1,1,0,0,0,1,0]
 => [5,2,3,1,4] => [1,4,2,3,5] => [5,2,4,3,1] => 4 = 3 + 1
[1,1,0,1,1,0,0,1,0,0]
 => [4,2,3,1,5] => [1,5,2,3,4] => [5,1,4,3,2] => 4 = 3 + 1
[1,1,0,1,1,0,1,0,0,0]
 => [3,2,4,1,5] => [1,5,2,4,3] => [5,1,4,2,3] => 3 = 2 + 1
[1,1,0,1,1,1,0,0,0,0]
 => [2,3,4,1,5] => [1,5,2,3,4] => [5,1,4,3,2] => 4 = 3 + 1
[1,1,0,1,0,0,1,0,1,0,1,0,1,0]
 => [7,6,5,4,2,1,3] => [1,3,2,4,5,6,7] => [7,5,6,4,3,2,1] => ? = 5 + 1
[1,1,0,1,0,0,1,0,1,0,1,1,0,0]
 => [6,7,5,4,2,1,3] => [1,3,2,4,5,6,7] => [7,5,6,4,3,2,1] => ? = 5 + 1
[1,1,0,1,0,0,1,0,1,1,0,0,1,0]
 => [7,5,6,4,2,1,3] => [1,3,2,4,5,6,7] => [7,5,6,4,3,2,1] => ? = 5 + 1
[1,1,0,1,0,0,1,0,1,1,1,0,0,0]
 => [5,6,7,4,2,1,3] => [1,3,2,4,5,6,7] => [7,5,6,4,3,2,1] => ? = 5 + 1
[1,1,0,1,0,0,1,1,0,0,1,0,1,0]
 => [7,6,4,5,2,1,3] => [1,3,2,4,5,6,7] => [7,5,6,4,3,2,1] => ? = 5 + 1
[1,1,0,1,0,0,1,1,0,0,1,1,0,0]
 => [6,7,4,5,2,1,3] => [1,3,2,4,5,6,7] => [7,5,6,4,3,2,1] => ? = 5 + 1
[1,1,0,1,0,0,1,1,1,0,0,0,1,0]
 => [7,4,5,6,2,1,3] => [1,3,2,4,5,6,7] => [7,5,6,4,3,2,1] => ? = 5 + 1
[1,1,0,1,0,0,1,1,1,1,0,0,0,0]
 => [4,5,6,7,2,1,3] => [1,3,2,4,5,6,7] => [7,5,6,4,3,2,1] => ? = 5 + 1
[1,1,0,1,0,1,0,0,1,0,1,0,1,0]
 => [7,6,5,3,2,1,4] => [1,4,2,3,5,6,7] => [7,4,6,5,3,2,1] => ? = 5 + 1
[1,1,0,1,0,1,0,0,1,0,1,1,0,0]
 => [6,7,5,3,2,1,4] => [1,4,2,3,5,6,7] => [7,4,6,5,3,2,1] => ? = 5 + 1
[1,1,0,1,0,1,0,0,1,1,0,0,1,0]
 => [7,5,6,3,2,1,4] => [1,4,2,3,5,6,7] => [7,4,6,5,3,2,1] => ? = 5 + 1
[1,1,0,1,0,1,0,0,1,1,1,0,0,0]
 => [5,6,7,3,2,1,4] => [1,4,2,3,5,6,7] => [7,4,6,5,3,2,1] => ? = 5 + 1
[1,1,0,1,0,1,0,1,0,0,1,0,1,0]
 => [7,6,4,3,2,1,5] => [1,5,2,3,4,6,7] => [7,3,6,5,4,2,1] => ? = 5 + 1
[1,1,0,1,0,1,0,1,0,0,1,1,0,0]
 => [6,7,4,3,2,1,5] => [1,5,2,3,4,6,7] => [7,3,6,5,4,2,1] => ? = 5 + 1
[1,1,0,1,0,1,0,1,0,1,0,0,1,0]
 => [7,5,4,3,2,1,6] => [1,6,2,3,4,5,7] => [7,2,6,5,4,3,1] => ? = 5 + 1
[1,1,0,1,0,1,0,1,1,0,0,0,1,0]
 => [7,4,5,3,2,1,6] => [1,6,2,3,4,5,7] => [7,2,6,5,4,3,1] => ? = 5 + 1
[1,1,0,1,0,1,1,0,0,0,1,0,1,0]
 => [7,6,3,4,2,1,5] => [1,5,2,3,4,6,7] => [7,3,6,5,4,2,1] => ? = 5 + 1
[1,1,0,1,0,1,1,0,0,0,1,1,0,0]
 => [6,7,3,4,2,1,5] => [1,5,2,3,4,6,7] => [7,3,6,5,4,2,1] => ? = 5 + 1
[1,1,0,1,0,1,1,0,0,1,0,0,1,0]
 => [7,5,3,4,2,1,6] => [1,6,2,3,4,5,7] => [7,2,6,5,4,3,1] => ? = 5 + 1
[1,1,0,1,0,1,1,1,0,0,0,0,1,0]
 => [7,3,4,5,2,1,6] => [1,6,2,3,4,5,7] => [7,2,6,5,4,3,1] => ? = 5 + 1
[1,1,0,1,1,0,0,0,1,0,1,0,1,0]
 => [7,6,5,2,3,1,4] => [1,4,2,3,5,6,7] => [7,4,6,5,3,2,1] => ? = 5 + 1
[1,1,0,1,1,0,0,0,1,0,1,1,0,0]
 => [6,7,5,2,3,1,4] => [1,4,2,3,5,6,7] => [7,4,6,5,3,2,1] => ? = 5 + 1
[1,1,0,1,1,0,0,0,1,1,0,0,1,0]
 => [7,5,6,2,3,1,4] => [1,4,2,3,5,6,7] => [7,4,6,5,3,2,1] => ? = 5 + 1
[1,1,0,1,1,0,0,0,1,1,1,0,0,0]
 => [5,6,7,2,3,1,4] => [1,4,2,3,5,6,7] => [7,4,6,5,3,2,1] => ? = 5 + 1
[1,1,0,1,1,0,0,1,0,0,1,0,1,0]
 => [7,6,4,2,3,1,5] => [1,5,2,3,4,6,7] => [7,3,6,5,4,2,1] => ? = 5 + 1
[1,1,0,1,1,0,0,1,0,0,1,1,0,0]
 => [6,7,4,2,3,1,5] => [1,5,2,3,4,6,7] => [7,3,6,5,4,2,1] => ? = 5 + 1
[1,1,0,1,1,0,0,1,0,1,0,0,1,0]
 => [7,5,4,2,3,1,6] => [1,6,2,3,4,5,7] => [7,2,6,5,4,3,1] => ? = 5 + 1
[1,1,0,1,1,0,0,1,1,0,0,0,1,0]
 => [7,4,5,2,3,1,6] => [1,6,2,3,4,5,7] => [7,2,6,5,4,3,1] => ? = 5 + 1
[1,1,0,1,1,1,0,0,0,0,1,0,1,0]
 => [7,6,2,3,4,1,5] => [1,5,2,3,4,6,7] => [7,3,6,5,4,2,1] => ? = 5 + 1
[1,1,0,1,1,1,0,0,0,0,1,1,0,0]
 => [6,7,2,3,4,1,5] => [1,5,2,3,4,6,7] => [7,3,6,5,4,2,1] => ? = 5 + 1
[1,1,0,1,1,1,0,0,0,1,0,0,1,0]
 => [7,5,2,3,4,1,6] => [1,6,2,3,4,5,7] => [7,2,6,5,4,3,1] => ? = 5 + 1
[1,1,0,1,1,1,1,0,0,0,0,0,1,0]
 => [7,2,3,4,5,1,6] => [1,6,2,3,4,5,7] => [7,2,6,5,4,3,1] => ? = 5 + 1
[1,1,1,0,1,0,0,0,1,0,1,0,1,0]
 => [7,6,5,2,1,3,4] => [1,3,4,2,5,6,7] => [7,5,4,6,3,2,1] => ? = 4 + 1
[1,1,1,0,1,0,0,0,1,0,1,1,0,0]
 => [6,7,5,2,1,3,4] => [1,3,4,2,5,6,7] => [7,5,4,6,3,2,1] => ? = 4 + 1
[1,1,1,0,1,0,0,0,1,1,0,0,1,0]
 => [7,5,6,2,1,3,4] => [1,3,4,2,5,6,7] => [7,5,4,6,3,2,1] => ? = 4 + 1
[1,1,1,0,1,0,0,0,1,1,1,0,0,0]
 => [5,6,7,2,1,3,4] => [1,3,4,2,5,6,7] => [7,5,4,6,3,2,1] => ? = 4 + 1
[1,1,1,0,1,0,1,0,0,0,1,0,1,0]
 => [7,6,3,2,1,4,5] => [1,4,5,2,3,6,7] => [7,4,3,6,5,2,1] => ? = 4 + 1
[1,1,1,0,1,0,1,0,0,0,1,1,0,0]
 => [6,7,3,2,1,4,5] => [1,4,5,2,3,6,7] => [7,4,3,6,5,2,1] => ? = 4 + 1
[1,1,1,0,1,1,0,0,0,0,1,0,1,0]
 => [7,6,2,3,1,4,5] => [1,4,5,2,3,6,7] => [7,4,3,6,5,2,1] => ? = 4 + 1
[1,1,1,0,1,1,0,0,0,0,1,1,0,0]
 => [6,7,2,3,1,4,5] => [1,4,5,2,3,6,7] => [7,4,3,6,5,2,1] => ? = 4 + 1
[1,1,1,1,0,1,0,0,0,0,1,0,1,0]
 => [7,6,2,1,3,4,5] => [1,3,4,5,2,6,7] => [7,5,4,3,6,2,1] => ? = 3 + 1
[1,1,1,1,0,1,0,0,0,0,1,1,0,0]
 => [6,7,2,1,3,4,5] => [1,3,4,5,2,6,7] => [7,5,4,3,6,2,1] => ? = 3 + 1
[1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [7,2,1,3,4,5,6] => [1,3,4,5,6,2,7] => [7,5,4,3,2,6,1] => ? = 2 + 1
[1,0,1,0,1,1,0,1,1,1,0,0,1,0,0,0]
 => [6,4,5,7,3,8,2,1] => [1,2,3,8,4,5,7,6] => [8,7,6,1,5,4,2,3] => ? = 5 + 1
[1,0,1,1,1,0,0,1,1,1,0,0,1,0,0,0]
 => [6,4,5,7,2,3,8,1] => [1,2,3,8,4,5,7,6] => [8,7,6,1,5,4,2,3] => ? = 5 + 1
[1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
 => [5,2,3,4,6,7,8,1] => [1,2,3,4,6,7,8,5] => [8,7,6,5,3,2,1,4] => ? = 4 + 1
[1,1,0,0,1,1,0,1,0,1,1,0,1,0,0,0]
 => [6,5,7,4,3,8,1,2] => [1,2,3,8,4,5,7,6] => [8,7,6,1,5,4,2,3] => ? = 5 + 1
[1,1,0,0,1,1,1,1,0,1,0,0,0,0,1,0]
 => [8,4,3,5,6,7,1,2] => [1,2,3,5,6,7,4,8] => [8,7,6,4,3,2,5,1] => ? = 4 + 1
[1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => [8,7,6,5,4,2,1,3] => [1,3,2,4,5,6,7,8] => [8,6,7,5,4,3,2,1] => ? = 6 + 1
[1,1,0,1,0,0,1,0,1,1,1,1,0,0,0,0]
 => [5,6,7,8,4,2,1,3] => [1,3,2,4,5,6,7,8] => [8,6,7,5,4,3,2,1] => ? = 6 + 1
Description
The number of saliances of the permutation. A saliance is a right-to-left maximum. This can be described as an occurrence of the mesh pattern $([1], {(1,1)})$, i.e., the upper right quadrant is shaded, see [1].
Matching statistic: St000203
Mapping
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
St000203: Binary trees ⟶ ℤResult quality: 64% ●values known / values provided: 64%●distinct values known / distinct values provided: 64%
Values
[1,0]
 => [1] => [1] => [.,.]
 => 1 = 0 + 1
[1,0,1,0]
 => [2,1] => [1,2] => [.,[.,.]]
 => 2 = 1 + 1
[1,1,0,0]
 => [1,2] => [1,2] => [.,[.,.]]
 => 2 = 1 + 1
[1,0,1,0,1,0]
 => [3,2,1] => [1,2,3] => [.,[.,[.,.]]]
 => 3 = 2 + 1
[1,0,1,1,0,0]
 => [2,3,1] => [1,2,3] => [.,[.,[.,.]]]
 => 3 = 2 + 1
[1,1,0,0,1,0]
 => [3,1,2] => [1,2,3] => [.,[.,[.,.]]]
 => 3 = 2 + 1
[1,1,0,1,0,0]
 => [2,1,3] => [1,3,2] => [.,[[.,.],.]]
 => 2 = 1 + 1
[1,1,1,0,0,0]
 => [1,2,3] => [1,2,3] => [.,[.,[.,.]]]
 => 3 = 2 + 1
[1,0,1,0,1,0,1,0]
 => [4,3,2,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
 => 4 = 3 + 1
[1,0,1,0,1,1,0,0]
 => [3,4,2,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
 => 4 = 3 + 1
[1,0,1,1,0,0,1,0]
 => [4,2,3,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
 => 4 = 3 + 1
[1,0,1,1,0,1,0,0]
 => [3,2,4,1] => [1,2,4,3] => [.,[.,[[.,.],.]]]
 => 3 = 2 + 1
[1,0,1,1,1,0,0,0]
 => [2,3,4,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
 => 4 = 3 + 1
[1,1,0,0,1,0,1,0]
 => [4,3,1,2] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
 => 4 = 3 + 1
[1,1,0,0,1,1,0,0]
 => [3,4,1,2] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
 => 4 = 3 + 1
[1,1,0,1,0,0,1,0]
 => [4,2,1,3] => [1,3,2,4] => [.,[[.,.],[.,.]]]
 => 3 = 2 + 1
[1,1,0,1,0,1,0,0]
 => [3,2,1,4] => [1,4,2,3] => [.,[[.,.],[.,.]]]
 => 3 = 2 + 1
[1,1,0,1,1,0,0,0]
 => [2,3,1,4] => [1,4,2,3] => [.,[[.,.],[.,.]]]
 => 3 = 2 + 1
[1,1,1,0,0,0,1,0]
 => [4,1,2,3] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
 => 4 = 3 + 1
[1,1,1,0,0,1,0,0]
 => [3,1,2,4] => [1,2,4,3] => [.,[.,[[.,.],.]]]
 => 3 = 2 + 1
[1,1,1,0,1,0,0,0]
 => [2,1,3,4] => [1,3,4,2] => [.,[[.,[.,.]],.]]
 => 2 = 1 + 1
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
 => 4 = 3 + 1
[1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
 => 5 = 4 + 1
[1,0,1,0,1,0,1,1,0,0]
 => [4,5,3,2,1] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
 => 5 = 4 + 1
[1,0,1,0,1,1,0,0,1,0]
 => [5,3,4,2,1] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
 => 5 = 4 + 1
[1,0,1,0,1,1,0,1,0,0]
 => [4,3,5,2,1] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
 => 4 = 3 + 1
[1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
 => 5 = 4 + 1
[1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,3,1] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
 => 5 = 4 + 1
[1,0,1,1,0,0,1,1,0,0]
 => [4,5,2,3,1] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
 => 5 = 4 + 1
[1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,4,1] => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
 => 4 = 3 + 1
[1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,5,1] => [1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
 => 4 = 3 + 1
[1,0,1,1,0,1,1,0,0,0]
 => [3,4,2,5,1] => [1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
 => 4 = 3 + 1
[1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
 => 5 = 4 + 1
[1,0,1,1,1,0,0,1,0,0]
 => [4,2,3,5,1] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
 => 4 = 3 + 1
[1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => [1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
 => 3 = 2 + 1
[1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
 => 5 = 4 + 1
[1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,2] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
 => 5 = 4 + 1
[1,1,0,0,1,0,1,1,0,0]
 => [4,5,3,1,2] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
 => 5 = 4 + 1
[1,1,0,0,1,1,0,0,1,0]
 => [5,3,4,1,2] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
 => 5 = 4 + 1
[1,1,0,0,1,1,0,1,0,0]
 => [4,3,5,1,2] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
 => 4 = 3 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
 => 5 = 4 + 1
[1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,3] => [1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
 => 4 = 3 + 1
[1,1,0,1,0,0,1,1,0,0]
 => [4,5,2,1,3] => [1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
 => 4 = 3 + 1
[1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,4] => [1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
 => 4 = 3 + 1
[1,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1,5] => [1,5,2,3,4] => [.,[[.,.],[.,[.,.]]]]
 => 4 = 3 + 1
[1,1,0,1,0,1,1,0,0,0]
 => [3,4,2,1,5] => [1,5,2,3,4] => [.,[[.,.],[.,[.,.]]]]
 => 4 = 3 + 1
[1,1,0,1,1,0,0,0,1,0]
 => [5,2,3,1,4] => [1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
 => 4 = 3 + 1
[1,1,0,1,1,0,0,1,0,0]
 => [4,2,3,1,5] => [1,5,2,3,4] => [.,[[.,.],[.,[.,.]]]]
 => 4 = 3 + 1
[1,1,0,1,1,0,1,0,0,0]
 => [3,2,4,1,5] => [1,5,2,4,3] => [.,[[.,.],[[.,.],.]]]
 => 3 = 2 + 1
[1,1,0,1,1,1,0,0,0,0]
 => [2,3,4,1,5] => [1,5,2,3,4] => [.,[[.,.],[.,[.,.]]]]
 => 4 = 3 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [8,7,6,5,4,3,2,1] => [1,2,3,4,5,6,7,8] => [.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
 => ? = 7 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [7,8,6,5,4,3,2,1] => [1,2,3,4,5,6,7,8] => [.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
 => ? = 7 + 1
[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [6,7,8,5,4,3,2,1] => [1,2,3,4,5,6,7,8] => [.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
 => ? = 7 + 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [7,8,5,6,4,3,2,1] => [1,2,3,4,5,6,7,8] => [.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
 => ? = 7 + 1
[1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [5,6,7,8,4,3,2,1] => [1,2,3,4,5,6,7,8] => [.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
 => ? = 7 + 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [8,7,6,4,5,3,2,1] => [1,2,3,4,5,6,7,8] => [.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
 => ? = 7 + 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [7,8,6,4,5,3,2,1] => [1,2,3,4,5,6,7,8] => [.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
 => ? = 7 + 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [8,6,7,4,5,3,2,1] => [1,2,3,4,5,6,7,8] => [.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
 => ? = 7 + 1
[1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [4,5,6,7,8,3,2,1] => [1,2,3,4,5,6,7,8] => [.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
 => ? = 7 + 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => [8,7,6,5,3,4,2,1] => [1,2,3,4,5,6,7,8] => [.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
 => ? = 7 + 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [7,8,6,5,3,4,2,1] => [1,2,3,4,5,6,7,8] => [.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
 => ? = 7 + 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => [8,6,7,5,3,4,2,1] => [1,2,3,4,5,6,7,8] => [.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
 => ? = 7 + 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [8,7,5,6,3,4,2,1] => [1,2,3,4,5,6,7,8] => [.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
 => ? = 7 + 1
[1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => [5,6,7,8,3,4,2,1] => [1,2,3,4,5,6,7,8] => [.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
 => ? = 7 + 1
[1,0,1,0,1,1,0,1,1,1,0,0,1,0,0,0]
 => [6,4,5,7,3,8,2,1] => [1,2,3,8,4,5,7,6] => [.,[.,[.,[[.,.],[.,[[.,.],.]]]]]]
 => ? = 5 + 1
[1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => [6,7,8,3,4,5,2,1] => [1,2,3,4,5,6,7,8] => [.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
 => ? = 7 + 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => [8,7,6,5,4,2,3,1] => [1,2,3,4,5,6,7,8] => [.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
 => ? = 7 + 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => [7,8,6,5,4,2,3,1] => [1,2,3,4,5,6,7,8] => [.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
 => ? = 7 + 1
[1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
 => [8,6,7,5,4,2,3,1] => [1,2,3,4,5,6,7,8] => [.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
 => ? = 7 + 1
[1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
 => [8,7,5,6,4,2,3,1] => [1,2,3,4,5,6,7,8] => [.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
 => ? = 7 + 1
[1,0,1,1,0,0,1,0,1,1,1,1,0,0,0,0]
 => [5,6,7,8,4,2,3,1] => [1,2,3,4,5,6,7,8] => [.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
 => ? = 7 + 1
[1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
 => [8,7,6,4,5,2,3,1] => [1,2,3,4,5,6,7,8] => [.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
 => ? = 7 + 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
 => [8,6,7,4,5,2,3,1] => [1,2,3,4,5,6,7,8] => [.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
 => ? = 7 + 1
[1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => [4,5,6,7,8,2,3,1] => [1,2,3,4,5,6,7,8] => [.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
 => ? = 7 + 1
[1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0]
 => [8,7,6,5,2,3,4,1] => [1,2,3,4,5,6,7,8] => [.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
 => ? = 7 + 1
[1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]
 => [6,7,8,5,2,3,4,1] => [1,2,3,4,5,6,7,8] => [.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
 => ? = 7 + 1
[1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
 => [8,5,6,7,2,3,4,1] => [1,2,3,4,5,6,7,8] => [.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
 => ? = 7 + 1
[1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => [5,6,7,8,2,3,4,1] => [1,2,3,4,5,6,7,8] => [.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
 => ? = 7 + 1
[1,0,1,1,1,0,0,1,1,1,0,0,1,0,0,0]
 => [6,4,5,7,2,3,8,1] => [1,2,3,8,4,5,7,6] => [.,[.,[.,[[.,.],[.,[[.,.],.]]]]]]
 => ? = 5 + 1
[1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
 => [5,2,3,4,6,7,8,1] => [1,2,3,4,6,7,8,5] => [.,[.,[.,[.,[[.,[.,[.,.]]],.]]]]]
 => ? = 4 + 1
[1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [2,3,4,5,6,7,8,1] => [1,2,3,4,5,6,7,8] => [.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
 => ? = 7 + 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [8,7,6,5,4,3,1,2] => [1,2,3,4,5,6,7,8] => [.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
 => ? = 7 + 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [7,8,6,5,4,3,1,2] => [1,2,3,4,5,6,7,8] => [.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
 => ? = 7 + 1
[1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [8,6,7,5,4,3,1,2] => [1,2,3,4,5,6,7,8] => [.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
 => ? = 7 + 1
[1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [8,7,5,6,4,3,1,2] => [1,2,3,4,5,6,7,8] => [.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
 => ? = 7 + 1
[1,1,0,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [5,6,7,8,4,3,1,2] => [1,2,3,4,5,6,7,8] => [.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
 => ? = 7 + 1
[1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [8,7,6,4,5,3,1,2] => [1,2,3,4,5,6,7,8] => [.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
 => ? = 7 + 1
[1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [7,8,6,4,5,3,1,2] => [1,2,3,4,5,6,7,8] => [.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
 => ? = 7 + 1
[1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [4,5,6,7,8,3,1,2] => [1,2,3,4,5,6,7,8] => [.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
 => ? = 7 + 1
[1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => [8,7,6,5,3,4,1,2] => [1,2,3,4,5,6,7,8] => [.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
 => ? = 7 + 1
[1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [8,7,5,6,3,4,1,2] => [1,2,3,4,5,6,7,8] => [.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
 => ? = 7 + 1
[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [7,8,5,6,3,4,1,2] => [1,2,3,4,5,6,7,8] => [.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
 => ? = 7 + 1
[1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => [5,6,7,8,3,4,1,2] => [1,2,3,4,5,6,7,8] => [.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
 => ? = 7 + 1
[1,1,0,0,1,1,0,1,0,1,1,0,1,0,0,0]
 => [6,5,7,4,3,8,1,2] => [1,2,3,8,4,5,7,6] => [.,[.,[.,[[.,.],[.,[[.,.],.]]]]]]
 => ? = 5 + 1
[1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => [7,8,3,4,5,6,1,2] => [1,2,3,4,5,6,7,8] => [.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
 => ? = 7 + 1
[1,1,0,0,1,1,1,1,0,1,0,0,0,0,1,0]
 => [8,4,3,5,6,7,1,2] => [1,2,3,5,6,7,4,8] => [.,[.,[.,[[.,[.,[.,.]]],[.,.]]]]]
 => ? = 4 + 1
[1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => [8,3,4,5,6,7,1,2] => [1,2,3,4,5,6,7,8] => [.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
 => ? = 7 + 1
[1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [3,4,5,6,7,8,1,2] => [1,2,3,4,5,6,7,8] => [.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
 => ? = 7 + 1
[1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => [8,7,6,5,4,2,1,3] => [1,3,2,4,5,6,7,8] => [.,[[.,.],[.,[.,[.,[.,[.,.]]]]]]]
 => ? = 6 + 1
[1,1,0,1,0,0,1,0,1,1,1,1,0,0,0,0]
 => [5,6,7,8,4,2,1,3] => [1,3,2,4,5,6,7,8] => [.,[[.,.],[.,[.,[.,[.,[.,.]]]]]]]
 => ? = 6 + 1
Description
The number of external nodes of a binary tree. That is, the number of nodes that can be reached from the root by only left steps or only right steps, plus $1$ for the root node itself. A counting formula for the number of external node in all binary trees of size $n$ can be found in [1].
Matching statistic: St000636
Mapping
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000636: Graphs ⟶ ℤResult quality: 64% ●values known / values provided: 64%●distinct values known / distinct values provided: 64%
Values
[1,0]
 => [1] => ([],1)
 => 1 = 0 + 1
[1,0,1,0]
 => [1,2] => ([],2)
 => 2 = 1 + 1
[1,1,0,0]
 => [2,1] => ([(0,1)],2)
 => 2 = 1 + 1
[1,0,1,0,1,0]
 => [1,2,3] => ([],3)
 => 3 = 2 + 1
[1,0,1,1,0,0]
 => [1,3,2] => ([(1,2)],3)
 => 3 = 2 + 1
[1,1,0,0,1,0]
 => [2,1,3] => ([(1,2)],3)
 => 3 = 2 + 1
[1,1,0,1,0,0]
 => [2,3,1] => ([(0,2),(1,2)],3)
 => 2 = 1 + 1
[1,1,1,0,0,0]
 => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => ([],4)
 => 4 = 3 + 1
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => ([(2,3)],4)
 => 4 = 3 + 1
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => ([(2,3)],4)
 => 4 = 3 + 1
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => ([(1,3),(2,3)],4)
 => 3 = 2 + 1
[1,0,1,1,1,0,0,0]
 => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => ([(2,3)],4)
 => 4 = 3 + 1
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => ([(0,3),(1,2)],4)
 => 4 = 3 + 1
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => ([(1,3),(2,3)],4)
 => 3 = 2 + 1
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 3 = 2 + 1
[1,1,0,1,1,0,0,0]
 => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[1,1,1,0,0,0,1,0]
 => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[1,1,1,0,0,1,0,0]
 => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[1,1,1,0,1,0,0,0]
 => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[1,1,1,1,0,0,0,0]
 => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => ([],5)
 => 5 = 4 + 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => ([(3,4)],5)
 => 5 = 4 + 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => ([(3,4)],5)
 => 5 = 4 + 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => ([(2,4),(3,4)],5)
 => 4 = 3 + 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => 5 = 4 + 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => ([(3,4)],5)
 => 5 = 4 + 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => ([(1,4),(2,3)],5)
 => 5 = 4 + 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => ([(2,4),(3,4)],5)
 => 4 = 3 + 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
 => 4 = 3 + 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => 5 = 4 + 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5 = 4 + 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => ([(3,4)],5)
 => 5 = 4 + 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => ([(1,4),(2,3)],5)
 => 5 = 4 + 1
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => ([(1,4),(2,3)],5)
 => 5 = 4 + 1
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
 => 4 = 3 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => 5 = 4 + 1
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => ([(2,4),(3,4)],5)
 => 4 = 3 + 1
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
 => 4 = 3 + 1
[1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
 => 4 = 3 + 1
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4 = 3 + 1
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[1,1,0,1,1,0,1,0,0,0]
 => [2,5,3,4,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => [3,2,1,7,6,5,4] => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 6 + 1
[1,1,1,0,1,0,0,0,1,1,1,0,0,0]
 => [4,2,3,1,7,6,5] => ([(0,1),(0,2),(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 4 + 1
[1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => [4,3,2,1,7,6,5] => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 6 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5,6,7,8] => ([],8)
 => ? = 7 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,4,5,6,8,7] => ([(6,7)],8)
 => ? = 7 + 1
[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,2,3,4,5,8,7,6] => ([(5,6),(5,7),(6,7)],8)
 => ? = 7 + 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,2,3,4,6,5,8,7] => ([(4,7),(5,6)],8)
 => ? = 7 + 1
[1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,2,3,4,8,7,6,5] => ([(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 7 + 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,2,3,5,4,6,7,8] => ([(6,7)],8)
 => ? = 7 + 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,2,3,5,4,6,8,7] => ([(4,7),(5,6)],8)
 => ? = 7 + 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,2,3,5,4,7,6,8] => ([(4,7),(5,6)],8)
 => ? = 7 + 1
[1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,2,3,8,7,6,5,4] => ([(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 7 + 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,2,4,3,5,6,7,8] => ([(6,7)],8)
 => ? = 7 + 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,2,4,3,5,6,8,7] => ([(4,7),(5,6)],8)
 => ? = 7 + 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5,7,6,8] => ([(4,7),(5,6)],8)
 => ? = 7 + 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,2,4,3,6,5,7,8] => ([(4,7),(5,6)],8)
 => ? = 7 + 1
[1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,2,4,3,8,7,6,5] => ([(2,3),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 7 + 1
[1,0,1,0,1,1,0,1,1,1,0,0,1,0,0,0]
 => [1,2,4,8,6,5,7,3] => ([(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,2,5,4,3,8,7,6] => ([(2,6),(2,7),(3,4),(3,5),(4,5),(6,7)],8)
 => ? = 7 + 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => [1,3,2,4,5,6,7,8] => ([(6,7)],8)
 => ? = 7 + 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => [1,3,2,4,5,6,8,7] => ([(4,7),(5,6)],8)
 => ? = 7 + 1
[1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
 => [1,3,2,4,5,7,6,8] => ([(4,7),(5,6)],8)
 => ? = 7 + 1
[1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,6,5,7,8] => ([(4,7),(5,6)],8)
 => ? = 7 + 1
[1,0,1,1,0,0,1,0,1,1,1,1,0,0,0,0]
 => [1,3,2,4,8,7,6,5] => ([(2,3),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 7 + 1
[1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
 => [1,3,2,5,4,6,7,8] => ([(4,7),(5,6)],8)
 => ? = 7 + 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
 => [1,3,2,5,4,7,6,8] => ([(2,7),(3,6),(4,5)],8)
 => ? = 7 + 1
[1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => [1,3,2,8,7,6,5,4] => ([(1,2),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 7 + 1
[1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0]
 => [1,4,3,2,5,6,7,8] => ([(5,6),(5,7),(6,7)],8)
 => ? = 7 + 1
[1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]
 => [1,4,3,2,5,8,7,6] => ([(2,6),(2,7),(3,4),(3,5),(4,5),(6,7)],8)
 => ? = 7 + 1
[1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
 => [1,4,3,2,7,6,5,8] => ([(2,6),(2,7),(3,4),(3,5),(4,5),(6,7)],8)
 => ? = 7 + 1
[1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => [1,4,3,2,8,7,6,5] => ([(1,2),(1,3),(2,3),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 7 + 1
[1,0,1,1,1,0,0,1,1,1,0,0,1,0,0,0]
 => [1,4,3,8,6,5,7,2] => ([(1,2),(1,7),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
 => [1,8,6,5,4,7,3,2] => ([(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,8,7,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 7 + 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [2,1,3,4,5,6,7,8] => ([(6,7)],8)
 => ? = 7 + 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [2,1,3,4,5,6,8,7] => ([(4,7),(5,6)],8)
 => ? = 7 + 1
[1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [2,1,3,4,5,7,6,8] => ([(4,7),(5,6)],8)
 => ? = 7 + 1
[1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [2,1,3,4,6,5,7,8] => ([(4,7),(5,6)],8)
 => ? = 7 + 1
[1,1,0,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [2,1,3,4,8,7,6,5] => ([(2,3),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 7 + 1
[1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,5,4,6,7,8] => ([(4,7),(5,6)],8)
 => ? = 7 + 1
[1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4,6,8,7] => ([(2,7),(3,6),(4,5)],8)
 => ? = 7 + 1
[1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,1,3,8,7,6,5,4] => ([(1,2),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 7 + 1
[1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => [2,1,4,3,5,6,7,8] => ([(4,7),(5,6)],8)
 => ? = 7 + 1
[1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [2,1,4,3,6,5,7,8] => ([(2,7),(3,6),(4,5)],8)
 => ? = 7 + 1
[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [2,1,4,3,6,5,8,7] => ([(0,7),(1,6),(2,5),(3,4)],8)
 => ? = 7 + 1
[1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => [2,1,4,3,8,7,6,5] => ([(0,3),(1,2),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 7 + 1
[1,1,0,0,1,1,0,1,0,1,1,0,1,0,0,0]
 => [2,1,4,5,8,6,7,3] => ([(0,1),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => [2,1,6,5,4,3,8,7] => ([(0,3),(1,2),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 7 + 1
[1,1,0,0,1,1,1,1,0,1,0,0,0,0,1,0]
 => [2,1,7,5,6,4,3,8] => ?
 => ? = 4 + 1
[1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => [2,1,7,6,5,4,3,8] => ([(1,2),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 7 + 1
Description
The hull number of a graph. The convex hull of a set of vertices $S$ of a graph is the smallest set $h(S)$ such that for any pair $u,v\in h(S)$ all vertices on a shortest path from $u$ to $v$ are also in $h(S)$. The hull number is the size of the smallest set $S$ such that $h(S)$ is the set of all vertices.
Matching statistic: St001654
(load all 2 compositions to match this statistic)
Mapping
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001654: Graphs ⟶ ℤResult quality: 64% ●values known / values provided: 64%●distinct values known / distinct values provided: 64%
Values
[1,0]
 => [1] => ([],1)
 => 1 = 0 + 1
[1,0,1,0]
 => [1,2] => ([],2)
 => 2 = 1 + 1
[1,1,0,0]
 => [2,1] => ([(0,1)],2)
 => 2 = 1 + 1
[1,0,1,0,1,0]
 => [1,2,3] => ([],3)
 => 3 = 2 + 1
[1,0,1,1,0,0]
 => [1,3,2] => ([(1,2)],3)
 => 3 = 2 + 1
[1,1,0,0,1,0]
 => [2,1,3] => ([(1,2)],3)
 => 3 = 2 + 1
[1,1,0,1,0,0]
 => [2,3,1] => ([(0,2),(1,2)],3)
 => 2 = 1 + 1
[1,1,1,0,0,0]
 => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => ([],4)
 => 4 = 3 + 1
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => ([(2,3)],4)
 => 4 = 3 + 1
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => ([(2,3)],4)
 => 4 = 3 + 1
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => ([(1,3),(2,3)],4)
 => 3 = 2 + 1
[1,0,1,1,1,0,0,0]
 => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => ([(2,3)],4)
 => 4 = 3 + 1
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => ([(0,3),(1,2)],4)
 => 4 = 3 + 1
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => ([(1,3),(2,3)],4)
 => 3 = 2 + 1
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 3 = 2 + 1
[1,1,0,1,1,0,0,0]
 => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[1,1,1,0,0,0,1,0]
 => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[1,1,1,0,0,1,0,0]
 => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[1,1,1,0,1,0,0,0]
 => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[1,1,1,1,0,0,0,0]
 => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => ([],5)
 => 5 = 4 + 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => ([(3,4)],5)
 => 5 = 4 + 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => ([(3,4)],5)
 => 5 = 4 + 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => ([(2,4),(3,4)],5)
 => 4 = 3 + 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => 5 = 4 + 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => ([(3,4)],5)
 => 5 = 4 + 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => ([(1,4),(2,3)],5)
 => 5 = 4 + 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => ([(2,4),(3,4)],5)
 => 4 = 3 + 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
 => 4 = 3 + 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => 5 = 4 + 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5 = 4 + 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => ([(3,4)],5)
 => 5 = 4 + 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => ([(1,4),(2,3)],5)
 => 5 = 4 + 1
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => ([(1,4),(2,3)],5)
 => 5 = 4 + 1
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
 => 4 = 3 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => 5 = 4 + 1
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => ([(2,4),(3,4)],5)
 => 4 = 3 + 1
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
 => 4 = 3 + 1
[1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
 => 4 = 3 + 1
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4 = 3 + 1
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[1,1,0,1,1,0,1,0,0,0]
 => [2,5,3,4,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => [3,2,1,7,6,5,4] => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 6 + 1
[1,1,1,0,1,0,0,0,1,1,1,0,0,0]
 => [4,2,3,1,7,6,5] => ([(0,1),(0,2),(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 4 + 1
[1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => [4,3,2,1,7,6,5] => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 6 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5,6,7,8] => ([],8)
 => ? = 7 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,4,5,6,8,7] => ([(6,7)],8)
 => ? = 7 + 1
[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,2,3,4,5,8,7,6] => ([(5,6),(5,7),(6,7)],8)
 => ? = 7 + 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,2,3,4,6,5,8,7] => ([(4,7),(5,6)],8)
 => ? = 7 + 1
[1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,2,3,4,8,7,6,5] => ([(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 7 + 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,2,3,5,4,6,7,8] => ([(6,7)],8)
 => ? = 7 + 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,2,3,5,4,6,8,7] => ([(4,7),(5,6)],8)
 => ? = 7 + 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,2,3,5,4,7,6,8] => ([(4,7),(5,6)],8)
 => ? = 7 + 1
[1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,2,3,8,7,6,5,4] => ([(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 7 + 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,2,4,3,5,6,7,8] => ([(6,7)],8)
 => ? = 7 + 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,2,4,3,5,6,8,7] => ([(4,7),(5,6)],8)
 => ? = 7 + 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5,7,6,8] => ([(4,7),(5,6)],8)
 => ? = 7 + 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,2,4,3,6,5,7,8] => ([(4,7),(5,6)],8)
 => ? = 7 + 1
[1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,2,4,3,8,7,6,5] => ([(2,3),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 7 + 1
[1,0,1,0,1,1,0,1,1,1,0,0,1,0,0,0]
 => [1,2,4,8,6,5,7,3] => ([(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,2,5,4,3,8,7,6] => ([(2,6),(2,7),(3,4),(3,5),(4,5),(6,7)],8)
 => ? = 7 + 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => [1,3,2,4,5,6,7,8] => ([(6,7)],8)
 => ? = 7 + 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => [1,3,2,4,5,6,8,7] => ([(4,7),(5,6)],8)
 => ? = 7 + 1
[1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
 => [1,3,2,4,5,7,6,8] => ([(4,7),(5,6)],8)
 => ? = 7 + 1
[1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,6,5,7,8] => ([(4,7),(5,6)],8)
 => ? = 7 + 1
[1,0,1,1,0,0,1,0,1,1,1,1,0,0,0,0]
 => [1,3,2,4,8,7,6,5] => ([(2,3),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 7 + 1
[1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
 => [1,3,2,5,4,6,7,8] => ([(4,7),(5,6)],8)
 => ? = 7 + 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
 => [1,3,2,5,4,7,6,8] => ([(2,7),(3,6),(4,5)],8)
 => ? = 7 + 1
[1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => [1,3,2,8,7,6,5,4] => ([(1,2),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 7 + 1
[1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0]
 => [1,4,3,2,5,6,7,8] => ([(5,6),(5,7),(6,7)],8)
 => ? = 7 + 1
[1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]
 => [1,4,3,2,5,8,7,6] => ([(2,6),(2,7),(3,4),(3,5),(4,5),(6,7)],8)
 => ? = 7 + 1
[1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
 => [1,4,3,2,7,6,5,8] => ([(2,6),(2,7),(3,4),(3,5),(4,5),(6,7)],8)
 => ? = 7 + 1
[1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => [1,4,3,2,8,7,6,5] => ([(1,2),(1,3),(2,3),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 7 + 1
[1,0,1,1,1,0,0,1,1,1,0,0,1,0,0,0]
 => [1,4,3,8,6,5,7,2] => ([(1,2),(1,7),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
 => [1,8,6,5,4,7,3,2] => ([(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,8,7,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 7 + 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [2,1,3,4,5,6,7,8] => ([(6,7)],8)
 => ? = 7 + 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [2,1,3,4,5,6,8,7] => ([(4,7),(5,6)],8)
 => ? = 7 + 1
[1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [2,1,3,4,5,7,6,8] => ([(4,7),(5,6)],8)
 => ? = 7 + 1
[1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [2,1,3,4,6,5,7,8] => ([(4,7),(5,6)],8)
 => ? = 7 + 1
[1,1,0,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [2,1,3,4,8,7,6,5] => ([(2,3),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 7 + 1
[1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,5,4,6,7,8] => ([(4,7),(5,6)],8)
 => ? = 7 + 1
[1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4,6,8,7] => ([(2,7),(3,6),(4,5)],8)
 => ? = 7 + 1
[1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,1,3,8,7,6,5,4] => ([(1,2),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 7 + 1
[1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => [2,1,4,3,5,6,7,8] => ([(4,7),(5,6)],8)
 => ? = 7 + 1
[1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [2,1,4,3,6,5,7,8] => ([(2,7),(3,6),(4,5)],8)
 => ? = 7 + 1
[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [2,1,4,3,6,5,8,7] => ([(0,7),(1,6),(2,5),(3,4)],8)
 => ? = 7 + 1
[1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => [2,1,4,3,8,7,6,5] => ([(0,3),(1,2),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 7 + 1
[1,1,0,0,1,1,0,1,0,1,1,0,1,0,0,0]
 => [2,1,4,5,8,6,7,3] => ([(0,1),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => [2,1,6,5,4,3,8,7] => ([(0,3),(1,2),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 7 + 1
[1,1,0,0,1,1,1,1,0,1,0,0,0,0,1,0]
 => [2,1,7,5,6,4,3,8] => ?
 => ? = 4 + 1
[1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => [2,1,7,6,5,4,3,8] => ([(1,2),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 7 + 1
Description
The monophonic hull number of a graph. The monophonic hull of a set of vertices $M$ of a graph $G$ is the set of vertices that lie on at least one induced path between vertices in $M$. The monophonic hull number is the size of the smallest set $M$ such that the monophonic hull of $M$ is all of $G$. For example, the monophonic hull number of a graph $G$ with $n$ vertices is $n$ if and only if $G$ is a disjoint union of complete graphs.
Matching statistic: St000144
(load all 14 compositions to match this statistic)
Mapping
St000144: Dyck paths ⟶ ℤResult quality: 42% ●values known / values provided: 42%●distinct values known / distinct values provided: 55%
Values
[1,0]
 => 1 = 0 + 1
[1,0,1,0]
 => 2 = 1 + 1
[1,1,0,0]
 => 2 = 1 + 1
[1,0,1,0,1,0]
 => 3 = 2 + 1
[1,0,1,1,0,0]
 => 3 = 2 + 1
[1,1,0,0,1,0]
 => 3 = 2 + 1
[1,1,0,1,0,0]
 => 2 = 1 + 1
[1,1,1,0,0,0]
 => 3 = 2 + 1
[1,0,1,0,1,0,1,0]
 => 4 = 3 + 1
[1,0,1,0,1,1,0,0]
 => 4 = 3 + 1
[1,0,1,1,0,0,1,0]
 => 4 = 3 + 1
[1,0,1,1,0,1,0,0]
 => 3 = 2 + 1
[1,0,1,1,1,0,0,0]
 => 4 = 3 + 1
[1,1,0,0,1,0,1,0]
 => 4 = 3 + 1
[1,1,0,0,1,1,0,0]
 => 4 = 3 + 1
[1,1,0,1,0,0,1,0]
 => 3 = 2 + 1
[1,1,0,1,0,1,0,0]
 => 3 = 2 + 1
[1,1,0,1,1,0,0,0]
 => 3 = 2 + 1
[1,1,1,0,0,0,1,0]
 => 4 = 3 + 1
[1,1,1,0,0,1,0,0]
 => 3 = 2 + 1
[1,1,1,0,1,0,0,0]
 => 2 = 1 + 1
[1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
[1,0,1,0,1,0,1,0,1,0]
 => 5 = 4 + 1
[1,0,1,0,1,0,1,1,0,0]
 => 5 = 4 + 1
[1,0,1,0,1,1,0,0,1,0]
 => 5 = 4 + 1
[1,0,1,0,1,1,0,1,0,0]
 => 4 = 3 + 1
[1,0,1,0,1,1,1,0,0,0]
 => 5 = 4 + 1
[1,0,1,1,0,0,1,0,1,0]
 => 5 = 4 + 1
[1,0,1,1,0,0,1,1,0,0]
 => 5 = 4 + 1
[1,0,1,1,0,1,0,0,1,0]
 => 4 = 3 + 1
[1,0,1,1,0,1,0,1,0,0]
 => 4 = 3 + 1
[1,0,1,1,0,1,1,0,0,0]
 => 4 = 3 + 1
[1,0,1,1,1,0,0,0,1,0]
 => 5 = 4 + 1
[1,0,1,1,1,0,0,1,0,0]
 => 4 = 3 + 1
[1,0,1,1,1,0,1,0,0,0]
 => 3 = 2 + 1
[1,0,1,1,1,1,0,0,0,0]
 => 5 = 4 + 1
[1,1,0,0,1,0,1,0,1,0]
 => 5 = 4 + 1
[1,1,0,0,1,0,1,1,0,0]
 => 5 = 4 + 1
[1,1,0,0,1,1,0,0,1,0]
 => 5 = 4 + 1
[1,1,0,0,1,1,0,1,0,0]
 => 4 = 3 + 1
[1,1,0,0,1,1,1,0,0,0]
 => 5 = 4 + 1
[1,1,0,1,0,0,1,0,1,0]
 => 4 = 3 + 1
[1,1,0,1,0,0,1,1,0,0]
 => 4 = 3 + 1
[1,1,0,1,0,1,0,0,1,0]
 => 4 = 3 + 1
[1,1,0,1,0,1,0,1,0,0]
 => 4 = 3 + 1
[1,1,0,1,0,1,1,0,0,0]
 => 4 = 3 + 1
[1,1,0,1,1,0,0,0,1,0]
 => 4 = 3 + 1
[1,1,0,1,1,0,0,1,0,0]
 => 4 = 3 + 1
[1,1,0,1,1,0,1,0,0,0]
 => 3 = 2 + 1
[1,1,0,1,1,1,0,0,0,0]
 => 4 = 3 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 6 + 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 6 + 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 6 + 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 6 + 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => ? = 6 + 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 6 + 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => ? = 6 + 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 6 + 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 6 + 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 6 + 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 6 + 1
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 6 + 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => ? = 6 + 1
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 6 + 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => ? = 6 + 1
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 6 + 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => ? = 6 + 1
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => ? = 6 + 1
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => ? = 6 + 1
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
 => ? = 6 + 1
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => ? = 6 + 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => ? = 6 + 1
[1,0,1,1,0,0,1,1,1,0,0,0,1,0]
 => ? = 6 + 1
[1,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => ? = 6 + 1
[1,0,1,1,1,0,0,0,1,0,1,0,1,0]
 => ? = 6 + 1
[1,0,1,1,1,0,0,0,1,0,1,1,0,0]
 => ? = 6 + 1
[1,0,1,1,1,0,0,0,1,1,0,0,1,0]
 => ? = 6 + 1
[1,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => ? = 6 + 1
[1,0,1,1,1,1,0,0,0,0,1,0,1,0]
 => ? = 6 + 1
[1,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => ? = 6 + 1
[1,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => ? = 6 + 1
[1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 6 + 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 6 + 1
[1,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 6 + 1
[1,1,0,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 6 + 1
[1,1,0,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 6 + 1
[1,1,0,0,1,0,1,1,0,0,1,0,1,0]
 => ? = 6 + 1
[1,1,0,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 6 + 1
[1,1,0,0,1,0,1,1,1,0,0,0,1,0]
 => ? = 6 + 1
[1,1,0,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 6 + 1
[1,1,0,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 6 + 1
[1,1,0,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 6 + 1
[1,1,0,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 6 + 1
[1,1,0,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 6 + 1
[1,1,0,0,1,1,1,0,0,0,1,0,1,0]
 => ? = 6 + 1
[1,1,0,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 6 + 1
[1,1,0,0,1,1,1,1,0,0,0,0,1,0]
 => ? = 6 + 1
[1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 6 + 1
[1,1,0,1,0,0,1,0,1,0,1,0,1,0]
 => ? = 5 + 1
[1,1,0,1,0,0,1,0,1,0,1,1,0,0]
 => ? = 5 + 1
Description
The pyramid weight of the Dyck path. The pyramid weight of a Dyck path is the sum of the lengths of the maximal pyramids (maximal sequences of the form $1^h0^h$) in the path. Maximal pyramids are called lower interactions by Le Borgne [2], see [[St000331]] and [[St000335]] for related statistics.
Matching statistic: St000991
(load all 6 compositions to match this statistic)
Mapping
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
St000991: Permutations ⟶ ℤResult quality: 42% ●values known / values provided: 42%●distinct values known / distinct values provided: 55%
Values
[1,0]
 => [1] => [1] => 1 = 0 + 1
[1,0,1,0]
 => [2,1] => [1,2] => 2 = 1 + 1
[1,1,0,0]
 => [1,2] => [1,2] => 2 = 1 + 1
[1,0,1,0,1,0]
 => [3,2,1] => [1,2,3] => 3 = 2 + 1
[1,0,1,1,0,0]
 => [2,3,1] => [1,2,3] => 3 = 2 + 1
[1,1,0,0,1,0]
 => [3,1,2] => [1,2,3] => 3 = 2 + 1
[1,1,0,1,0,0]
 => [2,1,3] => [1,3,2] => 2 = 1 + 1
[1,1,1,0,0,0]
 => [1,2,3] => [1,2,3] => 3 = 2 + 1
[1,0,1,0,1,0,1,0]
 => [4,3,2,1] => [1,2,3,4] => 4 = 3 + 1
[1,0,1,0,1,1,0,0]
 => [3,4,2,1] => [1,2,3,4] => 4 = 3 + 1
[1,0,1,1,0,0,1,0]
 => [4,2,3,1] => [1,2,3,4] => 4 = 3 + 1
[1,0,1,1,0,1,0,0]
 => [3,2,4,1] => [1,2,4,3] => 3 = 2 + 1
[1,0,1,1,1,0,0,0]
 => [2,3,4,1] => [1,2,3,4] => 4 = 3 + 1
[1,1,0,0,1,0,1,0]
 => [4,3,1,2] => [1,2,3,4] => 4 = 3 + 1
[1,1,0,0,1,1,0,0]
 => [3,4,1,2] => [1,2,3,4] => 4 = 3 + 1
[1,1,0,1,0,0,1,0]
 => [4,2,1,3] => [1,3,2,4] => 3 = 2 + 1
[1,1,0,1,0,1,0,0]
 => [3,2,1,4] => [1,4,2,3] => 3 = 2 + 1
[1,1,0,1,1,0,0,0]
 => [2,3,1,4] => [1,4,2,3] => 3 = 2 + 1
[1,1,1,0,0,0,1,0]
 => [4,1,2,3] => [1,2,3,4] => 4 = 3 + 1
[1,1,1,0,0,1,0,0]
 => [3,1,2,4] => [1,2,4,3] => 3 = 2 + 1
[1,1,1,0,1,0,0,0]
 => [2,1,3,4] => [1,3,4,2] => 2 = 1 + 1
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [1,2,3,4] => 4 = 3 + 1
[1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => [1,2,3,4,5] => 5 = 4 + 1
[1,0,1,0,1,0,1,1,0,0]
 => [4,5,3,2,1] => [1,2,3,4,5] => 5 = 4 + 1
[1,0,1,0,1,1,0,0,1,0]
 => [5,3,4,2,1] => [1,2,3,4,5] => 5 = 4 + 1
[1,0,1,0,1,1,0,1,0,0]
 => [4,3,5,2,1] => [1,2,3,5,4] => 4 = 3 + 1
[1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => [1,2,3,4,5] => 5 = 4 + 1
[1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,3,1] => [1,2,3,4,5] => 5 = 4 + 1
[1,0,1,1,0,0,1,1,0,0]
 => [4,5,2,3,1] => [1,2,3,4,5] => 5 = 4 + 1
[1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,4,1] => [1,2,4,3,5] => 4 = 3 + 1
[1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,5,1] => [1,2,5,3,4] => 4 = 3 + 1
[1,0,1,1,0,1,1,0,0,0]
 => [3,4,2,5,1] => [1,2,5,3,4] => 4 = 3 + 1
[1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => [1,2,3,4,5] => 5 = 4 + 1
[1,0,1,1,1,0,0,1,0,0]
 => [4,2,3,5,1] => [1,2,3,5,4] => 4 = 3 + 1
[1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => [1,2,4,5,3] => 3 = 2 + 1
[1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [1,2,3,4,5] => 5 = 4 + 1
[1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,2] => [1,2,3,4,5] => 5 = 4 + 1
[1,1,0,0,1,0,1,1,0,0]
 => [4,5,3,1,2] => [1,2,3,4,5] => 5 = 4 + 1
[1,1,0,0,1,1,0,0,1,0]
 => [5,3,4,1,2] => [1,2,3,4,5] => 5 = 4 + 1
[1,1,0,0,1,1,0,1,0,0]
 => [4,3,5,1,2] => [1,2,3,5,4] => 4 = 3 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => [1,2,3,4,5] => 5 = 4 + 1
[1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,3] => [1,3,2,4,5] => 4 = 3 + 1
[1,1,0,1,0,0,1,1,0,0]
 => [4,5,2,1,3] => [1,3,2,4,5] => 4 = 3 + 1
[1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,4] => [1,4,2,3,5] => 4 = 3 + 1
[1,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1,5] => [1,5,2,3,4] => 4 = 3 + 1
[1,1,0,1,0,1,1,0,0,0]
 => [3,4,2,1,5] => [1,5,2,3,4] => 4 = 3 + 1
[1,1,0,1,1,0,0,0,1,0]
 => [5,2,3,1,4] => [1,4,2,3,5] => 4 = 3 + 1
[1,1,0,1,1,0,0,1,0,0]
 => [4,2,3,1,5] => [1,5,2,3,4] => 4 = 3 + 1
[1,1,0,1,1,0,1,0,0,0]
 => [3,2,4,1,5] => [1,5,2,4,3] => 3 = 2 + 1
[1,1,0,1,1,1,0,0,0,0]
 => [2,3,4,1,5] => [1,5,2,3,4] => 4 = 3 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [7,6,5,4,3,2,1] => [1,2,3,4,5,6,7] => ? = 6 + 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [6,7,5,4,3,2,1] => [1,2,3,4,5,6,7] => ? = 6 + 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [7,5,6,4,3,2,1] => [1,2,3,4,5,6,7] => ? = 6 + 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [5,6,7,4,3,2,1] => [1,2,3,4,5,6,7] => ? = 6 + 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [7,6,4,5,3,2,1] => [1,2,3,4,5,6,7] => ? = 6 + 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [6,7,4,5,3,2,1] => [1,2,3,4,5,6,7] => ? = 6 + 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => [7,4,5,6,3,2,1] => [1,2,3,4,5,6,7] => ? = 6 + 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [4,5,6,7,3,2,1] => [1,2,3,4,5,6,7] => ? = 6 + 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [7,6,5,3,4,2,1] => [1,2,3,4,5,6,7] => ? = 6 + 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [6,7,5,3,4,2,1] => [1,2,3,4,5,6,7] => ? = 6 + 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [7,5,6,3,4,2,1] => [1,2,3,4,5,6,7] => ? = 6 + 1
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => [5,6,7,3,4,2,1] => [1,2,3,4,5,6,7] => ? = 6 + 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => [7,6,3,4,5,2,1] => [1,2,3,4,5,6,7] => ? = 6 + 1
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => [6,7,3,4,5,2,1] => [1,2,3,4,5,6,7] => ? = 6 + 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => [7,3,4,5,6,2,1] => [1,2,3,4,5,6,7] => ? = 6 + 1
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [3,4,5,6,7,2,1] => [1,2,3,4,5,6,7] => ? = 6 + 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => [7,6,5,4,2,3,1] => [1,2,3,4,5,6,7] => ? = 6 + 1
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [6,7,5,4,2,3,1] => [1,2,3,4,5,6,7] => ? = 6 + 1
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => [7,5,6,4,2,3,1] => [1,2,3,4,5,6,7] => ? = 6 + 1
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
 => [5,6,7,4,2,3,1] => [1,2,3,4,5,6,7] => ? = 6 + 1
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [7,6,4,5,2,3,1] => [1,2,3,4,5,6,7] => ? = 6 + 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [6,7,4,5,2,3,1] => [1,2,3,4,5,6,7] => ? = 6 + 1
[1,0,1,1,0,0,1,1,1,0,0,0,1,0]
 => [7,4,5,6,2,3,1] => [1,2,3,4,5,6,7] => ? = 6 + 1
[1,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => [4,5,6,7,2,3,1] => [1,2,3,4,5,6,7] => ? = 6 + 1
[1,0,1,1,1,0,0,0,1,0,1,0,1,0]
 => [7,6,5,2,3,4,1] => [1,2,3,4,5,6,7] => ? = 6 + 1
[1,0,1,1,1,0,0,0,1,0,1,1,0,0]
 => [6,7,5,2,3,4,1] => [1,2,3,4,5,6,7] => ? = 6 + 1
[1,0,1,1,1,0,0,0,1,1,0,0,1,0]
 => [7,5,6,2,3,4,1] => [1,2,3,4,5,6,7] => ? = 6 + 1
[1,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => [5,6,7,2,3,4,1] => [1,2,3,4,5,6,7] => ? = 6 + 1
[1,0,1,1,1,1,0,0,0,0,1,0,1,0]
 => [7,6,2,3,4,5,1] => [1,2,3,4,5,6,7] => ? = 6 + 1
[1,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => [6,7,2,3,4,5,1] => [1,2,3,4,5,6,7] => ? = 6 + 1
[1,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => [7,2,3,4,5,6,1] => [1,2,3,4,5,6,7] => ? = 6 + 1
[1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [2,3,4,5,6,7,1] => [1,2,3,4,5,6,7] => ? = 6 + 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => [7,6,5,4,3,1,2] => [1,2,3,4,5,6,7] => ? = 6 + 1
[1,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => [6,7,5,4,3,1,2] => [1,2,3,4,5,6,7] => ? = 6 + 1
[1,1,0,0,1,0,1,0,1,1,0,0,1,0]
 => [7,5,6,4,3,1,2] => [1,2,3,4,5,6,7] => ? = 6 + 1
[1,1,0,0,1,0,1,0,1,1,1,0,0,0]
 => [5,6,7,4,3,1,2] => [1,2,3,4,5,6,7] => ? = 6 + 1
[1,1,0,0,1,0,1,1,0,0,1,0,1,0]
 => [7,6,4,5,3,1,2] => [1,2,3,4,5,6,7] => ? = 6 + 1
[1,1,0,0,1,0,1,1,0,0,1,1,0,0]
 => [6,7,4,5,3,1,2] => [1,2,3,4,5,6,7] => ? = 6 + 1
[1,1,0,0,1,0,1,1,1,0,0,0,1,0]
 => [7,4,5,6,3,1,2] => [1,2,3,4,5,6,7] => ? = 6 + 1
[1,1,0,0,1,0,1,1,1,1,0,0,0,0]
 => [4,5,6,7,3,1,2] => [1,2,3,4,5,6,7] => ? = 6 + 1
[1,1,0,0,1,1,0,0,1,0,1,0,1,0]
 => [7,6,5,3,4,1,2] => [1,2,3,4,5,6,7] => ? = 6 + 1
[1,1,0,0,1,1,0,0,1,0,1,1,0,0]
 => [6,7,5,3,4,1,2] => [1,2,3,4,5,6,7] => ? = 6 + 1
[1,1,0,0,1,1,0,0,1,1,0,0,1,0]
 => [7,5,6,3,4,1,2] => [1,2,3,4,5,6,7] => ? = 6 + 1
[1,1,0,0,1,1,0,0,1,1,1,0,0,0]
 => [5,6,7,3,4,1,2] => [1,2,3,4,5,6,7] => ? = 6 + 1
[1,1,0,0,1,1,1,0,0,0,1,0,1,0]
 => [7,6,3,4,5,1,2] => [1,2,3,4,5,6,7] => ? = 6 + 1
[1,1,0,0,1,1,1,0,0,0,1,1,0,0]
 => [6,7,3,4,5,1,2] => [1,2,3,4,5,6,7] => ? = 6 + 1
[1,1,0,0,1,1,1,1,0,0,0,0,1,0]
 => [7,3,4,5,6,1,2] => [1,2,3,4,5,6,7] => ? = 6 + 1
[1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => [3,4,5,6,7,1,2] => [1,2,3,4,5,6,7] => ? = 6 + 1
[1,1,0,1,0,0,1,0,1,0,1,0,1,0]
 => [7,6,5,4,2,1,3] => [1,3,2,4,5,6,7] => ? = 5 + 1
[1,1,0,1,0,0,1,0,1,0,1,1,0,0]
 => [6,7,5,4,2,1,3] => [1,3,2,4,5,6,7] => ? = 5 + 1
Description
The number of right-to-left minima of a permutation. For the number of left-to-right maxima, see [[St000314]].
Matching statistic: St000314
Mapping
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000314: Permutations ⟶ ℤResult quality: 42% ●values known / values provided: 42%●distinct values known / distinct values provided: 55%
Values
[1,0]
 => [1] => [1] => [1] => 1 = 0 + 1
[1,0,1,0]
 => [2,1] => [1,2] => [1,2] => 2 = 1 + 1
[1,1,0,0]
 => [1,2] => [1,2] => [1,2] => 2 = 1 + 1
[1,0,1,0,1,0]
 => [3,2,1] => [1,2,3] => [1,2,3] => 3 = 2 + 1
[1,0,1,1,0,0]
 => [2,3,1] => [1,2,3] => [1,2,3] => 3 = 2 + 1
[1,1,0,0,1,0]
 => [3,1,2] => [1,2,3] => [1,2,3] => 3 = 2 + 1
[1,1,0,1,0,0]
 => [2,1,3] => [1,3,2] => [1,3,2] => 2 = 1 + 1
[1,1,1,0,0,0]
 => [1,2,3] => [1,2,3] => [1,2,3] => 3 = 2 + 1
[1,0,1,0,1,0,1,0]
 => [4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 4 = 3 + 1
[1,0,1,0,1,1,0,0]
 => [3,4,2,1] => [1,2,3,4] => [1,2,3,4] => 4 = 3 + 1
[1,0,1,1,0,0,1,0]
 => [4,2,3,1] => [1,2,3,4] => [1,2,3,4] => 4 = 3 + 1
[1,0,1,1,0,1,0,0]
 => [3,2,4,1] => [1,2,4,3] => [1,2,4,3] => 3 = 2 + 1
[1,0,1,1,1,0,0,0]
 => [2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 4 = 3 + 1
[1,1,0,0,1,0,1,0]
 => [4,3,1,2] => [1,2,3,4] => [1,2,3,4] => 4 = 3 + 1
[1,1,0,0,1,1,0,0]
 => [3,4,1,2] => [1,2,3,4] => [1,2,3,4] => 4 = 3 + 1
[1,1,0,1,0,0,1,0]
 => [4,2,1,3] => [1,3,2,4] => [1,3,2,4] => 3 = 2 + 1
[1,1,0,1,0,1,0,0]
 => [3,2,1,4] => [1,4,2,3] => [1,3,4,2] => 3 = 2 + 1
[1,1,0,1,1,0,0,0]
 => [2,3,1,4] => [1,4,2,3] => [1,3,4,2] => 3 = 2 + 1
[1,1,1,0,0,0,1,0]
 => [4,1,2,3] => [1,2,3,4] => [1,2,3,4] => 4 = 3 + 1
[1,1,1,0,0,1,0,0]
 => [3,1,2,4] => [1,2,4,3] => [1,2,4,3] => 3 = 2 + 1
[1,1,1,0,1,0,0,0]
 => [2,1,3,4] => [1,3,4,2] => [1,4,2,3] => 2 = 1 + 1
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 4 = 3 + 1
[1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => 5 = 4 + 1
[1,0,1,0,1,0,1,1,0,0]
 => [4,5,3,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => 5 = 4 + 1
[1,0,1,0,1,1,0,0,1,0]
 => [5,3,4,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => 5 = 4 + 1
[1,0,1,0,1,1,0,1,0,0]
 => [4,3,5,2,1] => [1,2,3,5,4] => [1,2,3,5,4] => 4 = 3 + 1
[1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => 5 = 4 + 1
[1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,3,1] => [1,2,3,4,5] => [1,2,3,4,5] => 5 = 4 + 1
[1,0,1,1,0,0,1,1,0,0]
 => [4,5,2,3,1] => [1,2,3,4,5] => [1,2,3,4,5] => 5 = 4 + 1
[1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,4,1] => [1,2,4,3,5] => [1,2,4,3,5] => 4 = 3 + 1
[1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,5,1] => [1,2,5,3,4] => [1,2,4,5,3] => 4 = 3 + 1
[1,0,1,1,0,1,1,0,0,0]
 => [3,4,2,5,1] => [1,2,5,3,4] => [1,2,4,5,3] => 4 = 3 + 1
[1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => [1,2,3,4,5] => [1,2,3,4,5] => 5 = 4 + 1
[1,0,1,1,1,0,0,1,0,0]
 => [4,2,3,5,1] => [1,2,3,5,4] => [1,2,3,5,4] => 4 = 3 + 1
[1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => [1,2,4,5,3] => [1,2,5,3,4] => 3 = 2 + 1
[1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => 5 = 4 + 1
[1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => 5 = 4 + 1
[1,1,0,0,1,0,1,1,0,0]
 => [4,5,3,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => 5 = 4 + 1
[1,1,0,0,1,1,0,0,1,0]
 => [5,3,4,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => 5 = 4 + 1
[1,1,0,0,1,1,0,1,0,0]
 => [4,3,5,1,2] => [1,2,3,5,4] => [1,2,3,5,4] => 4 = 3 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => 5 = 4 + 1
[1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,3] => [1,3,2,4,5] => [1,3,2,4,5] => 4 = 3 + 1
[1,1,0,1,0,0,1,1,0,0]
 => [4,5,2,1,3] => [1,3,2,4,5] => [1,3,2,4,5] => 4 = 3 + 1
[1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,4] => [1,4,2,3,5] => [1,3,4,2,5] => 4 = 3 + 1
[1,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1,5] => [1,5,2,3,4] => [1,3,4,5,2] => 4 = 3 + 1
[1,1,0,1,0,1,1,0,0,0]
 => [3,4,2,1,5] => [1,5,2,3,4] => [1,3,4,5,2] => 4 = 3 + 1
[1,1,0,1,1,0,0,0,1,0]
 => [5,2,3,1,4] => [1,4,2,3,5] => [1,3,4,2,5] => 4 = 3 + 1
[1,1,0,1,1,0,0,1,0,0]
 => [4,2,3,1,5] => [1,5,2,3,4] => [1,3,4,5,2] => 4 = 3 + 1
[1,1,0,1,1,0,1,0,0,0]
 => [3,2,4,1,5] => [1,5,2,4,3] => [1,3,5,4,2] => 3 = 2 + 1
[1,1,0,1,1,1,0,0,0,0]
 => [2,3,4,1,5] => [1,5,2,3,4] => [1,3,4,5,2] => 4 = 3 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [7,6,5,4,3,2,1] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 6 + 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [6,7,5,4,3,2,1] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 6 + 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [7,5,6,4,3,2,1] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 6 + 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [5,6,7,4,3,2,1] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 6 + 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [7,6,4,5,3,2,1] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 6 + 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [6,7,4,5,3,2,1] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 6 + 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => [7,4,5,6,3,2,1] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 6 + 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [4,5,6,7,3,2,1] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 6 + 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [7,6,5,3,4,2,1] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 6 + 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [6,7,5,3,4,2,1] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 6 + 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [7,5,6,3,4,2,1] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 6 + 1
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => [5,6,7,3,4,2,1] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 6 + 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => [7,6,3,4,5,2,1] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 6 + 1
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => [6,7,3,4,5,2,1] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 6 + 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => [7,3,4,5,6,2,1] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 6 + 1
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [3,4,5,6,7,2,1] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 6 + 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => [7,6,5,4,2,3,1] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 6 + 1
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [6,7,5,4,2,3,1] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 6 + 1
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => [7,5,6,4,2,3,1] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 6 + 1
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
 => [5,6,7,4,2,3,1] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 6 + 1
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [7,6,4,5,2,3,1] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 6 + 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [6,7,4,5,2,3,1] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 6 + 1
[1,0,1,1,0,0,1,1,1,0,0,0,1,0]
 => [7,4,5,6,2,3,1] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 6 + 1
[1,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => [4,5,6,7,2,3,1] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 6 + 1
[1,0,1,1,1,0,0,0,1,0,1,0,1,0]
 => [7,6,5,2,3,4,1] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 6 + 1
[1,0,1,1,1,0,0,0,1,0,1,1,0,0]
 => [6,7,5,2,3,4,1] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 6 + 1
[1,0,1,1,1,0,0,0,1,1,0,0,1,0]
 => [7,5,6,2,3,4,1] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 6 + 1
[1,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => [5,6,7,2,3,4,1] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 6 + 1
[1,0,1,1,1,1,0,0,0,0,1,0,1,0]
 => [7,6,2,3,4,5,1] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 6 + 1
[1,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => [6,7,2,3,4,5,1] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 6 + 1
[1,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => [7,2,3,4,5,6,1] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 6 + 1
[1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [2,3,4,5,6,7,1] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 6 + 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => [7,6,5,4,3,1,2] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 6 + 1
[1,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => [6,7,5,4,3,1,2] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 6 + 1
[1,1,0,0,1,0,1,0,1,1,0,0,1,0]
 => [7,5,6,4,3,1,2] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 6 + 1
[1,1,0,0,1,0,1,0,1,1,1,0,0,0]
 => [5,6,7,4,3,1,2] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 6 + 1
[1,1,0,0,1,0,1,1,0,0,1,0,1,0]
 => [7,6,4,5,3,1,2] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 6 + 1
[1,1,0,0,1,0,1,1,0,0,1,1,0,0]
 => [6,7,4,5,3,1,2] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 6 + 1
[1,1,0,0,1,0,1,1,1,0,0,0,1,0]
 => [7,4,5,6,3,1,2] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 6 + 1
[1,1,0,0,1,0,1,1,1,1,0,0,0,0]
 => [4,5,6,7,3,1,2] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 6 + 1
[1,1,0,0,1,1,0,0,1,0,1,0,1,0]
 => [7,6,5,3,4,1,2] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 6 + 1
[1,1,0,0,1,1,0,0,1,0,1,1,0,0]
 => [6,7,5,3,4,1,2] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 6 + 1
[1,1,0,0,1,1,0,0,1,1,0,0,1,0]
 => [7,5,6,3,4,1,2] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 6 + 1
[1,1,0,0,1,1,0,0,1,1,1,0,0,0]
 => [5,6,7,3,4,1,2] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 6 + 1
[1,1,0,0,1,1,1,0,0,0,1,0,1,0]
 => [7,6,3,4,5,1,2] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 6 + 1
[1,1,0,0,1,1,1,0,0,0,1,1,0,0]
 => [6,7,3,4,5,1,2] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 6 + 1
[1,1,0,0,1,1,1,1,0,0,0,0,1,0]
 => [7,3,4,5,6,1,2] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 6 + 1
[1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => [3,4,5,6,7,1,2] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 6 + 1
[1,1,0,1,0,0,1,0,1,0,1,0,1,0]
 => [7,6,5,4,2,1,3] => [1,3,2,4,5,6,7] => [1,3,2,4,5,6,7] => ? = 5 + 1
[1,1,0,1,0,0,1,0,1,0,1,1,0,0]
 => [6,7,5,4,2,1,3] => [1,3,2,4,5,6,7] => [1,3,2,4,5,6,7] => ? = 5 + 1
Description
The number of left-to-right-maxima of a permutation. An integer $\sigma_i$ in the one-line notation of a permutation $\sigma$ is a '''left-to-right-maximum''' if there does not exist a $j < i$ such that $\sigma_j > \sigma_i$. This is also the number of weak exceedences of a permutation that are not mid-points of a decreasing subsequence of length 3, see [1] for more on the later description.
Matching statistic: St000542
Mapping
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St000542: Permutations ⟶ ℤResult quality: 42% ●values known / values provided: 42%●distinct values known / distinct values provided: 55%
Values
[1,0]
 => [1] => [1] => [1] => 1 = 0 + 1
[1,0,1,0]
 => [2,1] => [1,2] => [2,1] => 2 = 1 + 1
[1,1,0,0]
 => [1,2] => [1,2] => [2,1] => 2 = 1 + 1
[1,0,1,0,1,0]
 => [3,2,1] => [1,2,3] => [3,2,1] => 3 = 2 + 1
[1,0,1,1,0,0]
 => [2,3,1] => [1,2,3] => [3,2,1] => 3 = 2 + 1
[1,1,0,0,1,0]
 => [3,1,2] => [1,2,3] => [3,2,1] => 3 = 2 + 1
[1,1,0,1,0,0]
 => [2,1,3] => [1,3,2] => [2,3,1] => 2 = 1 + 1
[1,1,1,0,0,0]
 => [1,2,3] => [1,2,3] => [3,2,1] => 3 = 2 + 1
[1,0,1,0,1,0,1,0]
 => [4,3,2,1] => [1,2,3,4] => [4,3,2,1] => 4 = 3 + 1
[1,0,1,0,1,1,0,0]
 => [3,4,2,1] => [1,2,3,4] => [4,3,2,1] => 4 = 3 + 1
[1,0,1,1,0,0,1,0]
 => [4,2,3,1] => [1,2,3,4] => [4,3,2,1] => 4 = 3 + 1
[1,0,1,1,0,1,0,0]
 => [3,2,4,1] => [1,2,4,3] => [3,4,2,1] => 3 = 2 + 1
[1,0,1,1,1,0,0,0]
 => [2,3,4,1] => [1,2,3,4] => [4,3,2,1] => 4 = 3 + 1
[1,1,0,0,1,0,1,0]
 => [4,3,1,2] => [1,2,3,4] => [4,3,2,1] => 4 = 3 + 1
[1,1,0,0,1,1,0,0]
 => [3,4,1,2] => [1,2,3,4] => [4,3,2,1] => 4 = 3 + 1
[1,1,0,1,0,0,1,0]
 => [4,2,1,3] => [1,3,2,4] => [4,2,3,1] => 3 = 2 + 1
[1,1,0,1,0,1,0,0]
 => [3,2,1,4] => [1,4,2,3] => [3,2,4,1] => 3 = 2 + 1
[1,1,0,1,1,0,0,0]
 => [2,3,1,4] => [1,4,2,3] => [3,2,4,1] => 3 = 2 + 1
[1,1,1,0,0,0,1,0]
 => [4,1,2,3] => [1,2,3,4] => [4,3,2,1] => 4 = 3 + 1
[1,1,1,0,0,1,0,0]
 => [3,1,2,4] => [1,2,4,3] => [3,4,2,1] => 3 = 2 + 1
[1,1,1,0,1,0,0,0]
 => [2,1,3,4] => [1,3,4,2] => [2,4,3,1] => 2 = 1 + 1
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 4 = 3 + 1
[1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => [1,2,3,4,5] => [5,4,3,2,1] => 5 = 4 + 1
[1,0,1,0,1,0,1,1,0,0]
 => [4,5,3,2,1] => [1,2,3,4,5] => [5,4,3,2,1] => 5 = 4 + 1
[1,0,1,0,1,1,0,0,1,0]
 => [5,3,4,2,1] => [1,2,3,4,5] => [5,4,3,2,1] => 5 = 4 + 1
[1,0,1,0,1,1,0,1,0,0]
 => [4,3,5,2,1] => [1,2,3,5,4] => [4,5,3,2,1] => 4 = 3 + 1
[1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => [1,2,3,4,5] => [5,4,3,2,1] => 5 = 4 + 1
[1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,3,1] => [1,2,3,4,5] => [5,4,3,2,1] => 5 = 4 + 1
[1,0,1,1,0,0,1,1,0,0]
 => [4,5,2,3,1] => [1,2,3,4,5] => [5,4,3,2,1] => 5 = 4 + 1
[1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,4,1] => [1,2,4,3,5] => [5,3,4,2,1] => 4 = 3 + 1
[1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,5,1] => [1,2,5,3,4] => [4,3,5,2,1] => 4 = 3 + 1
[1,0,1,1,0,1,1,0,0,0]
 => [3,4,2,5,1] => [1,2,5,3,4] => [4,3,5,2,1] => 4 = 3 + 1
[1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => [1,2,3,4,5] => [5,4,3,2,1] => 5 = 4 + 1
[1,0,1,1,1,0,0,1,0,0]
 => [4,2,3,5,1] => [1,2,3,5,4] => [4,5,3,2,1] => 4 = 3 + 1
[1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => [1,2,4,5,3] => [3,5,4,2,1] => 3 = 2 + 1
[1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [1,2,3,4,5] => [5,4,3,2,1] => 5 = 4 + 1
[1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,2] => [1,2,3,4,5] => [5,4,3,2,1] => 5 = 4 + 1
[1,1,0,0,1,0,1,1,0,0]
 => [4,5,3,1,2] => [1,2,3,4,5] => [5,4,3,2,1] => 5 = 4 + 1
[1,1,0,0,1,1,0,0,1,0]
 => [5,3,4,1,2] => [1,2,3,4,5] => [5,4,3,2,1] => 5 = 4 + 1
[1,1,0,0,1,1,0,1,0,0]
 => [4,3,5,1,2] => [1,2,3,5,4] => [4,5,3,2,1] => 4 = 3 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => [1,2,3,4,5] => [5,4,3,2,1] => 5 = 4 + 1
[1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,3] => [1,3,2,4,5] => [5,4,2,3,1] => 4 = 3 + 1
[1,1,0,1,0,0,1,1,0,0]
 => [4,5,2,1,3] => [1,3,2,4,5] => [5,4,2,3,1] => 4 = 3 + 1
[1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,4] => [1,4,2,3,5] => [5,3,2,4,1] => 4 = 3 + 1
[1,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1,5] => [1,5,2,3,4] => [4,3,2,5,1] => 4 = 3 + 1
[1,1,0,1,0,1,1,0,0,0]
 => [3,4,2,1,5] => [1,5,2,3,4] => [4,3,2,5,1] => 4 = 3 + 1
[1,1,0,1,1,0,0,0,1,0]
 => [5,2,3,1,4] => [1,4,2,3,5] => [5,3,2,4,1] => 4 = 3 + 1
[1,1,0,1,1,0,0,1,0,0]
 => [4,2,3,1,5] => [1,5,2,3,4] => [4,3,2,5,1] => 4 = 3 + 1
[1,1,0,1,1,0,1,0,0,0]
 => [3,2,4,1,5] => [1,5,2,4,3] => [3,4,2,5,1] => 3 = 2 + 1
[1,1,0,1,1,1,0,0,0,0]
 => [2,3,4,1,5] => [1,5,2,3,4] => [4,3,2,5,1] => 4 = 3 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [7,6,5,4,3,2,1] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 6 + 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [6,7,5,4,3,2,1] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 6 + 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [7,5,6,4,3,2,1] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 6 + 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [5,6,7,4,3,2,1] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 6 + 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [7,6,4,5,3,2,1] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 6 + 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [6,7,4,5,3,2,1] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 6 + 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => [7,4,5,6,3,2,1] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 6 + 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [4,5,6,7,3,2,1] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 6 + 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [7,6,5,3,4,2,1] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 6 + 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [6,7,5,3,4,2,1] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 6 + 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [7,5,6,3,4,2,1] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 6 + 1
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => [5,6,7,3,4,2,1] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 6 + 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => [7,6,3,4,5,2,1] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 6 + 1
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => [6,7,3,4,5,2,1] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 6 + 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => [7,3,4,5,6,2,1] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 6 + 1
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [3,4,5,6,7,2,1] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 6 + 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => [7,6,5,4,2,3,1] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 6 + 1
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [6,7,5,4,2,3,1] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 6 + 1
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => [7,5,6,4,2,3,1] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 6 + 1
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
 => [5,6,7,4,2,3,1] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 6 + 1
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [7,6,4,5,2,3,1] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 6 + 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [6,7,4,5,2,3,1] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 6 + 1
[1,0,1,1,0,0,1,1,1,0,0,0,1,0]
 => [7,4,5,6,2,3,1] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 6 + 1
[1,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => [4,5,6,7,2,3,1] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 6 + 1
[1,0,1,1,1,0,0,0,1,0,1,0,1,0]
 => [7,6,5,2,3,4,1] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 6 + 1
[1,0,1,1,1,0,0,0,1,0,1,1,0,0]
 => [6,7,5,2,3,4,1] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 6 + 1
[1,0,1,1,1,0,0,0,1,1,0,0,1,0]
 => [7,5,6,2,3,4,1] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 6 + 1
[1,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => [5,6,7,2,3,4,1] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 6 + 1
[1,0,1,1,1,1,0,0,0,0,1,0,1,0]
 => [7,6,2,3,4,5,1] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 6 + 1
[1,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => [6,7,2,3,4,5,1] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 6 + 1
[1,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => [7,2,3,4,5,6,1] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 6 + 1
[1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [2,3,4,5,6,7,1] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 6 + 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => [7,6,5,4,3,1,2] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 6 + 1
[1,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => [6,7,5,4,3,1,2] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 6 + 1
[1,1,0,0,1,0,1,0,1,1,0,0,1,0]
 => [7,5,6,4,3,1,2] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 6 + 1
[1,1,0,0,1,0,1,0,1,1,1,0,0,0]
 => [5,6,7,4,3,1,2] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 6 + 1
[1,1,0,0,1,0,1,1,0,0,1,0,1,0]
 => [7,6,4,5,3,1,2] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 6 + 1
[1,1,0,0,1,0,1,1,0,0,1,1,0,0]
 => [6,7,4,5,3,1,2] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 6 + 1
[1,1,0,0,1,0,1,1,1,0,0,0,1,0]
 => [7,4,5,6,3,1,2] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 6 + 1
[1,1,0,0,1,0,1,1,1,1,0,0,0,0]
 => [4,5,6,7,3,1,2] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 6 + 1
[1,1,0,0,1,1,0,0,1,0,1,0,1,0]
 => [7,6,5,3,4,1,2] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 6 + 1
[1,1,0,0,1,1,0,0,1,0,1,1,0,0]
 => [6,7,5,3,4,1,2] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 6 + 1
[1,1,0,0,1,1,0,0,1,1,0,0,1,0]
 => [7,5,6,3,4,1,2] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 6 + 1
[1,1,0,0,1,1,0,0,1,1,1,0,0,0]
 => [5,6,7,3,4,1,2] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 6 + 1
[1,1,0,0,1,1,1,0,0,0,1,0,1,0]
 => [7,6,3,4,5,1,2] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 6 + 1
[1,1,0,0,1,1,1,0,0,0,1,1,0,0]
 => [6,7,3,4,5,1,2] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 6 + 1
[1,1,0,0,1,1,1,1,0,0,0,0,1,0]
 => [7,3,4,5,6,1,2] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 6 + 1
[1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => [3,4,5,6,7,1,2] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 6 + 1
[1,1,0,1,0,0,1,0,1,0,1,0,1,0]
 => [7,6,5,4,2,1,3] => [1,3,2,4,5,6,7] => [7,6,5,4,2,3,1] => ? = 5 + 1
[1,1,0,1,0,0,1,0,1,0,1,1,0,0]
 => [6,7,5,4,2,1,3] => [1,3,2,4,5,6,7] => [7,6,5,4,2,3,1] => ? = 5 + 1
Description
The number of left-to-right-minima of a permutation. An integer $\sigma_i$ in the one-line notation of a permutation $\sigma$ is a left-to-right-minimum if there does not exist a j < i such that $\sigma_j < \sigma_i$.
The following 14 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001179Number of indecomposable injective modules with projective dimension at most 2 in the corresponding Nakayama algebra. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. 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. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000906The length of the shortest maximal chain in a poset. St000031The number of cycles in the cycle decomposition of a permutation. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000703The number of deficiencies of a permutation. St000994The number of cycle peaks and the number of cycle valleys of a 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 $K[x]/(x^n)$. St001935The number of ascents in a parking function. St000942The number of critical left to right maxima of the parking functions. St001773The number of minimal elements in Bruhat order not less than the signed permutation.