- FindStat

Your data matches 22 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000011
(load all 2 compositions to match this statistic)

Mapping

Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,0]
 => [1] => [1,0]
 => [1,0]
 => 1
[1,0,1,0]
 => [2,1] => [1,1,0,0]
 => [1,0,1,0]
 => 2
[1,1,0,0]
 => [1,2] => [1,0,1,0]
 => [1,1,0,0]
 => 1
[1,0,1,0,1,0]
 => [3,2,1] => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 3
[1,0,1,1,0,0]
 => [2,3,1] => [1,1,0,1,0,0]
 => [1,1,0,1,0,0]
 => 1
[1,1,0,0,1,0]
 => [3,1,2] => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 3
[1,1,0,1,0,0]
 => [2,1,3] => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 2
[1,1,1,0,0,0]
 => [1,2,3] => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 1
[1,0,1,0,1,0,1,0]
 => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 4
[1,0,1,0,1,1,0,0]
 => [3,4,2,1] => [1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => 1
[1,0,1,1,0,0,1,0]
 => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 4
[1,0,1,1,0,1,0,0]
 => [3,2,4,1] => [1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => 2
[1,0,1,1,1,0,0,0]
 => [2,3,4,1] => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 1
[1,1,0,0,1,0,1,0]
 => [4,3,1,2] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 4
[1,1,0,0,1,1,0,0]
 => [3,4,1,2] => [1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => 1
[1,1,0,1,0,0,1,0]
 => [4,2,1,3] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 4
[1,1,0,1,0,1,0,0]
 => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => 3
[1,1,0,1,1,0,0,0]
 => [2,3,1,4] => [1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 1
[1,1,1,0,0,0,1,0]
 => [4,1,2,3] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 4
[1,1,1,0,0,1,0,0]
 => [3,1,2,4] => [1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => 3
[1,1,1,0,1,0,0,0]
 => [2,1,3,4] => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => 2
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 1
[1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
[1,0,1,0,1,0,1,1,0,0]
 => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1
[1,0,1,0,1,1,0,0,1,0]
 => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
[1,0,1,0,1,1,0,1,0,0]
 => [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 2
[1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 1
[1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
[1,0,1,1,0,0,1,1,0,0]
 => [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1
[1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
[1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 3
[1,0,1,1,0,1,1,0,0,0]
 => [3,4,2,5,1] => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => 1
[1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
[1,0,1,1,1,0,0,1,0,0]
 => [4,2,3,5,1] => [1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 3
[1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => 2
[1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 1
[1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
[1,1,0,0,1,0,1,1,0,0]
 => [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1
[1,1,0,0,1,1,0,0,1,0]
 => [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
[1,1,0,0,1,1,0,1,0,0]
 => [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 2
[1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 1
[1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
[1,1,0,1,0,0,1,1,0,0]
 => [4,5,2,1,3] => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1
[1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
[1,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 4
[1,1,0,1,0,1,1,0,0,0]
 => [3,4,2,1,5] => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 1
[1,1,0,1,1,0,0,0,1,0]
 => [5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
[1,1,0,1,1,0,0,1,0,0]
 => [4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 4
[1,1,0,1,1,0,1,0,0,0]
 => [3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => 2
[1,1,0,1,1,1,0,0,0,0]
 => [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 1

Description

The number of touch points (or returns) of a Dyck path. This is the number of points, excluding the origin, where the Dyck path has height 0.

Matching statistic: St000439

Mapping

Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
St000439: Dyck paths ⟶ ℤResult quality: 85% ●values known / values provided: 90%●distinct values known / distinct values provided: 85%

Values

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

Description

The position of the first down step of a Dyck path.

Matching statistic: St000678

Mapping

Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St000678: Dyck paths ⟶ ℤResult quality: 62% ●values known / values provided: 75%●distinct values known / distinct values provided: 62%

Values

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

Description

The number of up steps after the last double rise of a Dyck path.

Matching statistic: St000645

Mapping

Mp00029: Dyck paths —to binary tree: left tree, up step, right tree, down step⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000645: Dyck paths ⟶ ℤResult quality: 74% ●values known / values provided: 74%●distinct values known / distinct values provided: 85%

Values

[1,0]
 => [.,.]
 => [1] => [1,0]
 => 0 = 1 - 1
[1,0,1,0]
 => [[.,.],.]
 => [1,2] => [1,0,1,0]
 => 1 = 2 - 1
[1,1,0,0]
 => [.,[.,.]]
 => [2,1] => [1,1,0,0]
 => 0 = 1 - 1
[1,0,1,0,1,0]
 => [[[.,.],.],.]
 => [1,2,3] => [1,0,1,0,1,0]
 => 2 = 3 - 1
[1,0,1,1,0,0]
 => [[.,.],[.,.]]
 => [3,1,2] => [1,1,1,0,0,0]
 => 0 = 1 - 1
[1,1,0,0,1,0]
 => [[.,[.,.]],.]
 => [2,1,3] => [1,1,0,0,1,0]
 => 2 = 3 - 1
[1,1,0,1,0,0]
 => [.,[[.,.],.]]
 => [2,3,1] => [1,1,0,1,0,0]
 => 1 = 2 - 1
[1,1,1,0,0,0]
 => [.,[.,[.,.]]]
 => [3,2,1] => [1,1,1,0,0,0]
 => 0 = 1 - 1
[1,0,1,0,1,0,1,0]
 => [[[[.,.],.],.],.]
 => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => 3 = 4 - 1
[1,0,1,0,1,1,0,0]
 => [[[.,.],.],[.,.]]
 => [4,1,2,3] => [1,1,1,1,0,0,0,0]
 => 0 = 1 - 1
[1,0,1,1,0,0,1,0]
 => [[[.,.],[.,.]],.]
 => [3,1,2,4] => [1,1,1,0,0,0,1,0]
 => 3 = 4 - 1
[1,0,1,1,0,1,0,0]
 => [[.,.],[[.,.],.]]
 => [3,4,1,2] => [1,1,1,0,1,0,0,0]
 => 1 = 2 - 1
[1,0,1,1,1,0,0,0]
 => [[.,.],[.,[.,.]]]
 => [4,3,1,2] => [1,1,1,1,0,0,0,0]
 => 0 = 1 - 1
[1,1,0,0,1,0,1,0]
 => [[[.,[.,.]],.],.]
 => [2,1,3,4] => [1,1,0,0,1,0,1,0]
 => 3 = 4 - 1
[1,1,0,0,1,1,0,0]
 => [[.,[.,.]],[.,.]]
 => [4,2,1,3] => [1,1,1,1,0,0,0,0]
 => 0 = 1 - 1
[1,1,0,1,0,0,1,0]
 => [[.,[[.,.],.]],.]
 => [2,3,1,4] => [1,1,0,1,0,0,1,0]
 => 3 = 4 - 1
[1,1,0,1,0,1,0,0]
 => [.,[[[.,.],.],.]]
 => [2,3,4,1] => [1,1,0,1,0,1,0,0]
 => 2 = 3 - 1
[1,1,0,1,1,0,0,0]
 => [.,[[.,.],[.,.]]]
 => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => 0 = 1 - 1
[1,1,1,0,0,0,1,0]
 => [[.,[.,[.,.]]],.]
 => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => 3 = 4 - 1
[1,1,1,0,0,1,0,0]
 => [.,[[.,[.,.]],.]]
 => [3,2,4,1] => [1,1,1,0,0,1,0,0]
 => 2 = 3 - 1
[1,1,1,0,1,0,0,0]
 => [.,[.,[[.,.],.]]]
 => [3,4,2,1] => [1,1,1,0,1,0,0,0]
 => 1 = 2 - 1
[1,1,1,1,0,0,0,0]
 => [.,[.,[.,[.,.]]]]
 => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
 => [[[[[.,.],.],.],.],.]
 => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => 4 = 5 - 1
[1,0,1,0,1,0,1,1,0,0]
 => [[[[.,.],.],.],[.,.]]
 => [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [[[[.,.],.],[.,.]],.]
 => [4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0]
 => 4 = 5 - 1
[1,0,1,0,1,1,0,1,0,0]
 => [[[.,.],.],[[.,.],.]]
 => [4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
 => 1 = 2 - 1
[1,0,1,0,1,1,1,0,0,0]
 => [[[.,.],.],[.,[.,.]]]
 => [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [[[[.,.],[.,.]],.],.]
 => [3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0]
 => 4 = 5 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [[[.,.],[.,.]],[.,.]]
 => [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[1,0,1,1,0,1,0,0,1,0]
 => [[[.,.],[[.,.],.]],.]
 => [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
 => 4 = 5 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [[.,.],[[[.,.],.],.]]
 => [3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
 => 2 = 3 - 1
[1,0,1,1,0,1,1,0,0,0]
 => [[.,.],[[.,.],[.,.]]]
 => [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[1,0,1,1,1,0,0,0,1,0]
 => [[[.,.],[.,[.,.]]],.]
 => [4,3,1,2,5] => [1,1,1,1,0,0,0,0,1,0]
 => 4 = 5 - 1
[1,0,1,1,1,0,0,1,0,0]
 => [[.,.],[[.,[.,.]],.]]
 => [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
 => 2 = 3 - 1
[1,0,1,1,1,0,1,0,0,0]
 => [[.,.],[.,[[.,.],.]]]
 => [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
 => 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
 => [[.,.],[.,[.,[.,.]]]]
 => [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[1,1,0,0,1,0,1,0,1,0]
 => [[[[.,[.,.]],.],.],.]
 => [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
 => 4 = 5 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [[[.,[.,.]],.],[.,.]]
 => [5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [[[.,[.,.]],[.,.]],.]
 => [4,2,1,3,5] => [1,1,1,1,0,0,0,0,1,0]
 => 4 = 5 - 1
[1,1,0,0,1,1,0,1,0,0]
 => [[.,[.,.]],[[.,.],.]]
 => [4,5,2,1,3] => [1,1,1,1,0,1,0,0,0,0]
 => 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [[.,[.,.]],[.,[.,.]]]
 => [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[1,1,0,1,0,0,1,0,1,0]
 => [[[.,[[.,.],.]],.],.]
 => [2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
 => 4 = 5 - 1
[1,1,0,1,0,0,1,1,0,0]
 => [[.,[[.,.],.]],[.,.]]
 => [5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[1,1,0,1,0,1,0,0,1,0]
 => [[.,[[[.,.],.],.]],.]
 => [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
 => 4 = 5 - 1
[1,1,0,1,0,1,0,1,0,0]
 => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
 => 3 = 4 - 1
[1,1,0,1,0,1,1,0,0,0]
 => [.,[[[.,.],.],[.,.]]]
 => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[1,1,0,1,1,0,0,0,1,0]
 => [[.,[[.,.],[.,.]]],.]
 => [4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0]
 => 4 = 5 - 1
[1,1,0,1,1,0,0,1,0,0]
 => [.,[[[.,.],[.,.]],.]]
 => [4,2,3,5,1] => [1,1,1,1,0,0,0,1,0,0]
 => 3 = 4 - 1
[1,1,0,1,1,0,1,0,0,0]
 => [.,[[.,.],[[.,.],.]]]
 => [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
 => 1 = 2 - 1
[1,1,0,1,1,1,0,0,0,0]
 => [.,[[.,.],[.,[.,.]]]]
 => [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
 => [[[[[[.,.],.],.],[[.,.],.]],.],.]
 => [5,6,1,2,3,4,7,8] => ?
 => ? = 8 - 1
[1,0,1,0,1,1,0,0,1,1,0,1,1,0,0,0]
 => [[[[.,.],.],[.,.]],[[.,.],[.,.]]]
 => [8,6,7,4,1,2,3,5] => ?
 => ? = 1 - 1
[1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
 => [[[[[.,.],.],[.,.]],[.,[.,.]]],.]
 => [7,6,4,1,2,3,5,8] => ?
 => ? = 8 - 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0]
 => [[[[[[.,.],.],[[.,.],.]],.],.],.]
 => [4,5,1,2,3,6,7,8] => [1,1,1,1,0,1,0,0,0,0,1,0,1,0,1,0]
 => ? = 8 - 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,0,1,0]
 => [[[[[.,.],.],[[[.,.],.],.]],.],.]
 => [4,5,6,1,2,3,7,8] => [1,1,1,1,0,1,0,1,0,0,0,0,1,0,1,0]
 => ? = 8 - 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,0,1,0]
 => [[[[.,.],.],[[[[.,.],.],.],.]],.]
 => [4,5,6,7,1,2,3,8] => [1,1,1,1,0,1,0,1,0,1,0,0,0,0,1,0]
 => ? = 8 - 1
[1,0,1,0,1,1,0,1,1,0,0,0,1,0,1,0]
 => [[[[[.,.],.],[[.,.],[.,.]]],.],.]
 => [6,4,5,1,2,3,7,8] => ?
 => ? = 8 - 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,0,1,0]
 => [[[[.,.],.],[[.,.],[[.,.],.]]],.]
 => [6,7,4,5,1,2,3,8] => ?
 => ? = 8 - 1
[1,0,1,0,1,1,1,0,0,0,1,1,0,0,1,0]
 => [[[[[.,.],.],[.,[.,.]]],[.,.]],.]
 => [7,5,4,1,2,3,6,8] => ?
 => ? = 8 - 1
[1,0,1,0,1,1,1,0,0,1,0,0,1,0,1,0]
 => [[[[[.,.],.],[[.,[.,.]],.]],.],.]
 => [5,4,6,1,2,3,7,8] => ?
 => ? = 8 - 1
[1,0,1,0,1,1,1,0,0,1,0,0,1,1,0,0]
 => [[[[.,.],.],[[.,[.,.]],.]],[.,.]]
 => [8,5,4,6,1,2,3,7] => ?
 => ? = 1 - 1
[1,0,1,0,1,1,1,0,1,0,0,0,1,0,1,0]
 => [[[[[.,.],.],[.,[[.,.],.]]],.],.]
 => [5,6,4,1,2,3,7,8] => ?
 => ? = 8 - 1
[1,0,1,0,1,1,1,0,1,0,0,1,0,0,1,0]
 => [[[[.,.],.],[[.,[[.,.],.]],.]],.]
 => [5,6,4,7,1,2,3,8] => [1,1,1,1,1,0,1,0,0,1,0,0,0,0,1,0]
 => ? = 8 - 1
[1,0,1,0,1,1,1,1,0,0,1,0,0,0,1,0]
 => [[[[.,.],.],[.,[[.,[.,.]],.]]],.]
 => [6,5,7,4,1,2,3,8] => ?
 => ? = 8 - 1
[1,0,1,0,1,1,1,1,0,1,0,0,0,0,1,0]
 => [[[[.,.],.],[.,[.,[[.,.],.]]]],.]
 => [6,7,5,4,1,2,3,8] => ?
 => ? = 8 - 1
[1,0,1,1,0,0,1,0,1,1,1,0,0,0,1,0]
 => [[[[[.,.],[.,.]],.],[.,[.,.]]],.]
 => [7,6,3,1,2,4,5,8] => ?
 => ? = 8 - 1
[1,0,1,1,0,0,1,1,0,1,0,0,1,0,1,0]
 => [[[[[.,.],[.,.]],[[.,.],.]],.],.]
 => [5,6,3,1,2,4,7,8] => ?
 => ? = 8 - 1
[1,0,1,1,0,0,1,1,1,0,0,0,1,0,1,0]
 => [[[[[.,.],[.,.]],[.,[.,.]]],.],.]
 => [6,5,3,1,2,4,7,8] => ?
 => ? = 8 - 1
[1,0,1,1,0,0,1,1,1,0,0,0,1,1,0,0]
 => [[[[.,.],[.,.]],[.,[.,.]]],[.,.]]
 => [8,6,5,3,1,2,4,7] => ?
 => ? = 1 - 1
[1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,0]
 => [[[[.,.],[.,.]],[[.,[.,.]],.]],.]
 => [6,5,7,3,1,2,4,8] => ?
 => ? = 8 - 1
[1,0,1,1,0,0,1,1,1,1,0,0,0,0,1,0]
 => [[[[.,.],[.,.]],[.,[.,[.,.]]]],.]
 => [7,6,5,3,1,2,4,8] => ?
 => ? = 8 - 1
[1,0,1,1,0,1,0,0,1,1,0,0,1,0,1,0]
 => [[[[[.,.],[[.,.],.]],[.,.]],.],.]
 => [6,3,4,1,2,5,7,8] => ?
 => ? = 8 - 1
[1,0,1,1,0,1,0,0,1,1,1,0,0,0,1,0]
 => [[[[.,.],[[.,.],.]],[.,[.,.]]],.]
 => [7,6,3,4,1,2,5,8] => ?
 => ? = 8 - 1
[1,0,1,1,0,1,0,1,0,0,1,1,0,0,1,0]
 => [[[[.,.],[[[.,.],.],.]],[.,.]],.]
 => [7,3,4,5,1,2,6,8] => ?
 => ? = 8 - 1
[1,0,1,1,0,1,0,1,0,1,1,0,0,0,1,0]
 => [[[.,.],[[[[.,.],.],.],[.,.]]],.]
 => [7,3,4,5,6,1,2,8] => ?
 => ? = 8 - 1
[1,0,1,1,0,1,0,1,1,0,0,1,1,0,0,0]
 => [[.,.],[[[[.,.],.],[.,.]],[.,.]]]
 => [8,6,3,4,5,7,1,2] => ?
 => ? = 1 - 1
[1,0,1,1,0,1,1,0,0,1,0,0,1,0,1,0]
 => [[[[.,.],[[[.,.],[.,.]],.]],.],.]
 => [5,3,4,6,1,2,7,8] => ?
 => ? = 8 - 1
[1,0,1,1,0,1,1,0,0,1,0,1,0,0,1,0]
 => [[[.,.],[[[[.,.],[.,.]],.],.]],.]
 => [5,3,4,6,7,1,2,8] => ?
 => ? = 8 - 1
[1,0,1,1,0,1,1,0,1,0,0,0,1,0,1,0]
 => [[[[.,.],[[.,.],[[.,.],.]]],.],.]
 => [5,6,3,4,1,2,7,8] => [1,1,1,1,1,0,1,0,0,0,0,0,1,0,1,0]
 => ? = 8 - 1
[1,0,1,1,0,1,1,0,1,0,0,1,0,0,1,0]
 => [[[.,.],[[[.,.],[[.,.],.]],.]],.]
 => [5,6,3,4,7,1,2,8] => [1,1,1,1,1,0,1,0,0,0,1,0,0,0,1,0]
 => ? = 8 - 1
[1,0,1,1,0,1,1,0,1,0,1,0,0,0,1,0]
 => [[[.,.],[[.,.],[[[.,.],.],.]]],.]
 => [5,6,7,3,4,1,2,8] => ?
 => ? = 8 - 1
[1,0,1,1,0,1,1,0,1,0,1,0,0,1,0,0]
 => [[.,.],[[[.,.],[[[.,.],.],.]],.]]
 => [5,6,7,3,4,8,1,2] => ?
 => ? = 6 - 1
[1,0,1,1,0,1,1,1,0,0,0,1,0,0,1,0]
 => [[[.,.],[[[.,.],[.,[.,.]]],.]],.]
 => [6,5,3,4,7,1,2,8] => ?
 => ? = 8 - 1
[1,0,1,1,0,1,1,1,0,0,0,1,0,1,0,0]
 => [[.,.],[[[[.,.],[.,[.,.]]],.],.]]
 => [6,5,3,4,7,8,1,2] => ?
 => ? = 6 - 1
[1,0,1,1,0,1,1,1,0,0,1,0,0,1,0,0]
 => [[.,.],[[[.,.],[[.,[.,.]],.]],.]]
 => [6,5,7,3,4,8,1,2] => ?
 => ? = 6 - 1
[1,0,1,1,0,1,1,1,0,1,0,0,0,0,1,0]
 => [[[.,.],[[.,.],[.,[[.,.],.]]]],.]
 => [6,7,5,3,4,1,2,8] => ?
 => ? = 8 - 1
[1,0,1,1,0,1,1,1,0,1,1,0,0,0,0,0]
 => [[.,.],[[.,.],[.,[[.,.],[.,.]]]]]
 => [8,6,7,5,3,4,1,2] => ?
 => ? = 1 - 1
[1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]
 => [[[[.,.],[.,[.,.]]],.],[.,[.,.]]]
 => [8,7,4,3,1,2,5,6] => ?
 => ? = 1 - 1
[1,0,1,1,1,0,0,0,1,1,0,1,0,0,1,0]
 => [[[[.,.],[.,[.,.]]],[[.,.],.]],.]
 => [6,7,4,3,1,2,5,8] => ?
 => ? = 8 - 1
[1,0,1,1,1,0,0,0,1,1,0,1,1,0,0,0]
 => [[[.,.],[.,[.,.]]],[[.,.],[.,.]]]
 => [8,6,7,4,3,1,2,5] => ?
 => ? = 1 - 1
[1,0,1,1,1,0,0,0,1,1,1,0,1,0,0,0]
 => [[[.,.],[.,[.,.]]],[.,[[.,.],.]]]
 => [7,8,6,4,3,1,2,5] => ?
 => ? = 2 - 1
[1,0,1,1,1,0,0,1,0,0,1,1,0,1,0,0]
 => [[[.,.],[[.,[.,.]],.]],[[.,.],.]]
 => [7,8,4,3,5,1,2,6] => ?
 => ? = 2 - 1
[1,0,1,1,1,0,0,1,0,0,1,1,1,0,0,0]
 => [[[.,.],[[.,[.,.]],.]],[.,[.,.]]]
 => [8,7,4,3,5,1,2,6] => ?
 => ? = 1 - 1
[1,0,1,1,1,0,0,1,0,1,0,0,1,0,1,0]
 => [[[[.,.],[[[.,[.,.]],.],.]],.],.]
 => [4,3,5,6,1,2,7,8] => [1,1,1,1,0,0,1,0,1,0,0,0,1,0,1,0]
 => ? = 8 - 1
[1,0,1,1,1,0,0,1,0,1,0,1,0,0,1,0]
 => [[[.,.],[[[[.,[.,.]],.],.],.]],.]
 => [4,3,5,6,7,1,2,8] => [1,1,1,1,0,0,1,0,1,0,1,0,0,0,1,0]
 => ? = 8 - 1
[1,0,1,1,1,0,0,1,0,1,1,0,0,0,1,0]
 => [[[.,.],[[[.,[.,.]],.],[.,.]]],.]
 => [7,4,3,5,6,1,2,8] => ?
 => ? = 8 - 1
[1,0,1,1,1,0,0,1,0,1,1,0,0,1,0,0]
 => [[.,.],[[[[.,[.,.]],.],[.,.]],.]]
 => [7,4,3,5,6,8,1,2] => ?
 => ? = 6 - 1
[1,0,1,1,1,0,0,1,1,0,0,1,1,0,0,0]
 => [[.,.],[[[.,[.,.]],[.,.]],[.,.]]]
 => [8,6,4,3,5,7,1,2] => ?
 => ? = 1 - 1
[1,0,1,1,1,0,0,1,1,0,1,0,0,1,0,0]
 => [[.,.],[[[.,[.,.]],[[.,.],.]],.]]
 => [6,7,4,3,5,8,1,2] => ?
 => ? = 6 - 1
[1,0,1,1,1,0,0,1,1,1,0,1,0,0,0,0]
 => [[.,.],[[.,[.,.]],[.,[[.,.],.]]]]
 => [7,8,6,4,3,5,1,2] => ?
 => ? = 2 - 1

Description

The sum of the areas of the rectangles formed by two consecutive peaks and the valley in between. For a Dyck path

$D = D_1 \cdots D_{2n}$ with peaks in positions

$i_1 < \ldots < i_k$ and valleys in positions

$j_1 < \ldots < j_{k-1}$ , this statistic is given by

$\sum_{a=1}^{k-1} (j_a-i_a)(i_{a+1}-j_a)$

Matching statistic: St000383

Mapping

Mp00028: Dyck paths —reverse⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00100: Dyck paths —touch composition⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 56% ●values known / values provided: 56%●distinct values known / distinct values provided: 77%

Values

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

Description

The last part of an integer composition.

Matching statistic: St000025

Mapping

Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
St000025: Dyck paths ⟶ ℤResult quality: 32% ●values known / values provided: 32%●distinct values known / distinct values provided: 62%

Values

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

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: St000026
(load all 2 compositions to match this statistic)

Mapping

Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
St000026: Dyck paths ⟶ ℤResult quality: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 54%

Values

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

Description

The position of the first return of a Dyck path.

Matching statistic: St001135

Mapping

Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St001135: Dyck paths ⟶ ℤResult quality: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 54%

Values

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

Description

The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path.

Matching statistic: St000234

Mapping

Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000234: Permutations ⟶ ℤResult quality: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 54%

Values

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

Description

The number of global ascents of a permutation. The global ascents are the integers

$i$ such that

$C(\pi)=\{i\in [n-1] \mid \forall 1 \leq j \leq i < k \leq n: \pi(j) < \pi(k)\}.$ Equivalently, by the pigeonhole principle,

$C(\pi)=\{i\in [n-1] \mid \forall 1 \leq j \leq i: \pi(j) \leq i \}.$ For

$n > 1$ it can also be described as an occurrence of the mesh pattern

$([1,2], \{(0,2),(1,0),(1,1),(2,0),(2,1) \})$ or equivalently

$([1,2], \{(0,1),(0,2),(1,1),(1,2),(2,0) \}),$ see [3]. According to [2], this is also the cardinality of the connectivity set of a permutation. The permutation is connected, when the connectivity set is empty. This gives [[oeis:A003319]].

Matching statistic: St001461

Mapping

Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St001461: Permutations ⟶ ℤResult quality: 23% ●values known / values provided: 23%●distinct values known / distinct values provided: 54%

Values

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

Description

The number of topologically connected components of the chord diagram of a permutation. The chord diagram of a permutation

$\pi\in\mathfrak S_n$ is obtained by placing labels

$1,\dots,n$ in cyclic order on a cycle and drawing a (straight) arc from

$i$ to

$\pi(i)$ for every label

$i$ . This statistic records the number of topologically connected components in the chord diagram. In particular, if two arcs cross, all four labels connected by the two arcs are in the same component. The permutation

$\pi\in\mathfrak S_n$ stabilizes an interval

$I=\{a,a+1,\dots,b\}$ if

$\pi(I)=I$ . It is stabilized-interval-free, if the only interval

$\pi$ stablizes is

$\{1,\dots,n\}$ . Thus, this statistic is

$1$ if

$\pi$ is stabilized-interval-free.

The following 12 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000501The size of the first part in the decomposition of a permutation. St000740The last entry of a permutation. St000335The difference of lower and upper interactions. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St000056The decomposition (or block) number of a permutation. St000990The first ascent of a permutation. St001201The grade of the simple module

$S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St000260The radius of a connected graph. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St000199The column of the unique '1' in the last row of the alternating sign matrix. St001713The difference of the first and last value in the first row of the Gelfand-Tsetlin pattern.