Your data matches 36 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000025
(load all 7 compositions to match this statistic)
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St000025: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1,0,1,0]
 => [1,1,0,0]
 => 2
[2]
 => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 1
[1,1]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 2
[3]
 => [1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => 1
[2,1]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 3
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => 2
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 2
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,0,0]
 => 3
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 2
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 1
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 2
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => 1
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 2
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 3
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 3
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0]
 => 2
[5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => 2
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => 1
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => 2
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 1
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 4
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => 2
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 1
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => 3
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => 3
[5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,0,1,1,0,1,0,1,1,0,0,0]
 => 1
[5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,1,0,0]
 => 2
[4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1
[4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 3
[4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => 2
[3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => 2
[3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1
[3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 4
[3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,0,1,1,0,1,0,1,0,0]
 => 2
[2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 3
[2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,1,1,0,0,1,0,1,0,0,1,0]
 => 3
[2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
 => 3
[5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,1,1,0,0,0]
 => 1
[5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [1,1,1,0,1,0,1,1,0,0,0,0]
 => 3
[5,1,1,1]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,0,1,0,1,1,0,0]
 => 2
[4,4]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0]
 => 1
[4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 2
[4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 1
[4,2,1,1]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 3
[4,1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,1,0,0,1,0,1,1,0,1,0,0]
 => 2
[3,3,2]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => 1
[3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 2
[3,2,2,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 4
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: St000678
(load all 6 compositions to match this statistic)
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00028: Dyck paths —reverse⟶ Dyck paths
St000678: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1,0,1,0]
 => [1,0,1,0]
 => 2
[2]
 => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 1
[1,1]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 2
[3]
 => [1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => 1
[2,1]
 => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => 3
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0]
 => 2
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => 2
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 3
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 2
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 1
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => 2
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => 1
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => 2
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => 3
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => 3
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 2
[5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => 2
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => 1
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => 2
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 1
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => 4
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => 2
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 1
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => 3
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => 3
[5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => 1
[5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => 2
[4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1
[4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 3
[4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => 2
[3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => 2
[3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => 1
[3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => 4
[3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => 2
[2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 3
[2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => 3
[2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => 3
[5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [1,0,1,1,0,1,1,1,0,0,0,0]
 => 1
[5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => 3
[5,1,1,1]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => 2
[4,4]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => 1
[4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 2
[4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 1
[4,2,1,1]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => 3
[4,1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => 2
[3,3,2]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => 1
[3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 2
[3,2,2,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => 4
Description
The number of up steps after the last double rise of a Dyck path.
Matching statistic: St000439
(load all 7 compositions to match this statistic)
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St000439: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1,0,1,0]
 => [1,1,0,0]
 => 3 = 2 + 1
[2]
 => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 2 = 1 + 1
[1,1]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 3 = 2 + 1
[3]
 => [1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => 2 = 1 + 1
[2,1]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 4 = 3 + 1
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => 3 = 2 + 1
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 2 = 1 + 1
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 3 = 2 + 1
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 2 = 1 + 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,0,0]
 => 4 = 3 + 1
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 3 = 2 + 1
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 2 = 1 + 1
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 3 = 2 + 1
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => 2 = 1 + 1
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 3 = 2 + 1
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 4 = 3 + 1
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 4 = 3 + 1
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0]
 => 3 = 2 + 1
[5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => 3 = 2 + 1
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => 2 = 1 + 1
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => 3 = 2 + 1
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 2 = 1 + 1
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 5 = 4 + 1
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => 3 = 2 + 1
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 2 = 1 + 1
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => 4 = 3 + 1
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => 4 = 3 + 1
[5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,0,1,1,0,1,0,1,1,0,0,0]
 => 2 = 1 + 1
[5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,1,0,0]
 => 3 = 2 + 1
[4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 2 = 1 + 1
[4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 4 = 3 + 1
[4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => 3 = 2 + 1
[3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => 3 = 2 + 1
[3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => 2 = 1 + 1
[3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 5 = 4 + 1
[3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,0,1,1,0,1,0,1,0,0]
 => 3 = 2 + 1
[2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 4 = 3 + 1
[2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,1,1,0,0,1,0,1,0,0,1,0]
 => 4 = 3 + 1
[2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
 => 4 = 3 + 1
[5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,1,1,0,0,0]
 => 2 = 1 + 1
[5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [1,1,1,0,1,0,1,1,0,0,0,0]
 => 4 = 3 + 1
[5,1,1,1]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,0,1,0,1,1,0,0]
 => 3 = 2 + 1
[4,4]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0]
 => 2 = 1 + 1
[4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 3 = 2 + 1
[4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 2 = 1 + 1
[4,2,1,1]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 4 = 3 + 1
[4,1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,1,0,0,1,0,1,1,0,1,0,0]
 => 3 = 2 + 1
[3,3,2]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => 2 = 1 + 1
[3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 3 = 2 + 1
[3,2,2,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 5 = 4 + 1
Description
The position of the first down step of a Dyck path.
Matching statistic: St000011
(load all 6 compositions to match this statistic)
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1,0,1,0]
 => [1,1,0,0]
 => [1,0,1,0]
 => 2
[2]
 => [1,1,0,0,1,0]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 1
[1,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => [1,1,0,0,1,0]
 => 2
[3]
 => [1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 1
[2,1]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 3
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 2
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 1
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 2
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0]
 => 3
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => 2
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => 1
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => 2
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 1
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 2
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => 3
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => 3
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0]
 => 2
[5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,1,0,0]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => 2
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 1
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 2
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 1
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 4
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => 2
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 1
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 3
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0,1,0]
 => 3
[5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,0,1,0]
 => [1,1,1,1,0,1,0,0,1,0,0,0]
 => 1
[5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,1,0,0]
 => [1,0,1,1,1,0,0,1,1,0,0,0]
 => 2
[4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 1
[4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 3
[4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => 2
[3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 2
[3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 1
[3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 4
[3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,1,0,1,0,0]
 => 2
[2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => 3
[2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,1,1,0,1,0,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0,1,1,0,0]
 => 3
[2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,1,0,1,0,0,1,0]
 => 3
[5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,1,0,0,0]
 => 1
[5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [1,1,0,0,1,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,1,0,0,0,1,0]
 => 3
[5,1,1,1]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => [1,0,1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => 2
[4,4]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => 1
[4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => 2
[4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 1
[4,2,1,1]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => 3
[4,1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,1,0,1,0,0]
 => 2
[3,3,2]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 1
[3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 2
[3,2,2,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 4
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: St000326
(load all 2 compositions to match this statistic)
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1,0,1,0]
 => [1,2] => 0 => 2
[2]
 => [1,1,0,0,1,0]
 => [2,1,3] => 10 => 1
[1,1]
 => [1,0,1,1,0,0]
 => [1,3,2] => 01 => 2
[3]
 => [1,1,1,0,0,0,1,0]
 => [3,2,1,4] => 110 => 1
[2,1]
 => [1,0,1,0,1,0]
 => [1,2,3] => 00 => 3
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,4,3,2] => 011 => 2
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [4,3,2,1,5] => 1110 => 1
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [2,3,1,4] => 010 => 2
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [2,1,4,3] => 101 => 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,3,4,2] => 001 => 3
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,5,4,3,2] => 0111 => 2
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [5,4,3,2,1,6] => 11110 => 1
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [3,4,2,1,5] => 0110 => 2
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [2,1,3,4] => 100 => 1
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,3,2,4] => 010 => 2
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => 001 => 3
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,3,2] => 0011 => 3
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,6,5,4,3,2] => 01111 => 2
[5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [4,5,3,2,1,6] => 01110 => 2
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [3,2,4,1,5] => 1010 => 1
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [2,4,3,1,5] => 0110 => 2
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [3,2,1,5,4] => 1101 => 1
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => 000 => 4
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,4,3,5,2] => 0101 => 2
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => 1011 => 1
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => 0011 => 3
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,5,6,4,3,2] => 00111 => 3
[5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [4,3,5,2,1,6] => 10110 => 1
[5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => [3,5,4,2,1,6] => 01110 => 2
[4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [3,2,1,4,5] => 1100 => 1
[4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => 0010 => 3
[4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,4,3,2,5] => 0110 => 2
[3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => 0101 => 2
[3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => 1001 => 1
[3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => 0001 => 4
[3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,5,4,6,3,2] => 01011 => 2
[2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => 0011 => 3
[2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,4,6,5,3,2] => 00111 => 3
[2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,6,7,5,4,3,2] => 001111 => 3
[5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [4,3,2,5,1,6] => 11010 => 1
[5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [3,4,5,2,1,6] => 00110 => 3
[5,1,1,1]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => [2,5,4,3,1,6] => 01110 => 2
[4,4]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [4,3,2,1,6,5] => 11101 => 1
[4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => 0100 => 2
[4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => 1010 => 1
[4,2,1,1]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => 0010 => 3
[4,1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,5,4,3,6,2] => 01101 => 2
[3,3,2]
 => [1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => 1001 => 1
[3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => 0101 => 2
[3,2,2,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => 0001 => 4
Description
The position of the first one in a binary word after appending a 1 at the end. Regarding the binary word as a subset of $\{1,\dots,n,n+1\}$ that contains $n+1$, this is the minimal element of the set.
Matching statistic: St000382
(load all 2 compositions to match this statistic)
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1,0,1,0]
 => [1,2] => [2] => 2
[2]
 => [1,1,0,0,1,0]
 => [2,1,3] => [1,2] => 1
[1,1]
 => [1,0,1,1,0,0]
 => [1,3,2] => [2,1] => 2
[3]
 => [1,1,1,0,0,0,1,0]
 => [3,2,1,4] => [1,1,2] => 1
[2,1]
 => [1,0,1,0,1,0]
 => [1,2,3] => [3] => 3
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,4,3,2] => [2,1,1] => 2
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [4,3,2,1,5] => [1,1,1,2] => 1
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [2,3,1,4] => [2,2] => 2
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [1,2,1] => 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [3,1] => 3
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,5,4,3,2] => [2,1,1,1] => 2
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [5,4,3,2,1,6] => [1,1,1,1,2] => 1
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [3,4,2,1,5] => [2,1,2] => 2
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [1,3] => 1
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [2,2] => 2
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [3,1] => 3
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,3,2] => [3,1,1] => 3
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,6,5,4,3,2] => [2,1,1,1,1] => 2
[5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [4,5,3,2,1,6] => [2,1,1,2] => 2
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [3,2,4,1,5] => [1,2,2] => 1
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [2,4,3,1,5] => [2,1,2] => 2
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [3,2,1,5,4] => [1,1,2,1] => 1
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [4] => 4
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,4,3,5,2] => [2,2,1] => 2
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => [1,2,1,1] => 1
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => [3,1,1] => 3
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,5,6,4,3,2] => [3,1,1,1] => 3
[5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [4,3,5,2,1,6] => [1,2,1,2] => 1
[5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => [3,5,4,2,1,6] => [2,1,1,2] => 2
[4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [3,2,1,4,5] => [1,1,3] => 1
[4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => [3,2] => 3
[4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,4,3,2,5] => [2,1,2] => 2
[3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => [2,2,1] => 2
[3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => [1,3,1] => 1
[3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => [4,1] => 4
[3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,5,4,6,3,2] => [2,2,1,1] => 2
[2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => [3,1,1] => 3
[2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,4,6,5,3,2] => [3,1,1,1] => 3
[2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,6,7,5,4,3,2] => [3,1,1,1,1] => 3
[5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [4,3,2,5,1,6] => [1,1,2,2] => 1
[5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [3,4,5,2,1,6] => [3,1,2] => 3
[5,1,1,1]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => [2,5,4,3,1,6] => [2,1,1,2] => 2
[4,4]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [4,3,2,1,6,5] => [1,1,1,2,1] => 1
[4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => [2,3] => 2
[4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => [1,2,2] => 1
[4,2,1,1]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => [3,2] => 3
[4,1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,5,4,3,6,2] => [2,1,2,1] => 2
[3,3,2]
 => [1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => [1,3,1] => 1
[3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => [2,2,1] => 2
[3,2,2,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => [4,1] => 4
Description
The first part of an integer composition.
Matching statistic: St000654
(load all 12 compositions to match this statistic)
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00137: Dyck paths —to symmetric ASM⟶ Alternating sign matrices
Mp00002: Alternating sign matrices —to left key permutation⟶ Permutations
St000654: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1,0,1,0]
 => [[1,0],[0,1]]
 => [1,2] => 2
[2]
 => [1,1,0,0,1,0]
 => [[0,1,0],[1,0,0],[0,0,1]]
 => [2,1,3] => 1
[1,1]
 => [1,0,1,1,0,0]
 => [[1,0,0],[0,0,1],[0,1,0]]
 => [1,3,2] => 2
[3]
 => [1,1,1,0,0,0,1,0]
 => [[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
 => [3,2,1,4] => 1
[2,1]
 => [1,0,1,0,1,0]
 => [[1,0,0],[0,1,0],[0,0,1]]
 => [1,2,3] => 3
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
 => [1,4,3,2] => 2
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1]]
 => [4,3,2,1,5] => 1
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
 => [1,3,2,4] => 2
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
 => [2,1,4,3] => 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
 => [1,2,4,3] => 3
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0]]
 => [1,5,4,3,2] => 2
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1]]
 => [5,4,3,2,1,6] => 1
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [[0,0,1,0,0],[0,1,-1,1,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
 => [1,4,3,2,5] => 2
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
 => [2,1,3,4] => 1
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
 => [1,3,2,4] => 2
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
 => [1,2,4,3] => 3
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,1,-1,1,0],[0,0,1,0,0]]
 => [1,2,5,4,3] => 3
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0]]
 => [1,6,5,4,3,2] => 2
[5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [[0,0,0,1,0,0],[0,0,1,-1,1,0],[0,1,-1,1,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1]]
 => [1,5,4,3,2,6] => 2
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [[0,0,1,0,0],[0,1,0,0,0],[1,0,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [2,1,4,3,5] => 1
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [[0,1,0,0,0],[1,-1,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
 => [1,4,3,2,5] => 2
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => [3,2,1,5,4] => 1
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
 => [1,2,3,4] => 4
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
 => [1,3,2,5,4] => 2
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0]]
 => [2,1,5,4,3] => 1
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,0,1],[0,0,0,1,0],[0,0,1,0,0]]
 => [1,2,5,4,3] => 3
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,-1,1],[0,0,1,-1,1,0],[0,1,-1,1,0,0],[0,0,1,0,0,0]]
 => [1,2,6,5,4,3] => 3
[5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,-1,1,0],[1,0,-1,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1]]
 => [2,1,5,4,3,6] => 1
[5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => [[0,0,1,0,0,0],[0,1,-1,0,1,0],[1,-1,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1]]
 => [1,5,4,3,2,6] => 2
[4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => [3,2,1,4,5] => 1
[4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [1,2,4,3,5] => 3
[4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
 => [1,4,3,2,5] => 2
[3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => [1,3,2,5,4] => 2
[3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [2,1,3,5,4] => 1
[3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [1,2,3,5,4] => 4
[3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,-1,1],[0,1,0,-1,1,0],[0,0,0,1,0,0]]
 => [1,3,2,6,5,4] => 2
[2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0]]
 => [1,2,5,4,3] => 3
[2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,-1,0,1],[0,1,-1,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]
 => [1,2,6,5,4,3] => 3
[2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [[1,0,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,-1,1],[0,0,0,1,-1,1,0],[0,0,1,-1,1,0,0],[0,1,-1,1,0,0,0],[0,0,1,0,0,0,0]]
 => [1,2,7,6,5,4,3] => 3
[5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,-1,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
 => [3,2,1,5,4,6] => 1
[5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [[0,0,1,0,0,0],[0,1,-1,1,0,0],[1,-1,1,-1,1,0],[0,1,-1,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1]]
 => [1,2,5,4,3,6] => 3
[5,1,1,1]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => [[0,1,0,0,0,0],[1,-1,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1]]
 => [1,5,4,3,2,6] => 2
[4,4]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
 => [4,3,2,1,6,5] => 1
[4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => [[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => [1,3,2,4,5] => 2
[4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [2,1,4,3,5] => 1
[4,2,1,1]
 => [1,0,1,1,0,1,0,0,1,0]
 => [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [1,2,4,3,5] => 3
[4,1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,-1,1],[0,0,0,0,1,0]]
 => [1,4,3,2,6,5] => 2
[3,3,2]
 => [1,1,0,0,1,0,1,1,0,0]
 => [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => [2,1,3,5,4] => 1
[3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,0]
 => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => [1,3,2,5,4] => 2
[3,2,2,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [1,2,3,5,4] => 4
Description
The first descent of a permutation. For a permutation $\pi$ of $\{1,\ldots,n\}$, this is the smallest index $0 < i \leq n$ such that $\pi(i) > \pi(i+1)$ where one considers $\pi(n+1)=0$.
Matching statistic: St000745
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
St000745: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1,0,1,0]
 => [2,1] => [[1],[2]]
 => 2
[2]
 => [1,1,0,0,1,0]
 => [3,1,2] => [[1,2],[3]]
 => 1
[1,1]
 => [1,0,1,1,0,0]
 => [2,3,1] => [[1,3],[2]]
 => 2
[3]
 => [1,1,1,0,0,0,1,0]
 => [4,1,2,3] => [[1,2,3],[4]]
 => 1
[2,1]
 => [1,0,1,0,1,0]
 => [3,2,1] => [[1],[2],[3]]
 => 3
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => [[1,3,4],[2]]
 => 2
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [5,1,2,3,4] => [[1,2,3,4],[5]]
 => 1
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [4,2,1,3] => [[1,3],[2],[4]]
 => 2
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [3,4,1,2] => [[1,2],[3,4]]
 => 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [3,2,4,1] => [[1,4],[2],[3]]
 => 3
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [[1,3,4,5],[2]]
 => 2
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [6,1,2,3,4,5] => [[1,2,3,4,5],[6]]
 => 1
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [5,2,1,3,4] => [[1,3,4],[2],[5]]
 => 2
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [4,3,1,2] => [[1,2],[3],[4]]
 => 1
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [4,2,3,1] => [[1,3],[2],[4]]
 => 2
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [3,4,2,1] => [[1,4],[2],[3]]
 => 3
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => [[1,4,5],[2],[3]]
 => 3
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => [[1,3,4,5,6],[2]]
 => 2
[5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [6,2,1,3,4,5] => [[1,3,4,5],[2],[6]]
 => 2
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [5,3,1,2,4] => [[1,2,4],[3],[5]]
 => 1
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [5,2,3,1,4] => [[1,3,4],[2],[5]]
 => 2
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [4,5,1,2,3] => [[1,2,3],[4,5]]
 => 1
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [4,3,2,1] => [[1],[2],[3],[4]]
 => 4
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [4,2,3,5,1] => [[1,3,5],[2],[4]]
 => 2
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => [[1,2,5],[3,4]]
 => 1
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [3,4,2,5,1] => [[1,4,5],[2],[3]]
 => 3
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [3,2,4,5,6,1] => [[1,4,5,6],[2],[3]]
 => 3
[5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [6,3,1,2,4,5] => [[1,2,4,5],[3],[6]]
 => 1
[5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => [6,2,3,1,4,5] => [[1,3,4,5],[2],[6]]
 => 2
[4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [5,4,1,2,3] => [[1,2,3],[4],[5]]
 => 1
[4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,4] => [[1,4],[2],[3],[5]]
 => 3
[4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => [[1,3,4],[2],[5]]
 => 2
[3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [4,5,2,1,3] => [[1,3],[2,5],[4]]
 => 2
[3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [4,3,5,1,2] => [[1,2],[3,5],[4]]
 => 1
[3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,5,1] => [[1,5],[2],[3],[4]]
 => 4
[3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [4,2,3,5,6,1] => [[1,3,5,6],[2],[4]]
 => 2
[2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => [[1,4,5],[2],[3]]
 => 3
[2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [3,4,2,5,6,1] => [[1,4,5,6],[2],[3]]
 => 3
[2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [3,2,4,5,6,7,1] => [[1,4,5,6,7],[2],[3]]
 => 3
[5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [6,4,1,2,3,5] => [[1,2,3,5],[4],[6]]
 => 1
[5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [6,3,2,1,4,5] => [[1,4,5],[2],[3],[6]]
 => 3
[5,1,1,1]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => [6,2,3,4,1,5] => [[1,3,4,5],[2],[6]]
 => 2
[4,4]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [5,6,1,2,3,4] => [[1,2,3,4],[5,6]]
 => 1
[4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,3] => [[1,3],[2],[4],[5]]
 => 2
[4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [5,3,4,1,2] => [[1,2],[3,4],[5]]
 => 1
[4,2,1,1]
 => [1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,4,1] => [[1,4],[2],[3],[5]]
 => 3
[4,1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [5,2,3,4,6,1] => [[1,3,4,6],[2],[5]]
 => 2
[3,3,2]
 => [1,1,0,0,1,0,1,1,0,0]
 => [4,5,3,1,2] => [[1,2],[3,5],[4]]
 => 1
[3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,0]
 => [4,5,2,3,1] => [[1,3],[2,5],[4]]
 => 2
[3,2,2,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [4,3,5,2,1] => [[1,5],[2],[3],[4]]
 => 4
Description
The index of the last row whose first entry is the row number in a standard Young tableau.
Matching statistic: St000971
(load all 2 compositions to match this statistic)
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000971: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1,0,1,0]
 => [1,1,0,0]
 => {{1,2}}
 => 2
[2]
 => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => {{1},{2,3}}
 => 1
[1,1]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => {{1,2},{3}}
 => 2
[3]
 => [1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => {{1},{2},{3,4}}
 => 1
[2,1]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => {{1,2,3}}
 => 3
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => {{1,2},{3},{4}}
 => 2
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => {{1},{2},{3},{4,5}}
 => 1
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => {{1,3,4},{2}}
 => 2
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => {{1},{2,3},{4}}
 => 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,0,0]
 => {{1,4},{2,3}}
 => 3
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => {{1,2},{3},{4},{5}}
 => 2
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => {{1},{2},{3},{4},{5,6}}
 => 1
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => {{1,4,5},{2},{3}}
 => 2
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => {{1},{2,3,4}}
 => 1
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => {{1,2},{3,4}}
 => 2
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => {{1,2,3},{4}}
 => 3
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => {{1,5},{2,3},{4}}
 => 3
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0]
 => {{1,2},{3},{4},{5},{6}}
 => 2
[5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => {{1,5,6},{2},{3},{4}}
 => 2
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => {{1},{2,4,5},{3}}
 => 1
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => {{1,3},{2},{4,5}}
 => 2
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => {{1},{2},{3,4},{5}}
 => 1
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => {{1,2,3,4}}
 => 4
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => {{1,2},{3,5},{4}}
 => 2
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => {{1},{2,3},{4},{5}}
 => 1
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => {{1,4},{2,3},{5}}
 => 3
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => {{1,6},{2,3},{4},{5}}
 => 3
[5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,0,1,1,0,1,0,1,1,0,0,0]
 => {{1},{2,5,6},{3},{4}}
 => 1
[5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,1,0,0]
 => {{1,4},{2},{3},{5,6}}
 => 2
[4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => {{1},{2},{3,4,5}}
 => 1
[4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => {{1,2,4,5},{3}}
 => 3
[4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => {{1,2},{3},{4,5}}
 => 2
[3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => {{1,3,4},{2},{5}}
 => 2
[3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => {{1},{2,5},{3,4}}
 => 1
[3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => {{1,2,5},{3,4}}
 => 4
[3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,0,1,1,0,1,0,1,0,0]
 => {{1,2},{3,6},{4},{5}}
 => 2
[2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => {{1,2,3},{4},{5}}
 => 3
[2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,1,1,0,0,1,0,1,0,0,1,0]
 => {{1,5},{2,3},{4},{6}}
 => 3
[2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
 => {{1,7},{2,3},{4},{5},{6}}
 => 3
[5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,1,1,0,0,0]
 => {{1},{2},{3,5,6},{4}}
 => 1
[5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [1,1,1,0,1,0,1,1,0,0,0,0]
 => {{1,2,5,6},{3},{4}}
 => 3
[5,1,1,1]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,0,1,0,1,1,0,0]
 => {{1,3},{2},{4},{5,6}}
 => 2
[4,4]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0]
 => {{1},{2},{3},{4,5},{6}}
 => 1
[4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => {{1,3,4,5},{2}}
 => 2
[4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => {{1},{2,3},{4,5}}
 => 1
[4,2,1,1]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => {{1,4,5},{2,3}}
 => 3
[4,1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,1,0,0,1,0,1,1,0,1,0,0]
 => {{1,2},{3},{4,6},{5}}
 => 2
[3,3,2]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => {{1},{2,3,4},{5}}
 => 1
[3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => {{1,2},{3,4},{5}}
 => 2
[3,2,2,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => {{1,5},{2,3,4}}
 => 4
Description
The smallest closer of a set partition. A closer (or right hand endpoint) of a set partition is a number that is maximal in its block. For this statistic, singletons are considered as closers. In other words, this is the smallest among the maximal elements of the blocks.
Matching statistic: St001050
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00028: Dyck paths —reverse⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St001050: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1,0,1,0]
 => [1,0,1,0]
 => {{1},{2}}
 => 2
[2]
 => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => {{1},{2,3}}
 => 1
[1,1]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => {{1,2},{3}}
 => 2
[3]
 => [1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => {{1},{2,3,4}}
 => 1
[2,1]
 => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => {{1},{2},{3}}
 => 3
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0]
 => {{1,2,3},{4}}
 => 2
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => {{1},{2,3,4,5}}
 => 1
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => {{1},{2,4},{3}}
 => 2
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => {{1,2},{3,4}}
 => 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => {{1,3},{2},{4}}
 => 3
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => {{1,2,3,4},{5}}
 => 2
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => {{1},{2,3,4,5,6}}
 => 1
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => {{1},{2,3,5},{4}}
 => 2
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => {{1},{2},{3,4}}
 => 1
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => {{1},{2,3},{4}}
 => 2
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => {{1,2},{3},{4}}
 => 3
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => {{1,2,4},{3},{5}}
 => 3
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => {{1,2,3,4,5},{6}}
 => 2
[5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => {{1},{2,3,4,6},{5}}
 => 2
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => {{1},{2,4,5},{3}}
 => 1
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => {{1},{2,5},{3,4}}
 => 2
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => {{1,2},{3,4,5}}
 => 1
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => {{1},{2},{3},{4}}
 => 4
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => {{1,3,4},{2},{5}}
 => 2
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => {{1,2,3},{4,5}}
 => 1
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => {{1,4},{2,3},{5}}
 => 3
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => {{1,2,3,5},{4},{6}}
 => 3
[5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => {{1},{2,3,5,6},{4}}
 => 1
[5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => {{1},{2,3,6},{4,5}}
 => 2
[4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => {{1},{2},{3,4,5}}
 => 1
[4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => {{1},{2,5},{3},{4}}
 => 3
[4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => {{1},{2,3,4},{5}}
 => 2
[3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => {{1,2},{3,5},{4}}
 => 2
[3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => {{1,3},{2},{4,5}}
 => 1
[3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => {{1,4},{2},{3},{5}}
 => 4
[3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => {{1,2,4,5},{3},{6}}
 => 2
[2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => {{1,2,3},{4},{5}}
 => 3
[2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => {{1,2,5},{3,4},{6}}
 => 3
[2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => {{1,2,3,4,6},{5},{7}}
 => 3
[5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [1,0,1,1,0,1,1,1,0,0,0,0]
 => {{1},{2,4,5,6},{3}}
 => 1
[5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => {{1},{2,3,6},{4},{5}}
 => 3
[5,1,1,1]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => {{1},{2,6},{3,4,5}}
 => 2
[4,4]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => {{1,2},{3,4,5,6}}
 => 1
[4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => {{1},{2},{3,5},{4}}
 => 2
[4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => {{1},{2,3},{4,5}}
 => 1
[4,2,1,1]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => {{1},{2,4},{3},{5}}
 => 3
[4,1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => {{1,3,4,5},{2},{6}}
 => 2
[3,3,2]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => {{1,2},{3},{4,5}}
 => 1
[3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => {{1,2},{3,4},{5}}
 => 2
[3,2,2,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => {{1,3},{2},{4},{5}}
 => 4
Description
The number of terminal closers of a set partition. A closer of a set partition is a number that is maximal in its block. In particular, a singleton is a closer. This statistic counts the number of terminal closers. In other words, this is the number of closers such that all larger elements are also closers.
The following 26 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000069The number of maximal elements of a poset. St000068The number of minimal elements in a poset. St000759The smallest missing part in an integer partition. St000740The last entry of a permutation. St000989The number of final rises of a permutation. St000054The first entry of the permutation. St000007The number of saliances of the permutation. St000066The column of the unique '1' in the first row of the alternating sign matrix. St000297The number of leading ones in a binary word. St000738The first entry in the last row of a standard tableau. St000838The number of terminal right-hand endpoints when the vertices are written in order. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St000061The number of nodes on the left branch of a binary tree. St000314The number of left-to-right-maxima of a permutation. St000542The number of left-to-right-minima of a permutation. St000990The first ascent of a permutation. St000991The number of right-to-left minima of a permutation. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St000051The size of the left subtree of a binary tree. St000133The "bounce" of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000338The number of pixed points of a permutation.