Your data matches 58 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000674
(load all 47 compositions to match this statistic)
Mapping
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00142: Dyck paths —promotion⟶ Dyck paths
St000674: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1,1,0,0]
 => [1,0,1,0]
 => 2
[1,0,1,0]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => 3
[1,1,0,0]
 => [1,1,1,0,0,0]
 => [1,0,1,1,0,0]
 => 1
[1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => 4
[1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => 2
[1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 2
[1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [1,0,1,1,0,1,0,0]
 => 1
[1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => 1
[1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
[1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 3
[1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 3
[1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 2
[1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 2
[1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 3
[1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 1
[1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => 2
[1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 1
[1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => 1
[1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => 2
[1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1
[1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => 1
[1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 6
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 4
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0]
 => 4
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => 3
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 3
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => 4
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => 2
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,0,1,0]
 => 3
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => 2
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,1,1,0,0,0]
 => 2
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => [1,0,1,0,1,1,1,0,0,0,1,0]
 => 3
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => 2
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => 2
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => 2
[1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0]
 => 4
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => 2
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => 2
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,1,0,1,0,0,0]
 => [1,0,1,1,0,0,1,1,0,1,0,0]
 => 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,1,1,0,0,0]
 => 1
[1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0,1,0]
 => 3
[1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0,1,1,0,0]
 => 1
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => [1,0,1,1,0,1,0,1,0,0,1,0]
 => 2
[1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => 1
[1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,1,0,1,1,0,0,0,0]
 => [1,0,1,1,0,1,0,1,1,0,0,0]
 => 1
[1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => [1,0,1,1,0,1,1,0,0,0,1,0]
 => 2
[1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,1,0,0,0]
 => [1,0,1,1,0,1,1,0,0,1,0,0]
 => 1
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => [1,0,1,1,0,1,1,0,1,0,0,0]
 => 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,0,1,1,0,1,1,1,0,0,0,0]
 => 1
Description
The number of hills of a Dyck path. A hill is a peak with up step starting and down step ending at height zero.
Matching statistic: St000475
(load all 5 compositions to match this statistic)
Mapping
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000475: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => [1]
 => 1 = 2 - 1
[1,0,1,0]
 => [1,1] => [1,1]
 => 2 = 3 - 1
[1,1,0,0]
 => [2] => [2]
 => 0 = 1 - 1
[1,0,1,0,1,0]
 => [1,1,1] => [1,1,1]
 => 3 = 4 - 1
[1,0,1,1,0,0]
 => [1,2] => [2,1]
 => 1 = 2 - 1
[1,1,0,0,1,0]
 => [2,1] => [2,1]
 => 1 = 2 - 1
[1,1,0,1,0,0]
 => [3] => [3]
 => 0 = 1 - 1
[1,1,1,0,0,0]
 => [3] => [3]
 => 0 = 1 - 1
[1,0,1,0,1,0,1,0]
 => [1,1,1,1] => [1,1,1,1]
 => 4 = 5 - 1
[1,0,1,0,1,1,0,0]
 => [1,1,2] => [2,1,1]
 => 2 = 3 - 1
[1,0,1,1,0,0,1,0]
 => [1,2,1] => [2,1,1]
 => 2 = 3 - 1
[1,0,1,1,0,1,0,0]
 => [1,3] => [3,1]
 => 1 = 2 - 1
[1,0,1,1,1,0,0,0]
 => [1,3] => [3,1]
 => 1 = 2 - 1
[1,1,0,0,1,0,1,0]
 => [2,1,1] => [2,1,1]
 => 2 = 3 - 1
[1,1,0,0,1,1,0,0]
 => [2,2] => [2,2]
 => 0 = 1 - 1
[1,1,0,1,0,0,1,0]
 => [3,1] => [3,1]
 => 1 = 2 - 1
[1,1,0,1,0,1,0,0]
 => [4] => [4]
 => 0 = 1 - 1
[1,1,0,1,1,0,0,0]
 => [4] => [4]
 => 0 = 1 - 1
[1,1,1,0,0,0,1,0]
 => [3,1] => [3,1]
 => 1 = 2 - 1
[1,1,1,0,0,1,0,0]
 => [4] => [4]
 => 0 = 1 - 1
[1,1,1,0,1,0,0,0]
 => [4] => [4]
 => 0 = 1 - 1
[1,1,1,1,0,0,0,0]
 => [4] => [4]
 => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1] => [1,1,1,1,1]
 => 5 = 6 - 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,2] => [2,1,1,1]
 => 3 = 4 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,2,1] => [2,1,1,1]
 => 3 = 4 - 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,3] => [3,1,1]
 => 2 = 3 - 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,3] => [3,1,1]
 => 2 = 3 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,2,1,1] => [2,1,1,1]
 => 3 = 4 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,2,2] => [2,2,1]
 => 1 = 2 - 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,1] => [3,1,1]
 => 2 = 3 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,4] => [4,1]
 => 1 = 2 - 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,4] => [4,1]
 => 1 = 2 - 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,3,1] => [3,1,1]
 => 2 = 3 - 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,4] => [4,1]
 => 1 = 2 - 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,4] => [4,1]
 => 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,4] => [4,1]
 => 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,1,1] => [2,1,1,1]
 => 3 = 4 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,2] => [2,2,1]
 => 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [2,2,1] => [2,2,1]
 => 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
 => [2,3] => [3,2]
 => 0 = 1 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,3] => [3,2]
 => 0 = 1 - 1
[1,1,0,1,0,0,1,0,1,0]
 => [3,1,1] => [3,1,1]
 => 2 = 3 - 1
[1,1,0,1,0,0,1,1,0,0]
 => [3,2] => [3,2]
 => 0 = 1 - 1
[1,1,0,1,0,1,0,0,1,0]
 => [4,1] => [4,1]
 => 1 = 2 - 1
[1,1,0,1,0,1,0,1,0,0]
 => [5] => [5]
 => 0 = 1 - 1
[1,1,0,1,0,1,1,0,0,0]
 => [5] => [5]
 => 0 = 1 - 1
[1,1,0,1,1,0,0,0,1,0]
 => [4,1] => [4,1]
 => 1 = 2 - 1
[1,1,0,1,1,0,0,1,0,0]
 => [5] => [5]
 => 0 = 1 - 1
[1,1,0,1,1,0,1,0,0,0]
 => [5] => [5]
 => 0 = 1 - 1
[1,1,0,1,1,1,0,0,0,0]
 => [5] => [5]
 => 0 = 1 - 1
Description
The number of parts equal to 1 in a partition.
Matching statistic: St000678
(load all 6 compositions to match this statistic)
Mapping
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
St000678: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1,1,0,0]
 => [1,0,1,0]
 => [1,0,1,0]
 => 2
[1,0,1,0]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => 3
[1,1,0,0]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => 1
[1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => 4
[1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => 2
[1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => 2
[1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => 1
[1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => 1
[1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
[1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 3
[1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => 3
[1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => 2
[1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 2
[1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 3
[1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => 1
[1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => 2
[1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1
[1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 1
[1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 2
[1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => 1
[1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1
[1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 6
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => 4
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,1,0,0,1,0]
 => 4
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => 3
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => 3
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,1,0,1,0,0,1,0,1,0]
 => 4
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,1,0,1,1,0,0,1,0,0]
 => 2
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,1,0,0]
 => [1,0,1,1,1,0,1,0,0,0,1,0]
 => [1,0,1,1,1,0,1,0,0,0,1,0]
 => 3
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => 2
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => 2
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => [1,0,1,1,0,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0,1,0]
 => 3
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => [1,0,1,1,0,1,1,0,1,0,0,0]
 => [1,0,1,1,0,1,1,0,1,0,0,0]
 => 2
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => 2
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => 2
[1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0]
 => 4
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,1,0,0,0]
 => [1,1,0,1,0,0,1,1,0,1,0,0]
 => [1,1,0,1,0,0,1,1,0,1,0,0]
 => 2
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0,1,0]
 => 2
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => [1,0,1,1,1,0,0,1,1,0,0,0]
 => 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0,1,1,0,0]
 => 1
[1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0,1,0]
 => 3
[1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,1,0,0]
 => 1
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [1,0,1,1,1,1,0,0,0,0,1,0]
 => 2
[1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 1
[1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,1,0,1,1,0,0,0,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => [1,0,1,1,1,0,0,0,1,1,0,0]
 => 1
[1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => 2
[1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,1,0,0,0]
 => [1,1,1,0,1,1,0,0,1,0,0,0]
 => [1,1,0,1,0,0,1,1,1,0,0,0]
 => 1
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,1,1,1,0,1,0,0,1,0,0,0]
 => [1,0,1,1,0,0,1,1,1,0,0,0]
 => 1
Description
The number of up steps after the last double rise of a Dyck path.
Matching statistic: St000445
(load all 5 compositions to match this statistic)
Mapping
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000445: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => [1] => [1,0]
 => 1 = 2 - 1
[1,0,1,0]
 => [1,2] => [1,2] => [1,0,1,0]
 => 2 = 3 - 1
[1,1,0,0]
 => [2,1] => [2,1] => [1,1,0,0]
 => 0 = 1 - 1
[1,0,1,0,1,0]
 => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
 => 3 = 4 - 1
[1,0,1,1,0,0]
 => [1,3,2] => [1,3,2] => [1,0,1,1,0,0]
 => 1 = 2 - 1
[1,1,0,0,1,0]
 => [2,1,3] => [2,1,3] => [1,1,0,0,1,0]
 => 1 = 2 - 1
[1,1,0,1,0,0]
 => [2,3,1] => [3,1,2] => [1,1,1,0,0,0]
 => 0 = 1 - 1
[1,1,1,0,0,0]
 => [3,2,1] => [3,2,1] => [1,1,1,0,0,0]
 => 0 = 1 - 1
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => 4 = 5 - 1
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
 => 2 = 3 - 1
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
 => 2 = 3 - 1
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
 => 1 = 2 - 1
[1,0,1,1,1,0,0,0]
 => [1,4,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => 1 = 2 - 1
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
 => 2 = 3 - 1
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
 => 0 = 1 - 1
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
 => 1 = 2 - 1
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
 => 0 = 1 - 1
[1,1,0,1,1,0,0,0]
 => [2,4,3,1] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
 => 0 = 1 - 1
[1,1,1,0,0,0,1,0]
 => [3,2,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => 1 = 2 - 1
[1,1,1,0,0,1,0,0]
 => [3,2,4,1] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
 => 0 = 1 - 1
[1,1,1,0,1,0,0,0]
 => [4,2,3,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => 0 = 1 - 1
[1,1,1,1,0,0,0,0]
 => [4,3,2,1] => [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] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => 5 = 6 - 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
 => 3 = 4 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
 => 3 = 4 - 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
 => 2 = 3 - 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
 => 2 = 3 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
 => 3 = 4 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
 => 1 = 2 - 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => [1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
 => 2 = 3 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => [1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
 => 1 = 2 - 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => [1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
 => 1 = 2 - 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,3,2,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
 => 2 = 3 - 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,3,5,2] => [1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
 => 1 = 2 - 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,5,3,4,2] => [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
 => 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,4,3,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
 => 3 = 4 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
 => 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
 => 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => [2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0]
 => 0 = 1 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
 => 0 = 1 - 1
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => [3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0]
 => 2 = 3 - 1
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => [3,1,2,5,4] => [1,1,1,0,0,0,1,1,0,0]
 => 0 = 1 - 1
[1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => [4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0]
 => 1 = 2 - 1
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,4,1] => [5,1,2,4,3] => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,3,1,5] => [4,1,3,2,5] => [1,1,1,1,0,0,0,0,1,0]
 => 1 = 2 - 1
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,3,5,1] => [5,1,3,2,4] => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[1,1,0,1,1,0,1,0,0,0]
 => [2,5,3,4,1] => [5,1,3,4,2] => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,4,3,1] => [5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
Description
The number of rises of length 1 of a Dyck path.
Matching statistic: St000986
(load all 2 compositions to match this statistic)
Mapping
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000986: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => [1] => ([],1)
 => 1 = 2 - 1
[1,0,1,0]
 => [1,1] => [2] => ([],2)
 => 2 = 3 - 1
[1,1,0,0]
 => [2] => [1,1] => ([(0,1)],2)
 => 0 = 1 - 1
[1,0,1,0,1,0]
 => [1,1,1] => [3] => ([],3)
 => 3 = 4 - 1
[1,0,1,1,0,0]
 => [1,2] => [1,2] => ([(1,2)],3)
 => 1 = 2 - 1
[1,1,0,0,1,0]
 => [2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 1 = 2 - 1
[1,1,0,1,0,0]
 => [3] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 0 = 1 - 1
[1,1,1,0,0,0]
 => [3] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 0 = 1 - 1
[1,0,1,0,1,0,1,0]
 => [1,1,1,1] => [4] => ([],4)
 => 4 = 5 - 1
[1,0,1,0,1,1,0,0]
 => [1,1,2] => [1,3] => ([(2,3)],4)
 => 2 = 3 - 1
[1,0,1,1,0,0,1,0]
 => [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
 => 2 = 3 - 1
[1,0,1,1,0,1,0,0]
 => [1,3] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[1,0,1,1,1,0,0,0]
 => [1,3] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[1,1,0,0,1,0,1,0]
 => [2,1,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 3 - 1
[1,1,0,0,1,1,0,0]
 => [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 0 = 1 - 1
[1,1,0,1,0,0,1,0]
 => [3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[1,1,0,1,0,1,0,0]
 => [4] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0 = 1 - 1
[1,1,0,1,1,0,0,0]
 => [4] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0 = 1 - 1
[1,1,1,0,0,0,1,0]
 => [3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[1,1,1,0,0,1,0,0]
 => [4] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0 = 1 - 1
[1,1,1,0,1,0,0,0]
 => [4] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0 = 1 - 1
[1,1,1,1,0,0,0,0]
 => [4] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1] => [5] => ([],5)
 => 5 = 6 - 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,2] => [1,4] => ([(3,4)],5)
 => 3 = 4 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,2,1] => [2,3] => ([(2,4),(3,4)],5)
 => 3 = 4 - 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,3] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => 2 = 3 - 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,3] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => 2 = 3 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,2,1,1] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3 = 4 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,2,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 2 - 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 3 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,4] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 2 - 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,4] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 2 - 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,3,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 3 - 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,4] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 2 - 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,4] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,4] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3 = 4 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
 => [2,3] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0 = 1 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,3] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0 = 1 - 1
[1,1,0,1,0,0,1,0,1,0]
 => [3,1,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 3 - 1
[1,1,0,1,0,0,1,1,0,0]
 => [3,2] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0 = 1 - 1
[1,1,0,1,0,1,0,0,1,0]
 => [4,1] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 2 - 1
[1,1,0,1,0,1,0,1,0,0]
 => [5] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0 = 1 - 1
[1,1,0,1,0,1,1,0,0,0]
 => [5] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0 = 1 - 1
[1,1,0,1,1,0,0,0,1,0]
 => [4,1] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 2 - 1
[1,1,0,1,1,0,0,1,0,0]
 => [5] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0 = 1 - 1
[1,1,0,1,1,0,1,0,0,0]
 => [5] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0 = 1 - 1
[1,1,0,1,1,1,0,0,0,0]
 => [5] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0 = 1 - 1
Description
The multiplicity of the eigenvalue zero of the adjacency matrix of the graph.
Matching statistic: St001126
(load all 2 compositions to match this statistic)
Mapping
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001126: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => [1] => [1,0]
 => 1 = 2 - 1
[1,0,1,0]
 => [1,1] => [2] => [1,1,0,0]
 => 2 = 3 - 1
[1,1,0,0]
 => [2] => [1,1] => [1,0,1,0]
 => 0 = 1 - 1
[1,0,1,0,1,0]
 => [1,1,1] => [3] => [1,1,1,0,0,0]
 => 3 = 4 - 1
[1,0,1,1,0,0]
 => [1,2] => [1,2] => [1,0,1,1,0,0]
 => 1 = 2 - 1
[1,1,0,0,1,0]
 => [2,1] => [2,1] => [1,1,0,0,1,0]
 => 1 = 2 - 1
[1,1,0,1,0,0]
 => [3] => [1,1,1] => [1,0,1,0,1,0]
 => 0 = 1 - 1
[1,1,1,0,0,0]
 => [3] => [1,1,1] => [1,0,1,0,1,0]
 => 0 = 1 - 1
[1,0,1,0,1,0,1,0]
 => [1,1,1,1] => [4] => [1,1,1,1,0,0,0,0]
 => 4 = 5 - 1
[1,0,1,0,1,1,0,0]
 => [1,1,2] => [1,3] => [1,0,1,1,1,0,0,0]
 => 2 = 3 - 1
[1,0,1,1,0,0,1,0]
 => [1,2,1] => [2,2] => [1,1,0,0,1,1,0,0]
 => 2 = 3 - 1
[1,0,1,1,0,1,0,0]
 => [1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 1 = 2 - 1
[1,0,1,1,1,0,0,0]
 => [1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 1 = 2 - 1
[1,1,0,0,1,0,1,0]
 => [2,1,1] => [3,1] => [1,1,1,0,0,0,1,0]
 => 2 = 3 - 1
[1,1,0,0,1,1,0,0]
 => [2,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 0 = 1 - 1
[1,1,0,1,0,0,1,0]
 => [3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 1 = 2 - 1
[1,1,0,1,0,1,0,0]
 => [4] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[1,1,0,1,1,0,0,0]
 => [4] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[1,1,1,0,0,0,1,0]
 => [3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 1 = 2 - 1
[1,1,1,0,0,1,0,0]
 => [4] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[1,1,1,0,1,0,0,0]
 => [4] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[1,1,1,1,0,0,0,0]
 => [4] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1] => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 5 = 6 - 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,2] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 3 = 4 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,2,1] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 3 = 4 - 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,3] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 2 = 3 - 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,3] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 2 = 3 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,2,1,1] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 3 = 4 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,2,2] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 1 = 2 - 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 2 = 3 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,4] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 1 = 2 - 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,4] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 1 = 2 - 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,3,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 2 = 3 - 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,4] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 1 = 2 - 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,4] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,4] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,1,1] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 3 = 4 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [2,2,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
 => [2,3] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 0 = 1 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,3] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 0 = 1 - 1
[1,1,0,1,0,0,1,0,1,0]
 => [3,1,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 2 = 3 - 1
[1,1,0,1,0,0,1,1,0,0]
 => [3,2] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 0 = 1 - 1
[1,1,0,1,0,1,0,0,1,0]
 => [4,1] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 1 = 2 - 1
[1,1,0,1,0,1,0,1,0,0]
 => [5] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[1,1,0,1,0,1,1,0,0,0]
 => [5] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[1,1,0,1,1,0,0,0,1,0]
 => [4,1] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 1 = 2 - 1
[1,1,0,1,1,0,0,1,0,0]
 => [5] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[1,1,0,1,1,0,1,0,0,0]
 => [5] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[1,1,0,1,1,1,0,0,0,0]
 => [5] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
Description
Number of simple module that are 1-regular in the corresponding Nakayama algebra.
Matching statistic: St000247
(load all 9 compositions to match this statistic)
Mapping
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
St000247: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => {{1}}
 => ? = 2 - 1
[1,0,1,0]
 => [1,2] => {{1},{2}}
 => 2 = 3 - 1
[1,1,0,0]
 => [2,1] => {{1,2}}
 => 0 = 1 - 1
[1,0,1,0,1,0]
 => [1,2,3] => {{1},{2},{3}}
 => 3 = 4 - 1
[1,0,1,1,0,0]
 => [1,3,2] => {{1},{2,3}}
 => 1 = 2 - 1
[1,1,0,0,1,0]
 => [2,1,3] => {{1,2},{3}}
 => 1 = 2 - 1
[1,1,0,1,0,0]
 => [2,3,1] => {{1,2,3}}
 => 0 = 1 - 1
[1,1,1,0,0,0]
 => [3,1,2] => {{1,2,3}}
 => 0 = 1 - 1
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => {{1},{2},{3},{4}}
 => 4 = 5 - 1
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => {{1},{2},{3,4}}
 => 2 = 3 - 1
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => {{1},{2,3},{4}}
 => 2 = 3 - 1
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => {{1},{2,3,4}}
 => 1 = 2 - 1
[1,0,1,1,1,0,0,0]
 => [1,4,2,3] => {{1},{2,3,4}}
 => 1 = 2 - 1
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => {{1,2},{3},{4}}
 => 2 = 3 - 1
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => {{1,2},{3,4}}
 => 0 = 1 - 1
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => {{1,2,3},{4}}
 => 1 = 2 - 1
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => {{1,2,3,4}}
 => 0 = 1 - 1
[1,1,0,1,1,0,0,0]
 => [2,4,1,3] => {{1,2,3,4}}
 => 0 = 1 - 1
[1,1,1,0,0,0,1,0]
 => [3,1,2,4] => {{1,2,3},{4}}
 => 1 = 2 - 1
[1,1,1,0,0,1,0,0]
 => [3,1,4,2] => {{1,2,3,4}}
 => 0 = 1 - 1
[1,1,1,0,1,0,0,0]
 => [3,4,1,2] => {{1,3},{2,4}}
 => 0 = 1 - 1
[1,1,1,1,0,0,0,0]
 => [4,1,2,3] => {{1,2,3,4}}
 => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
 => 5 = 6 - 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => {{1},{2},{3},{4,5}}
 => 3 = 4 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => {{1},{2},{3,4},{5}}
 => 3 = 4 - 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => {{1},{2},{3,4,5}}
 => 2 = 3 - 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,3,4] => {{1},{2},{3,4,5}}
 => 2 = 3 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => {{1},{2,3},{4},{5}}
 => 3 = 4 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => {{1},{2,3},{4,5}}
 => 1 = 2 - 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => {{1},{2,3,4},{5}}
 => 2 = 3 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => {{1},{2,3,4,5}}
 => 1 = 2 - 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,2,4] => {{1},{2,3,4,5}}
 => 1 = 2 - 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,2,3,5] => {{1},{2,3,4},{5}}
 => 2 = 3 - 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,2,5,3] => {{1},{2,3,4,5}}
 => 1 = 2 - 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,2,3] => {{1},{2,4},{3,5}}
 => 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,2,3,4] => {{1},{2,3,4,5}}
 => 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => {{1,2},{3},{4},{5}}
 => 3 = 4 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => {{1,2},{3},{4,5}}
 => 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => {{1,2},{3,4},{5}}
 => 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => {{1,2},{3,4,5}}
 => 0 = 1 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,3,4] => {{1,2},{3,4,5}}
 => 0 = 1 - 1
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => {{1,2,3},{4},{5}}
 => 2 = 3 - 1
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => {{1,2,3},{4,5}}
 => 0 = 1 - 1
[1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => {{1,2,3,4},{5}}
 => 1 = 2 - 1
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => {{1,2,3,4,5}}
 => 0 = 1 - 1
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,1,4] => {{1,2,3,4,5}}
 => 0 = 1 - 1
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,1,3,5] => {{1,2,3,4},{5}}
 => 1 = 2 - 1
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,1,5,3] => {{1,2,3,4,5}}
 => 0 = 1 - 1
[1,1,0,1,1,0,1,0,0,0]
 => [2,4,5,1,3] => {{1,2,4},{3,5}}
 => 0 = 1 - 1
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,1,3,4] => {{1,2,3,4,5}}
 => 0 = 1 - 1
[1,1,1,0,0,0,1,0,1,0]
 => [3,1,2,4,5] => {{1,2,3},{4},{5}}
 => 2 = 3 - 1
Description
The number of singleton blocks of a set partition.
Matching statistic: St000248
Mapping
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
Mp00221: Set partitions —conjugate⟶ Set partitions
St000248: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => {{1}}
 => {{1}}
 => ? = 2 - 1
[1,0,1,0]
 => [1,2] => {{1},{2}}
 => {{1,2}}
 => 2 = 3 - 1
[1,1,0,0]
 => [2,1] => {{1,2}}
 => {{1},{2}}
 => 0 = 1 - 1
[1,0,1,0,1,0]
 => [1,2,3] => {{1},{2},{3}}
 => {{1,2,3}}
 => 3 = 4 - 1
[1,0,1,1,0,0]
 => [1,3,2] => {{1},{2,3}}
 => {{1,3},{2}}
 => 1 = 2 - 1
[1,1,0,0,1,0]
 => [2,1,3] => {{1,2},{3}}
 => {{1,2},{3}}
 => 1 = 2 - 1
[1,1,0,1,0,0]
 => [2,3,1] => {{1,2,3}}
 => {{1},{2},{3}}
 => 0 = 1 - 1
[1,1,1,0,0,0]
 => [3,1,2] => {{1,2,3}}
 => {{1},{2},{3}}
 => 0 = 1 - 1
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => {{1},{2},{3},{4}}
 => {{1,2,3,4}}
 => 4 = 5 - 1
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => {{1},{2},{3,4}}
 => {{1,3,4},{2}}
 => 2 = 3 - 1
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => {{1},{2,3},{4}}
 => {{1,2,4},{3}}
 => 2 = 3 - 1
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => {{1},{2,3,4}}
 => {{1,4},{2},{3}}
 => 1 = 2 - 1
[1,0,1,1,1,0,0,0]
 => [1,4,2,3] => {{1},{2,3,4}}
 => {{1,4},{2},{3}}
 => 1 = 2 - 1
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => {{1,2},{3},{4}}
 => {{1,2,3},{4}}
 => 2 = 3 - 1
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => {{1,2},{3,4}}
 => {{1,3},{2},{4}}
 => 0 = 1 - 1
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => {{1,2,3},{4}}
 => {{1,2},{3},{4}}
 => 1 = 2 - 1
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => {{1,2,3,4}}
 => {{1},{2},{3},{4}}
 => 0 = 1 - 1
[1,1,0,1,1,0,0,0]
 => [2,4,1,3] => {{1,2,3,4}}
 => {{1},{2},{3},{4}}
 => 0 = 1 - 1
[1,1,1,0,0,0,1,0]
 => [3,1,2,4] => {{1,2,3},{4}}
 => {{1,2},{3},{4}}
 => 1 = 2 - 1
[1,1,1,0,0,1,0,0]
 => [3,1,4,2] => {{1,2,3,4}}
 => {{1},{2},{3},{4}}
 => 0 = 1 - 1
[1,1,1,0,1,0,0,0]
 => [3,4,1,2] => {{1,3},{2,4}}
 => {{1,3},{2,4}}
 => 0 = 1 - 1
[1,1,1,1,0,0,0,0]
 => [4,1,2,3] => {{1,2,3,4}}
 => {{1},{2},{3},{4}}
 => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
 => {{1,2,3,4,5}}
 => 5 = 6 - 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => {{1},{2},{3},{4,5}}
 => {{1,3,4,5},{2}}
 => 3 = 4 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => {{1},{2},{3,4},{5}}
 => {{1,2,4,5},{3}}
 => 3 = 4 - 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => {{1},{2},{3,4,5}}
 => {{1,4,5},{2},{3}}
 => 2 = 3 - 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,3,4] => {{1},{2},{3,4,5}}
 => {{1,4,5},{2},{3}}
 => 2 = 3 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => {{1},{2,3},{4},{5}}
 => {{1,2,3,5},{4}}
 => 3 = 4 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => {{1},{2,3},{4,5}}
 => {{1,3,5},{2},{4}}
 => 1 = 2 - 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => {{1},{2,3,4},{5}}
 => {{1,2,5},{3},{4}}
 => 2 = 3 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => {{1},{2,3,4,5}}
 => {{1,5},{2},{3},{4}}
 => 1 = 2 - 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,2,4] => {{1},{2,3,4,5}}
 => {{1,5},{2},{3},{4}}
 => 1 = 2 - 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,2,3,5] => {{1},{2,3,4},{5}}
 => {{1,2,5},{3},{4}}
 => 2 = 3 - 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,2,5,3] => {{1},{2,3,4,5}}
 => {{1,5},{2},{3},{4}}
 => 1 = 2 - 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,2,3] => {{1},{2,4},{3,5}}
 => {{1,3,5},{2,4}}
 => 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,2,3,4] => {{1},{2,3,4,5}}
 => {{1,5},{2},{3},{4}}
 => 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => {{1,2},{3},{4},{5}}
 => {{1,2,3,4},{5}}
 => 3 = 4 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => {{1,2},{3},{4,5}}
 => {{1,3,4},{2},{5}}
 => 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => {{1,2},{3,4},{5}}
 => {{1,2,4},{3},{5}}
 => 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => {{1,2},{3,4,5}}
 => {{1,4},{2},{3},{5}}
 => 0 = 1 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,3,4] => {{1,2},{3,4,5}}
 => {{1,4},{2},{3},{5}}
 => 0 = 1 - 1
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => {{1,2,3},{4},{5}}
 => {{1,2,3},{4},{5}}
 => 2 = 3 - 1
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => {{1,2,3},{4,5}}
 => {{1,3},{2},{4},{5}}
 => 0 = 1 - 1
[1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => {{1,2,3,4},{5}}
 => {{1,2},{3},{4},{5}}
 => 1 = 2 - 1
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => {{1,2,3,4,5}}
 => {{1},{2},{3},{4},{5}}
 => 0 = 1 - 1
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,1,4] => {{1,2,3,4,5}}
 => {{1},{2},{3},{4},{5}}
 => 0 = 1 - 1
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,1,3,5] => {{1,2,3,4},{5}}
 => {{1,2},{3},{4},{5}}
 => 1 = 2 - 1
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,1,5,3] => {{1,2,3,4,5}}
 => {{1},{2},{3},{4},{5}}
 => 0 = 1 - 1
[1,1,0,1,1,0,1,0,0,0]
 => [2,4,5,1,3] => {{1,2,4},{3,5}}
 => {{1,3},{2,4},{5}}
 => 0 = 1 - 1
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,1,3,4] => {{1,2,3,4,5}}
 => {{1},{2},{3},{4},{5}}
 => 0 = 1 - 1
[1,1,1,0,0,0,1,0,1,0]
 => [3,1,2,4,5] => {{1,2,3},{4},{5}}
 => {{1,2,3},{4},{5}}
 => 2 = 3 - 1
Description
The number of anti-singletons of a set partition. An anti-singleton of a set partition $S$ is an index $i$ such that $i$ and $i+1$ (considered cyclically) are both in the same block of $S$. For noncrossing set partitions, this is also the number of singletons of the image of $S$ under the Kreweras complement.
Matching statistic: St001691
(load all 2 compositions to match this statistic)
Mapping
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001691: Graphs ⟶ ℤResult quality: 96% ●values known / values provided: 96%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => [1] => ([],1)
 => 1 = 2 - 1
[1,0,1,0]
 => [1,2] => [1,2] => ([],2)
 => 2 = 3 - 1
[1,1,0,0]
 => [2,1] => [2,1] => ([(0,1)],2)
 => 0 = 1 - 1
[1,0,1,0,1,0]
 => [1,2,3] => [1,2,3] => ([],3)
 => 3 = 4 - 1
[1,0,1,1,0,0]
 => [1,3,2] => [1,3,2] => ([(1,2)],3)
 => 1 = 2 - 1
[1,1,0,0,1,0]
 => [2,1,3] => [2,1,3] => ([(1,2)],3)
 => 1 = 2 - 1
[1,1,0,1,0,0]
 => [2,3,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 0 = 1 - 1
[1,1,1,0,0,0]
 => [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 0 = 1 - 1
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [1,2,3,4] => ([],4)
 => 4 = 5 - 1
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [1,2,4,3] => ([(2,3)],4)
 => 2 = 3 - 1
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
 => 2 = 3 - 1
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[1,0,1,1,1,0,0,0]
 => [1,4,3,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
 => 2 = 3 - 1
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
 => 0 = 1 - 1
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0 = 1 - 1
[1,1,0,1,1,0,0,0]
 => [2,4,3,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0 = 1 - 1
[1,1,1,0,0,0,1,0]
 => [3,2,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[1,1,1,0,0,1,0,0]
 => [3,2,4,1] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0 = 1 - 1
[1,1,1,0,1,0,0,0]
 => [3,4,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0 = 1 - 1
[1,1,1,1,0,0,0,0]
 => [4,3,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
 => 5 = 6 - 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => [1,2,3,5,4] => ([(3,4)],5)
 => 3 = 4 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => [1,2,4,3,5] => ([(3,4)],5)
 => 3 = 4 - 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => 2 = 3 - 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => 2 = 3 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => [1,3,2,4,5] => ([(3,4)],5)
 => 3 = 4 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
 => 1 = 2 - 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => 2 = 3 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => [1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 2 - 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 2 - 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,3,2,5] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => 2 = 3 - 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,3,5,2] => [1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 2 - 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,3,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,4,3,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => [2,1,3,4,5] => ([(3,4)],5)
 => 3 = 4 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => [2,1,3,5,4] => ([(1,4),(2,3)],5)
 => 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => [2,1,4,3,5] => ([(1,4),(2,3)],5)
 => 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => 0 = 1 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => 0 = 1 - 1
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
 => 2 = 3 - 1
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => [3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => 0 = 1 - 1
[1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => [4,2,3,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 2 - 1
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0 = 1 - 1
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,4,1] => [5,2,4,3,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0 = 1 - 1
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,3,1,5] => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 2 - 1
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,3,5,1] => [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0 = 1 - 1
[1,1,0,1,1,0,1,0,0,0]
 => [2,4,5,3,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0 = 1 - 1
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,4,3,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0 = 1 - 1
[1,1,0,1,0,0,1,1,0,1,0,1,0,0]
 => [2,3,1,5,6,7,4] => [3,2,1,7,5,6,4] => ([(0,1),(0,2),(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 - 1
[1,1,0,1,0,1,0,1,1,0,1,0,0,0]
 => [2,3,4,6,7,5,1] => [7,2,3,6,5,4,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 - 1
[1,1,0,1,0,1,0,1,1,1,0,0,0,0]
 => [2,3,4,7,6,5,1] => [7,2,3,6,5,4,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 - 1
[1,1,0,1,1,1,0,0,0,1,0,1,0,0]
 => [2,5,4,3,6,7,1] => [7,4,3,2,5,6,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 - 1
[1,1,1,0,0,0,1,1,0,1,0,1,0,0]
 => [3,2,1,5,6,7,4] => [3,2,1,7,5,6,4] => ([(0,1),(0,2),(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 - 1
[1,1,1,0,0,0,1,1,0,1,1,0,0,0]
 => [3,2,1,5,7,6,4] => [3,2,1,7,6,5,4] => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 - 1
[1,1,1,0,0,0,1,1,1,0,0,1,0,0]
 => [3,2,1,6,5,7,4] => [3,2,1,7,5,6,4] => ([(0,1),(0,2),(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 - 1
[1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => [3,2,1,7,6,5,4] => [3,2,1,7,6,5,4] => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 - 1
[1,1,1,0,0,1,0,0,1,1,0,1,0,0]
 => [3,2,4,1,6,7,5] => [4,2,3,1,7,6,5] => ([(0,1),(0,2),(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 - 1
[1,1,1,0,0,1,0,1,1,0,1,0,0,0]
 => [3,2,4,6,7,5,1] => [7,2,3,6,5,4,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 - 1
[1,1,1,0,0,1,0,1,1,1,0,0,0,0]
 => [3,2,4,7,6,5,1] => [7,2,3,6,5,4,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 - 1
[1,1,1,0,0,1,1,0,1,0,0,1,0,0]
 => [3,2,5,6,4,7,1] => [7,2,5,4,3,6,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 - 1
[1,1,1,0,0,1,1,1,0,0,0,1,0,0]
 => [3,2,6,5,4,7,1] => [7,2,5,4,3,6,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 - 1
[1,1,1,0,1,0,0,0,1,1,0,1,0,0]
 => [3,4,2,1,6,7,5] => [4,3,2,1,7,6,5] => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 - 1
[1,1,1,0,1,0,1,0,0,1,0,1,0,0]
 => [3,4,5,2,6,7,1] => [7,4,3,2,5,6,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 - 1
[1,1,1,0,1,1,0,0,0,1,0,1,0,0]
 => [3,5,4,2,6,7,1] => [7,4,3,2,5,6,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 - 1
[1,1,1,1,0,0,1,0,0,1,0,1,0,0]
 => [4,3,5,2,6,7,1] => [7,4,3,2,5,6,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 - 1
[1,1,1,1,0,1,0,0,0,1,0,1,0,0]
 => [4,5,3,2,6,7,1] => [7,4,3,2,5,6,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 - 1
[1,1,1,1,1,0,0,0,0,1,0,1,0,0]
 => [5,4,3,2,6,7,1] => [7,4,3,2,5,6,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 - 1
Description
The number of kings in a graph. A vertex of a graph is a king, if all its neighbours have smaller degree. In particular, an isolated vertex is a king.
Matching statistic: St000439
(load all 3 compositions to match this statistic)
Mapping
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St000439: Dyck paths ⟶ ℤResult quality: 75% ●values known / values provided: 75%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1,1,0,0]
 => [1,0,1,0]
 => [1,1,0,0]
 => 3 = 2 + 1
[1,0,1,0]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 4 = 3 + 1
[1,1,0,0]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => 2 = 1 + 1
[1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 5 = 4 + 1
[1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 3 = 2 + 1
[1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 3 = 2 + 1
[1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => 2 = 1 + 1
[1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => 2 = 1 + 1
[1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 6 = 5 + 1
[1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 4 = 3 + 1
[1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 4 = 3 + 1
[1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 3 = 2 + 1
[1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 3 = 2 + 1
[1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 4 = 3 + 1
[1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => 2 = 1 + 1
[1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => 3 = 2 + 1
[1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 2 = 1 + 1
[1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 2 = 1 + 1
[1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 3 = 2 + 1
[1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => 2 = 1 + 1
[1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 2 = 1 + 1
[1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 2 = 1 + 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 7 = 6 + 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,0,1,1,0,0,0,0,0]
 => 5 = 4 + 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,1,1,1,0,0,1,1,0,0,0,0]
 => 5 = 4 + 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,1,1,0,0,0,0]
 => 4 = 3 + 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => 4 = 3 + 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => 5 = 4 + 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,0,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => 3 = 2 + 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,1,0,0]
 => [1,0,1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,0,0,1,0,1,1,0,0,0]
 => 4 = 3 + 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => 3 = 2 + 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => 3 = 2 + 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => [1,0,1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => 4 = 3 + 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => [1,0,1,1,0,1,1,0,1,0,0,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => 3 = 2 + 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => 3 = 2 + 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => 3 = 2 + 1
[1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => 5 = 4 + 1
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,1,0,0,0]
 => [1,1,0,1,0,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,1,0,0]
 => 3 = 2 + 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0,1,0]
 => [1,1,0,0,1,1,1,0,0,1,0,0]
 => 3 = 2 + 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => [1,0,1,1,0,1,1,0,0,1,0,0]
 => 2 = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,1,0,0]
 => 4 = 3 + 1
[1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,1,0,0]
 => 2 = 1 + 1
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,1,0,0]
 => 3 = 2 + 1
[1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 2 = 1 + 1
[1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,1,0,1,1,0,0,0,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => 2 = 1 + 1
[1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,1,0,0]
 => 3 = 2 + 1
[1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,1,0,0,0]
 => [1,1,1,0,1,1,0,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => 2 = 1 + 1
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [1,0,1,1,1,0,0,0,1,1,0,0]
 => 2 = 1 + 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,1,1,1,0,1,0,0,1,0,0,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => 2 = 1 + 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,1,1,0,0,0]
 => [1,0,1,0,1,0,1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,1,0,1,1,1,0,0,1,0,0,0,0,0]
 => ? = 4 + 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,0,1,0,1,1,0,1,0,0,1,0,0]
 => [1,0,1,0,1,0,1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,1,1,0,0,1,0,1,1,0,0,0,0,0]
 => ? = 5 + 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0]
 => [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,1,0,1,0,1,0,1,1,0,0,0,0,0]
 => ? = 4 + 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,0,1,0,1,1,0,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,1,0,1,1,0,0,1,1,0,0,0,0,0]
 => ? = 4 + 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,1,0,1,0,0]
 => [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]
 => ? = 6 + 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,1,0,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0]
 => [1,1,1,1,0,1,1,0,0,0,1,1,0,0,0,0]
 => ? = 4 + 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,1,0,0,1,0,0]
 => [1,0,1,0,1,1,0,1,1,0,0,1,0,0,1,0]
 => [1,1,1,1,0,0,1,1,1,0,0,1,0,0,0,0]
 => ? = 4 + 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,0,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,0,1,1,1,0,0,1,0,0,0]
 => [1,1,1,0,1,1,0,1,1,0,0,1,0,0,0,0]
 => ? = 3 + 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
 => [1,1,0,1,0,1,1,0,1,0,0,1,0,1,0,0]
 => [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]
 => ? = 5 + 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,0,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,1,1,0,1,0,0,0,0]
 => ? = 3 + 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,1,0,1,0,0,1,0,0]
 => [1,0,1,0,1,1,1,1,0,1,0,0,0,0,1,0]
 => [1,1,1,1,0,0,1,0,1,0,1,1,0,0,0,0]
 => ? = 4 + 1
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
 => [1,1,0,1,0,1,1,0,1,0,1,1,0,0,0,0]
 => [1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,1,0,1,1,0,0,0,0]
 => ? = 3 + 1
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,1,1,0,0,0,1,0,0]
 => [1,0,1,0,1,1,1,0,1,0,0,1,0,0,1,0]
 => [1,1,1,1,0,0,1,1,0,0,1,1,0,0,0,0]
 => ? = 4 + 1
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,1,0,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,1,1,0,0,1,0,0,0]
 => [1,1,1,0,1,0,1,1,1,0,0,1,0,0,0,0]
 => ? = 3 + 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,1,0,1,1,0,1,0,0,0,0]
 => [1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]
 => [1,1,1,0,1,1,1,0,0,0,1,1,0,0,0,0]
 => ? = 3 + 1
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,1,0,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,1,1,1,0,1,0,0,1,0,0,0]
 => [1,1,1,0,1,0,1,1,0,0,1,1,0,0,0,0]
 => ? = 3 + 1
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,1,0,1,1,1,0,0,1,0,0,1,0,0]
 => [1,0,1,0,1,1,0,1,1,0,1,0,0,0,1,0]
 => [1,1,1,1,0,0,1,1,0,1,1,0,0,0,0,0]
 => ? = 4 + 1
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,1,0,1,0,1,1,1,0,0,1,0,1,0,0,0]
 => [1,0,1,0,1,1,0,1,1,1,0,1,0,0,0,0]
 => [1,1,1,0,1,1,0,1,0,1,1,0,0,0,0,0]
 => ? = 3 + 1
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,1,0,1,0,0,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,0,1,0]
 => [1,1,1,1,0,0,1,1,1,1,0,0,0,0,0,0]
 => ? = 4 + 1
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,1,0,1,0,1,1,1,0,1,0,0,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,1,1,0,1,1,0,0,0,0,0,0]
 => ? = 3 + 1
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,1,0,1,0,1,1,1,0,1,1,0,0,0,0,0]
 => [1,0,1,0,1,1,0,1,1,0,1,0,1,0,0,0]
 => [1,1,1,0,1,1,0,1,1,1,0,0,0,0,0,0]
 => ? = 3 + 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,1,1,1,0,0,0,0,1,0,0]
 => [1,0,1,0,1,1,1,0,1,0,1,0,0,0,1,0]
 => [1,1,1,1,0,0,1,0,1,1,1,0,0,0,0,0]
 => ? = 4 + 1
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,1,0,1,0,1,1,1,1,0,0,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0]
 => [1,1,1,0,1,0,1,1,0,1,1,0,0,0,0,0]
 => ? = 3 + 1
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,1,0,1,1,1,1,0,1,0,0,0,0,0]
 => [1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]
 => [1,1,1,0,1,0,1,0,1,1,1,0,0,0,0,0]
 => ? = 3 + 1
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,0,1,0,1,1,1,1,1,0,0,0,0,0,0]
 => [1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,0,1,0,1,1,1,1,0,0,0,0,0,0]
 => ? = 3 + 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,1,0,1,0,0]
 => [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]
 => ? = 6 + 1
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,0,1,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,0,1,1,0,0,0,0,1,1,0,0,0]
 => ? = 4 + 1
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,1,0,0,1,0,0]
 => [1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,0]
 => [1,1,1,1,0,0,1,1,0,0,0,1,1,0,0,0]
 => ? = 4 + 1
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,1,0,1,1,0,1,0,0,0]
 => [1,0,1,1,0,1,0,0,1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,1,1,0,0,0,1,1,0,0,0]
 => ? = 3 + 1
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,0,0,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,0,1,0,0,1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,1,1,0,0,0,0,1,1,0,0,0]
 => ? = 3 + 1
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,1,0,0,1,0,1,0,0]
 => [1,0,1,1,0,1,1,0,0,1,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,1,1,0,0,1,0,0,0]
 => ? = 4 + 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,0,1,1,0,0,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,1,0,0,1,0,0,1,0,0,0]
 => ? = 2 + 1
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,1,0,1,0,0,1,0,0]
 => [1,0,1,1,0,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,1,0,0,1,1,0,1,1,0,0,1,0,0,0]
 => ? = 3 + 1
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,1,0,1,0,0,0]
 => [1,0,1,1,0,1,1,1,1,0,0,1,0,0,0,0]
 => [1,1,0,1,1,0,1,0,1,1,0,0,1,0,0,0]
 => ? = 2 + 1
[1,0,1,1,0,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,0,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,0,1,1,1,0,0,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,1,0,0,1,0,0,0]
 => ? = 2 + 1
[1,0,1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,1,1,1,0,0,0,1,0,0]
 => [1,0,1,1,0,1,1,0,0,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,1,1,1,0,0,0,1,0,0,0]
 => ? = 3 + 1
[1,0,1,1,0,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,1,1,0,0,1,0,0,0]
 => [1,0,1,1,0,1,1,0,0,1,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,1,1,0,0,0,1,0,0,0]
 => ? = 2 + 1
[1,0,1,1,0,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,0,0,1,1,1,0,1,0,0,0,0]
 => [1,0,1,1,0,1,1,0,0,1,0,1,0,1,0,0]
 => [1,1,0,1,1,1,1,1,0,0,0,0,1,0,0,0]
 => ? = 2 + 1
[1,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,0,0,1,1,1,1,0,0,0,0,0]
 => [1,0,1,1,0,1,1,1,0,0,1,0,1,0,0,0]
 => [1,1,0,1,1,0,1,1,1,0,0,0,1,0,0,0]
 => ? = 2 + 1
[1,0,1,1,0,1,0,0,1,0,1,0,1,0]
 => [1,1,0,1,1,0,1,0,0,1,0,1,0,1,0,0]
 => [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]
 => ? = 5 + 1
[1,0,1,1,0,1,0,0,1,0,1,1,0,0]
 => [1,1,0,1,1,0,1,0,0,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,1,0,0,0,1,1,0,1,0,0]
 => [1,1,1,0,1,1,0,0,0,1,0,1,1,0,0,0]
 => ? = 3 + 1
[1,0,1,1,0,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,1,1,0,0,0,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,1,0,1,0,0,0]
 => ? = 3 + 1
[1,0,1,1,0,1,0,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,1,1,0,1,0,0,0]
 => [1,0,1,1,1,0,1,1,1,0,0,0,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,1,0,1,0,0,0]
 => ? = 2 + 1
[1,0,1,1,0,1,0,0,1,1,1,0,0,0]
 => [1,1,0,1,1,0,1,0,0,1,1,1,0,0,0,0]
 => [1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
 => [1,1,0,1,1,1,0,0,0,1,1,0,1,0,0,0]
 => ? = 2 + 1
[1,0,1,1,0,1,0,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,1,0,1,0,0,1,0,1,0,0]
 => [1,0,1,1,1,1,0,1,0,0,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,1,0,1,1,0,0,0]
 => ? = 4 + 1
[1,0,1,1,0,1,0,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,1,0,1,0,0,1,1,0,0,0]
 => [1,0,1,1,1,1,0,1,1,0,0,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,0,1,1,0,1,0,0,0]
 => ? = 2 + 1
[1,0,1,1,0,1,0,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,1,0,1,0,0,1,0,0]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,1,1,0,0,0]
 => ? = 3 + 1
[1,0,1,1,0,1,0,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,0,1,0,1,1,0,0,0,0]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,0,1,0,1,1,0,0,0]
 => ? = 2 + 1
[1,0,1,1,0,1,0,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,1,0,1,1,0,0,0,1,0,0]
 => [1,0,1,1,1,1,0,1,0,0,0,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,1,1,0,0,0]
 => ? = 3 + 1
[1,0,1,1,0,1,0,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,1,0,1,1,0,0,1,0,0,0]
 => [1,0,1,1,1,1,0,1,1,0,0,0,1,0,0,0]
 => [1,1,0,1,0,1,1,0,0,1,1,0,1,0,0,0]
 => ? = 2 + 1
Description
The position of the first down step of a Dyck path.
The following 48 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000011The number of touch points (or returns) of a Dyck path. St000675The number of centered multitunnels of a Dyck path. St000022The number of fixed points of a permutation. St000895The number of ones on the main diagonal of an alternating sign matrix. St000502The number of successions of a set partitions. St000117The number of centered tunnels of a Dyck path. St000221The number of strong fixed points of a permutation. St001008Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St000164The number of short pairs. St000241The number of cyclical small excedances. St000315The number of isolated vertices of a graph. St000025The number of initial rises of a Dyck path. St000234The number of global ascents of a permutation. St001479The number of bridges of a graph. St001672The restrained domination number of a graph. St001826The maximal number of leaves on a vertex of a graph. St000237The number of small exceedances. St000456The monochromatic index of a connected graph. St000441The number of successions of a permutation. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001342The number of vertices in the center of a graph. St001368The number of vertices of maximal degree in a graph. St000007The number of saliances of the permutation. St000546The number of global descents of a permutation. St001545The second Elser number of a connected graph. St000214The number of adjacencies of a permutation. St000894The trace of an alternating sign matrix. St000717The number of ordinal summands of a poset. St001461The number of topologically connected components of the chord diagram of a permutation. St000843The decomposition number of a perfect matching. St000884The number of isolated descents of a permutation. St000313The number of degree 2 vertices of a graph. St000239The number of small weak excedances. 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. St000061The number of nodes on the left branch of a binary tree. St000084The number of subtrees. St000991The number of right-to-left minima of a permutation. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St000648The number of 2-excedences of a permutation. St000153The number of adjacent cycles of a permutation. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St001631The number of simple modules $S$ with $dim Ext^1(S,A)=1$ in the incidence algebra $A$ of the poset. St001903The number of fixed points of a parking function. St000732The number of double deficiencies of a permutation. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra