Your data matches 32 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000475
Mapping
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00102: Dyck paths —rise 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]
 => [1,1,0,0]
 => [2] => [2]
 => 0
[1,0,1,0]
 => [1,1,0,1,0,0]
 => [2,1] => [2,1]
 => 1
[1,1,0,0]
 => [1,1,1,0,0,0]
 => [3] => [3]
 => 0
[1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [2,1,1] => [2,1,1]
 => 2
[1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [2,2] => [2,2]
 => 0
[1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [3,1] => [3,1]
 => 1
[1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [3,1] => [3,1]
 => 1
[1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [4] => [4]
 => 0
[1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [2,1,1,1] => [2,1,1,1]
 => 3
[1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [2,1,2] => [2,2,1]
 => 1
[1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [2,2,1] => [2,2,1]
 => 1
[1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [2,2,1] => [2,2,1]
 => 1
[1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [2,3] => [3,2]
 => 0
[1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [3,1,1] => [3,1,1]
 => 2
[1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [3,2] => [3,2]
 => 0
[1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [3,1,1] => [3,1,1]
 => 2
[1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [3,1,1] => [3,1,1]
 => 2
[1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [3,2] => [3,2]
 => 0
[1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [4,1] => [4,1]
 => 1
[1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [4,1] => [4,1]
 => 1
[1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [4,1] => [4,1]
 => 1
[1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [5] => [5]
 => 0
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [2,1,1,1,1] => [2,1,1,1,1]
 => 4
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => [2,1,1,2] => [2,2,1,1]
 => 2
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => [2,1,2,1] => [2,2,1,1]
 => 2
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,0,0]
 => [2,1,2,1] => [2,2,1,1]
 => 2
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => [2,1,3] => [3,2,1]
 => 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => [2,2,1,1] => [2,2,1,1]
 => 2
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => [2,2,2] => [2,2,2]
 => 0
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,1,0,0]
 => [2,2,1,1] => [2,2,1,1]
 => 2
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => [2,2,1,1] => [2,2,1,1]
 => 2
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => [2,2,2] => [2,2,2]
 => 0
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => [2,3,1] => [3,2,1]
 => 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => [2,3,1] => [3,2,1]
 => 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => [2,3,1] => [3,2,1]
 => 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [2,4] => [4,2]
 => 0
[1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [3,1,1,1] => [3,1,1,1]
 => 3
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,1,0,0,0]
 => [3,1,2] => [3,2,1]
 => 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => [3,2,1] => [3,2,1]
 => 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,1,0,1,0,0,0]
 => [3,2,1] => [3,2,1]
 => 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => [3,3] => [3,3]
 => 0
[1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [3,1,1,1] => [3,1,1,1]
 => 3
[1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,1,0,0,0]
 => [3,1,2] => [3,2,1]
 => 1
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => [3,1,1,1] => [3,1,1,1]
 => 3
[1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [3,1,1,1] => [3,1,1,1]
 => 3
[1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,1,0,1,1,0,0,0,0]
 => [3,1,2] => [3,2,1]
 => 1
[1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => [3,2,1] => [3,2,1]
 => 1
[1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,1,0,0,0]
 => [3,2,1] => [3,2,1]
 => 1
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => [3,2,1] => [3,2,1]
 => 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => [3,3] => [3,3]
 => 0
Description
The number of parts equal to 1 in a partition.
Matching statistic: St000932
(load all 48 compositions to match this statistic)
Mapping
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00038: Integer compositions —reverse⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000932: Dyck paths ⟶ ℤResult quality: 71% ●values known / values provided: 71%●distinct values known / distinct values provided: 80%
Values
[1,0]
 => [1] => [1] => [1,0]
 => ? = 0
[1,0,1,0]
 => [1,1] => [1,1] => [1,0,1,0]
 => 1
[1,1,0,0]
 => [2] => [2] => [1,1,0,0]
 => 0
[1,0,1,0,1,0]
 => [1,1,1] => [1,1,1] => [1,0,1,0,1,0]
 => 2
[1,0,1,1,0,0]
 => [1,2] => [2,1] => [1,1,0,0,1,0]
 => 0
[1,1,0,0,1,0]
 => [2,1] => [1,2] => [1,0,1,1,0,0]
 => 1
[1,1,0,1,0,0]
 => [2,1] => [1,2] => [1,0,1,1,0,0]
 => 1
[1,1,1,0,0,0]
 => [3] => [3] => [1,1,1,0,0,0]
 => 0
[1,0,1,0,1,0,1,0]
 => [1,1,1,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 3
[1,0,1,0,1,1,0,0]
 => [1,1,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 1
[1,0,1,1,0,0,1,0]
 => [1,2,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 1
[1,0,1,1,0,1,0,0]
 => [1,2,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 1
[1,0,1,1,1,0,0,0]
 => [1,3] => [3,1] => [1,1,1,0,0,0,1,0]
 => 0
[1,1,0,0,1,0,1,0]
 => [2,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 2
[1,1,0,0,1,1,0,0]
 => [2,2] => [2,2] => [1,1,0,0,1,1,0,0]
 => 0
[1,1,0,1,0,0,1,0]
 => [2,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 2
[1,1,0,1,0,1,0,0]
 => [2,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 2
[1,1,0,1,1,0,0,0]
 => [2,2] => [2,2] => [1,1,0,0,1,1,0,0]
 => 0
[1,1,1,0,0,0,1,0]
 => [3,1] => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[1,1,1,0,0,1,0,0]
 => [3,1] => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[1,1,1,0,1,0,0,0]
 => [3,1] => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[1,1,1,1,0,0,0,0]
 => [4] => [4] => [1,1,1,1,0,0,0,0]
 => 0
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 4
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,2] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 2
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,2,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 2
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,2,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 2
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,2,1,1] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 2
[1,0,1,1,0,0,1,1,0,0]
 => [1,2,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 0
[1,0,1,1,0,1,0,0,1,0]
 => [1,2,1,1] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 2
[1,0,1,1,0,1,0,1,0,0]
 => [1,2,1,1] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 2
[1,0,1,1,0,1,1,0,0,0]
 => [1,2,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 0
[1,0,1,1,1,0,0,0,1,0]
 => [1,3,1] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,3,1] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,3,1] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 0
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,1,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 3
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,2] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 1
[1,1,0,0,1,1,0,0,1,0]
 => [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 1
[1,1,0,0,1,1,0,1,0,0]
 => [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 0
[1,1,0,1,0,0,1,0,1,0]
 => [2,1,1,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 3
[1,1,0,1,0,0,1,1,0,0]
 => [2,1,2] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 1
[1,1,0,1,0,1,0,0,1,0]
 => [2,1,1,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 3
[1,1,0,1,0,1,0,1,0,0]
 => [2,1,1,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 3
[1,1,0,1,0,1,1,0,0,0]
 => [2,1,2] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 1
[1,1,0,1,1,0,0,0,1,0]
 => [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 1
[1,1,0,1,1,0,0,1,0,0]
 => [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 1
[1,1,0,1,1,0,1,0,0,0]
 => [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 1
[1,1,0,1,1,1,0,0,0,0]
 => [2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 0
[1,1,1,0,0,0,1,0,1,0]
 => [3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 2
[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,1,3] => [3,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 4
[1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,4] => [4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
 => ? = 3
[1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,2,3] => [3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 2
[1,0,1,0,1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,2,3] => [3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 2
[1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,5] => [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
 => ? = 2
[1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,1,2,4] => [4,2,1,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
 => ? = 1
[1,0,1,0,1,1,0,1,0,0,1,1,1,0,0,0]
 => [1,1,2,1,3] => [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
 => ? = 2
[1,0,1,0,1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,1,2,4] => [4,2,1,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
 => ? = 1
[1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,6] => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => ? = 1
[1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
 => [1,2,2,3] => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 0
[1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => [1,2,5] => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
 => ? = 0
[1,0,1,1,0,1,0,0,1,1,1,1,0,0,0,0]
 => [1,2,1,4] => [4,1,2,1] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
 => ? = 1
[1,0,1,1,0,1,0,1,0,1,1,1,0,0,0,0]
 => [1,2,1,1,3] => [3,1,1,2,1] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 2
[1,0,1,1,0,1,0,1,1,1,1,0,0,0,0,0]
 => [1,2,1,4] => [4,1,2,1] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
 => ? = 1
[1,0,1,1,0,1,1,1,1,1,0,0,0,0,0,0]
 => [1,2,5] => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
 => ? = 0
[1,0,1,1,1,0,1,0,0,0,1,1,1,0,0,0]
 => [1,3,1,3] => [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
 => ? = 1
[1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => [1,4,3] => [3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
 => ? = 0
[1,0,1,1,1,1,0,0,0,1,1,1,0,0,0,0]
 => [1,4,3] => [3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
 => ? = 0
[1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
 => [1,4,3] => [3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
 => ? = 0
[1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,7] => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 0
[1,1,0,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [2,1,1,1,3] => [3,1,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 3
[1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,1,5] => [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
 => ? = 1
[1,1,0,0,1,1,0,1,0,1,1,1,0,0,0,0]
 => [2,2,1,3] => [3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 1
[1,1,0,0,1,1,0,1,1,1,1,0,0,0,0,0]
 => [2,2,4] => [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
 => ? = 0
[1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [2,6] => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => ? = 0
[1,1,0,1,0,0,1,0,1,1,1,1,0,0,0,0]
 => [2,1,1,4] => [4,1,1,2] => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
 => ? = 2
[1,1,0,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => [2,1,5] => [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
 => ? = 1
[1,1,0,1,0,1,0,0,1,0,1,1,1,0,0,0]
 => [2,1,1,1,3] => [3,1,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 3
[1,1,0,1,0,1,0,0,1,1,1,1,0,0,0,0]
 => [2,1,1,4] => [4,1,1,2] => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
 => ? = 2
[1,1,0,1,0,1,0,1,0,0,1,1,1,0,0,0]
 => [2,1,1,1,3] => [3,1,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 3
[1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
 => [2,1,1,1,3] => [3,1,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 3
[1,1,0,1,0,1,0,1,1,1,1,0,0,0,0,0]
 => [2,1,1,4] => [4,1,1,2] => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
 => ? = 2
[1,1,0,1,0,1,1,1,1,1,0,0,0,0,0,0]
 => [2,1,5] => [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
 => ? = 1
[1,1,0,1,1,0,1,1,1,1,0,0,0,0,0,0]
 => [2,2,4] => [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
 => ? = 0
[1,1,0,1,1,1,0,0,0,1,1,1,0,0,0,0]
 => [2,3,3] => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 0
[1,1,0,1,1,1,0,1,1,1,0,0,0,0,0,0]
 => [2,3,3] => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 0
[1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [2,6] => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => ? = 0
[1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
 => [3,1,1,3] => [3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 2
[1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
 => [3,1,4] => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
 => ? = 1
[1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
 => [3,2,3] => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 0
[1,1,1,0,0,0,1,1,0,1,1,1,0,0,0,0]
 => [3,2,3] => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 0
[1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
 => [3,5] => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
 => ? = 0
[1,1,1,0,0,1,0,0,1,1,1,1,0,0,0,0]
 => [3,1,4] => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
 => ? = 1
[1,1,1,0,0,1,0,1,0,1,1,1,0,0,0,0]
 => [3,1,1,3] => [3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 2
[1,1,1,0,0,1,1,1,1,1,0,0,0,0,0,0]
 => [3,5] => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
 => ? = 0
[1,1,1,0,1,0,0,0,1,1,1,1,0,0,0,0]
 => [3,1,4] => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
 => ? = 1
[1,1,1,0,1,0,0,1,1,1,1,0,0,0,0,0]
 => [3,1,4] => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
 => ? = 1
[1,1,1,0,1,0,1,0,0,0,1,1,1,0,0,0]
 => [3,1,1,3] => [3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 2
[1,1,1,0,1,0,1,0,0,1,1,1,0,0,0,0]
 => [3,1,1,3] => [3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 2
Description
The number of occurrences of the pattern UDU in a Dyck path. The number of Dyck paths with statistic value 0 are counted by the Motzkin numbers [1].
Matching statistic: St001067
(load all 33 compositions to match this statistic)
Mapping
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00028: Dyck paths —reverse⟶ Dyck paths
St001067: Dyck paths ⟶ ℤResult quality: 61% ●values known / values provided: 61%●distinct values known / distinct values provided: 70%
Values
[1,0]
 => []
 => []
 => []
 => ? = 0
[1,0,1,0]
 => [1]
 => [1,0,1,0]
 => [1,0,1,0]
 => 1
[1,1,0,0]
 => []
 => []
 => []
 => ? = 0
[1,0,1,0,1,0]
 => [2,1]
 => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => 2
[1,0,1,1,0,0]
 => [1,1]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 0
[1,1,0,0,1,0]
 => [2]
 => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 1
[1,1,0,1,0,0]
 => [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]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => 3
[1,0,1,0,1,1,0,0]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => 1
[1,0,1,1,0,0,1,0]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => 1
[1,0,1,1,0,1,0,0]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 1
[1,0,1,1,1,0,0,0]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0]
 => 0
[1,1,0,0,1,0,1,0]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => 2
[1,1,0,0,1,1,0,0]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 0
[1,1,0,1,0,0,1,0]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => 2
[1,1,0,1,0,1,0,0]
 => [2,1]
 => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => 2
[1,1,0,1,1,0,0,0]
 => [1,1]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 0
[1,1,1,0,0,0,1,0]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => 1
[1,1,1,0,0,1,0,0]
 => [2]
 => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 1
[1,1,1,0,1,0,0,0]
 => [1]
 => [1,0,1,0]
 => [1,0,1,0]
 => 1
[1,1,1,1,0,0,0,0]
 => []
 => []
 => []
 => ? = 0
[1,0,1,0,1,0,1,0,1,0]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 4
[1,0,1,0,1,0,1,1,0,0]
 => [3,3,2,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 2
[1,0,1,0,1,1,0,0,1,0]
 => [4,2,2,1]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 2
[1,0,1,0,1,1,0,1,0,0]
 => [3,2,2,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => 2
[1,0,1,0,1,1,1,0,0,0]
 => [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
[1,0,1,1,0,0,1,0,1,0]
 => [4,3,1,1]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 2
[1,0,1,1,0,0,1,1,0,0]
 => [3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 0
[1,0,1,1,0,1,0,0,1,0]
 => [4,2,1,1]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => 2
[1,0,1,1,0,1,0,1,0,0]
 => [3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => 2
[1,0,1,1,0,1,1,0,0,0]
 => [2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => 0
[1,0,1,1,1,0,0,0,1,0]
 => [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
[1,0,1,1,1,0,0,1,0,0]
 => [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
[1,0,1,1,1,0,1,0,0,0]
 => [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
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 0
[1,1,0,0,1,0,1,0,1,0]
 => [4,3,2]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 3
[1,1,0,0,1,0,1,1,0,0]
 => [3,3,2]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => 1
[1,1,0,0,1,1,0,0,1,0]
 => [4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 1
[1,1,0,0,1,1,0,1,0,0]
 => [3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 0
[1,1,0,1,0,0,1,0,1,0]
 => [4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 3
[1,1,0,1,0,0,1,1,0,0]
 => [3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => 1
[1,1,0,1,0,1,0,0,1,0]
 => [4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 3
[1,1,0,1,0,1,0,1,0,0]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => 3
[1,1,0,1,0,1,1,0,0,0]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => 1
[1,1,0,1,1,0,0,0,1,0]
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1
[1,1,0,1,1,0,0,1,0,0]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => 1
[1,1,0,1,1,0,1,0,0,0]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0]
 => 0
[1,1,1,0,0,0,1,0,1,0]
 => [4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 2
[1,1,1,0,0,0,1,1,0,0]
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 0
[1,1,1,0,0,1,0,0,1,0]
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => 2
[1,1,1,0,0,1,0,1,0,0]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => 2
[1,1,1,1,1,0,0,0,0,0]
 => []
 => []
 => []
 => ? = 0
[1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => []
 => []
 => ? = 0
[1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => []
 => []
 => []
 => ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [7,6,5,4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 7
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [6,6,5,4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 5
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [7,5,5,4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 5
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [6,5,5,4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 5
[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [5,5,5,4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 4
[1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [7,6,4,4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => ? = 5
[1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [6,6,4,4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => ? = 3
[1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
 => [7,5,4,4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,0,1,0,1,0,1,0,1,0]
 => ? = 5
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [6,5,4,4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
 => ? = 5
[1,0,1,0,1,0,1,0,1,1,0,1,1,0,0,0]
 => [5,5,4,4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0,1,0,1,0,1,0,1,0]
 => ? = 3
[1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => [7,4,4,4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0]
 => ? = 4
[1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [6,4,4,4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0,1,0,1,0,1,0]
 => ? = 4
[1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => [5,4,4,4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0,1,0,1,0,1,0,1,0]
 => ? = 4
[1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [4,4,4,4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
 => ? = 3
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [7,6,5,3,3,2,1]
 => ?
 => ?
 => ? = 5
[1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [6,6,5,3,3,2,1]
 => [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 3
[1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => [5,5,5,3,3,2,1]
 => ?
 => ?
 => ? = 2
[1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
 => [7,6,4,3,3,2,1]
 => ?
 => ?
 => ? = 5
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [6,5,4,3,3,2,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
 => ? = 5
[1,0,1,0,1,0,1,1,0,1,0,1,1,0,0,0]
 => [5,5,4,3,3,2,1]
 => ?
 => ?
 => ? = 3
[1,0,1,0,1,0,1,1,0,1,1,1,0,0,0,0]
 => [4,4,4,3,3,2,1]
 => ?
 => ?
 => ? = 2
[1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => [7,6,3,3,3,2,1]
 => ?
 => ?
 => ? = 4
[1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [6,5,3,3,3,2,1]
 => [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0,1,0,1,0,1,0]
 => ? = 4
[1,0,1,0,1,0,1,1,1,0,1,0,0,0,1,0]
 => [7,4,3,3,3,2,1]
 => ?
 => ?
 => ? = 4
[1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => [6,4,3,3,3,2,1]
 => [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0,1,0,1,0,1,0]
 => ? = 4
[1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
 => [5,4,3,3,3,2,1]
 => [1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0,1,0,1,0,1,0]
 => ? = 4
[1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => [7,3,3,3,3,2,1]
 => ?
 => ?
 => ? = 3
[1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => [6,3,3,3,3,2,1]
 => [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,0,0,1,0,1,0,1,0]
 => ? = 3
[1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
 => [4,3,3,3,3,2,1]
 => ?
 => ?
 => ? = 3
[1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [3,3,3,3,3,2,1]
 => [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
 => ? = 2
[1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => [7,6,5,4,2,2,1]
 => ?
 => ?
 => ? = 5
[1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [6,6,5,4,2,2,1]
 => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => ? = 3
[1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => [7,5,5,4,2,2,1]
 => ?
 => ?
 => ? = 3
[1,0,1,0,1,1,0,0,1,0,1,1,0,1,0,0]
 => [6,5,5,4,2,2,1]
 => [1,0,1,0,1,1,0,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0,1,1,0,0,1,0,1,0]
 => ? = 3
[1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [7,6,4,4,2,2,1]
 => ?
 => ?
 => ? = 3
[1,0,1,0,1,1,0,0,1,1,0,1,0,0,1,0]
 => [7,5,4,4,2,2,1]
 => ?
 => ?
 => ? = 3
[1,0,1,0,1,1,0,0,1,1,0,1,0,1,0,0]
 => [6,5,4,4,2,2,1]
 => [1,0,1,0,1,1,0,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,0,1,1,0,0,1,0,1,0]
 => ? = 3
[1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => [4,4,4,4,2,2,1]
 => ?
 => ?
 => ? = 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0]
 => [7,6,5,3,2,2,1]
 => ?
 => ?
 => ? = 5
[1,0,1,0,1,1,0,1,0,0,1,1,1,0,0,0]
 => [5,5,5,3,2,2,1]
 => ?
 => ?
 => ? = 2
[1,0,1,0,1,1,0,1,0,1,0,0,1,0,1,0]
 => [7,6,4,3,2,2,1]
 => ?
 => ?
 => ? = 5
[1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [6,5,4,3,2,2,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
 => ? = 5
[1,0,1,0,1,1,0,1,0,1,0,1,1,0,0,0]
 => [5,5,4,3,2,2,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0,1,0,1,0]
 => ? = 3
Description
The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra.
Matching statistic: St001484
Mapping
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St001484: Integer partitions ⟶ ℤResult quality: 57% ●values known / values provided: 57%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => [1,0]
 => []
 => 0
[1,0,1,0]
 => [1,1] => [1,0,1,0]
 => [1]
 => 1
[1,1,0,0]
 => [2] => [1,1,0,0]
 => []
 => 0
[1,0,1,0,1,0]
 => [1,1,1] => [1,0,1,0,1,0]
 => [2,1]
 => 2
[1,0,1,1,0,0]
 => [1,2] => [1,0,1,1,0,0]
 => [1,1]
 => 0
[1,1,0,0,1,0]
 => [2,1] => [1,1,0,0,1,0]
 => [2]
 => 1
[1,1,0,1,0,0]
 => [2,1] => [1,1,0,0,1,0]
 => [2]
 => 1
[1,1,1,0,0,0]
 => [3] => [1,1,1,0,0,0]
 => []
 => 0
[1,0,1,0,1,0,1,0]
 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [3,2,1]
 => 3
[1,0,1,0,1,1,0,0]
 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [2,2,1]
 => 1
[1,0,1,1,0,0,1,0]
 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [3,1,1]
 => 1
[1,0,1,1,0,1,0,0]
 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [3,1,1]
 => 1
[1,0,1,1,1,0,0,0]
 => [1,3] => [1,0,1,1,1,0,0,0]
 => [1,1,1]
 => 0
[1,1,0,0,1,0,1,0]
 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [3,2]
 => 2
[1,1,0,0,1,1,0,0]
 => [2,2] => [1,1,0,0,1,1,0,0]
 => [2,2]
 => 0
[1,1,0,1,0,0,1,0]
 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [3,2]
 => 2
[1,1,0,1,0,1,0,0]
 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [3,2]
 => 2
[1,1,0,1,1,0,0,0]
 => [2,2] => [1,1,0,0,1,1,0,0]
 => [2,2]
 => 0
[1,1,1,0,0,0,1,0]
 => [3,1] => [1,1,1,0,0,0,1,0]
 => [3]
 => 1
[1,1,1,0,0,1,0,0]
 => [3,1] => [1,1,1,0,0,0,1,0]
 => [3]
 => 1
[1,1,1,0,1,0,0,0]
 => [3,1] => [1,1,1,0,0,0,1,0]
 => [3]
 => 1
[1,1,1,1,0,0,0,0]
 => [4] => [1,1,1,1,0,0,0,0]
 => []
 => 0
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [4,3,2,1]
 => 4
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [3,3,2,1]
 => 2
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [4,2,2,1]
 => 2
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [4,2,2,1]
 => 2
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => [2,2,2,1]
 => 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [4,3,1,1]
 => 2
[1,0,1,1,0,0,1,1,0,0]
 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [3,3,1,1]
 => 0
[1,0,1,1,0,1,0,0,1,0]
 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [4,3,1,1]
 => 2
[1,0,1,1,0,1,0,1,0,0]
 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [4,3,1,1]
 => 2
[1,0,1,1,0,1,1,0,0,0]
 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [3,3,1,1]
 => 0
[1,0,1,1,1,0,0,0,1,0]
 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [4,1,1,1]
 => 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [4,1,1,1]
 => 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [4,1,1,1]
 => 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 0
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [4,3,2]
 => 3
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [3,3,2]
 => 1
[1,1,0,0,1,1,0,0,1,0]
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [4,2,2]
 => 1
[1,1,0,0,1,1,0,1,0,0]
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [4,2,2]
 => 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [2,2,2]
 => 0
[1,1,0,1,0,0,1,0,1,0]
 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [4,3,2]
 => 3
[1,1,0,1,0,0,1,1,0,0]
 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [3,3,2]
 => 1
[1,1,0,1,0,1,0,0,1,0]
 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [4,3,2]
 => 3
[1,1,0,1,0,1,0,1,0,0]
 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [4,3,2]
 => 3
[1,1,0,1,0,1,1,0,0,0]
 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [3,3,2]
 => 1
[1,1,0,1,1,0,0,0,1,0]
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [4,2,2]
 => 1
[1,1,0,1,1,0,0,1,0,0]
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [4,2,2]
 => 1
[1,1,0,1,1,0,1,0,0,0]
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [4,2,2]
 => 1
[1,1,0,1,1,1,0,0,0,0]
 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [2,2,2]
 => 0
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [7,5,5,4,3,2,1]
 => ? = 5
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [7,5,5,4,3,2,1]
 => ? = 5
[1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,1,2,1,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [7,6,4,4,3,2,1]
 => ? = 5
[1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,1,2,2] => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [6,6,4,4,3,2,1]
 => ? = 3
[1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,1,1,1,2,1,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [7,6,4,4,3,2,1]
 => ? = 5
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,2,1,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [7,6,4,4,3,2,1]
 => ? = 5
[1,0,1,0,1,0,1,0,1,1,0,1,1,0,0,0]
 => [1,1,1,1,2,2] => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [6,6,4,4,3,2,1]
 => ? = 3
[1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,1,3,1] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => [7,4,4,4,3,2,1]
 => ? = 4
[1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,1,3,1] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => [7,4,4,4,3,2,1]
 => ? = 4
[1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,1,3,1] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => [7,4,4,4,3,2,1]
 => ? = 4
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,2,1,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [7,6,5,3,3,2,1]
 => ? = 5
[1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,2,1,2] => [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [6,6,5,3,3,2,1]
 => ? = 3
[1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,2,3] => [1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => [5,5,5,3,3,2,1]
 => ? = 2
[1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,2,1,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [7,6,5,3,3,2,1]
 => ? = 5
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,2,1,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [7,6,5,3,3,2,1]
 => ? = 5
[1,0,1,0,1,0,1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,2,1,2] => [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [6,6,5,3,3,2,1]
 => ? = 3
[1,0,1,0,1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,2,3] => [1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => [5,5,5,3,3,2,1]
 => ? = 2
[1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,3,1,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => [7,6,3,3,3,2,1]
 => ? = 4
[1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,3,1,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => [7,6,3,3,3,2,1]
 => ? = 4
[1,0,1,0,1,0,1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,3,1,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => [7,6,3,3,3,2,1]
 => ? = 4
[1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,3,1,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => [7,6,3,3,3,2,1]
 => ? = 4
[1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,3,1,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => [7,6,3,3,3,2,1]
 => ? = 4
[1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,4,1] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => [7,3,3,3,3,2,1]
 => ? = 3
[1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,4,1] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => [7,3,3,3,3,2,1]
 => ? = 3
[1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,4,1] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => [7,3,3,3,3,2,1]
 => ? = 3
[1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,1,2,1,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => [7,6,5,4,2,2,1]
 => ? = 5
[1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,1,2,1,1,2] => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [6,6,5,4,2,2,1]
 => ? = 3
[1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,1,2,1,2,1] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => [7,5,5,4,2,2,1]
 => ? = 3
[1,0,1,0,1,1,0,0,1,0,1,1,0,1,0,0]
 => [1,1,2,1,2,1] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => [7,5,5,4,2,2,1]
 => ? = 3
[1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,2,2,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [7,6,4,4,2,2,1]
 => ? = 3
[1,0,1,0,1,1,0,0,1,1,0,1,0,0,1,0]
 => [1,1,2,2,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [7,6,4,4,2,2,1]
 => ? = 3
[1,0,1,0,1,1,0,0,1,1,0,1,0,1,0,0]
 => [1,1,2,2,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [7,6,4,4,2,2,1]
 => ? = 3
[1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,1,2,4] => [1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => [4,4,4,4,2,2,1]
 => ? = 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0]
 => [1,1,2,1,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => [7,6,5,4,2,2,1]
 => ? = 5
[1,0,1,0,1,1,0,1,0,0,1,1,1,0,0,0]
 => [1,1,2,1,3] => [1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0]
 => [5,5,5,4,2,2,1]
 => ? = 2
[1,0,1,0,1,1,0,1,0,1,0,0,1,0,1,0]
 => [1,1,2,1,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => [7,6,5,4,2,2,1]
 => ? = 5
[1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,2,1,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => [7,6,5,4,2,2,1]
 => ? = 5
[1,0,1,0,1,1,0,1,0,1,0,1,1,0,0,0]
 => [1,1,2,1,1,2] => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [6,6,5,4,2,2,1]
 => ? = 3
[1,0,1,0,1,1,0,1,1,0,0,0,1,0,1,0]
 => [1,1,2,2,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [7,6,4,4,2,2,1]
 => ? = 3
[1,0,1,0,1,1,0,1,1,0,1,0,1,0,0,0]
 => [1,1,2,2,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [7,6,4,4,2,2,1]
 => ? = 3
[1,0,1,0,1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,1,2,4] => [1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => [4,4,4,4,2,2,1]
 => ? = 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]
 => [1,1,3,1,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]
 => [7,6,5,2,2,2,1]
 => ? = 4
[1,0,1,0,1,1,1,0,0,1,0,0,1,0,1,0]
 => [1,1,3,1,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]
 => [7,6,5,2,2,2,1]
 => ? = 4
[1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,1,3,1,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]
 => [7,6,5,2,2,2,1]
 => ? = 4
[1,0,1,0,1,1,1,0,1,0,0,0,1,0,1,0]
 => [1,1,3,1,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]
 => [7,6,5,2,2,2,1]
 => ? = 4
[1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]
 => [1,1,3,1,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]
 => [7,6,5,2,2,2,1]
 => ? = 4
[1,0,1,0,1,1,1,0,1,0,1,0,0,1,0,0]
 => [1,1,3,1,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]
 => [7,6,5,2,2,2,1]
 => ? = 4
[1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,3,1,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]
 => [7,6,5,2,2,2,1]
 => ? = 4
[1,0,1,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,1,4,2] => [1,0,1,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => [6,6,2,2,2,2,1]
 => ? = 1
[1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0]
 => [1,1,4,1,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]
 => [7,6,2,2,2,2,1]
 => ? = 3
Description
The number of singletons of an integer partition. A singleton in an integer partition is a part that appear precisely once.
Matching statistic: St000445
(load all 7 compositions to match this statistic)
Mapping
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000445: Dyck paths ⟶ ℤResult quality: 57% ●values known / values provided: 57%●distinct values known / distinct values provided: 90%
Values
[1,0]
 => [1,1,0,0]
 => [2] => [1,1,0,0]
 => 0
[1,0,1,0]
 => [1,1,0,1,0,0]
 => [2,1] => [1,1,0,0,1,0]
 => 1
[1,1,0,0]
 => [1,1,1,0,0,0]
 => [3] => [1,1,1,0,0,0]
 => 0
[1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2
[1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [2,2] => [1,1,0,0,1,1,0,0]
 => 0
[1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [3,1] => [1,1,1,0,0,0,1,0]
 => 1
[1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [3,1] => [1,1,1,0,0,0,1,0]
 => 1
[1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [4] => [1,1,1,1,0,0,0,0]
 => 0
[1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 3
[1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 1
[1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 1
[1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 1
[1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 0
[1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 2
[1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 0
[1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 2
[1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 2
[1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 0
[1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 1
[1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 1
[1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 1
[1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 0
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => 4
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => 2
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => 2
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,0,0]
 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => 2
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
 => 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
 => 2
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => 0
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,1,0,0]
 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
 => 2
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
 => 2
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => 0
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
 => 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
 => 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
 => 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => 0
[1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
 => 3
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,1,0,0,0]
 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,1,0,1,0,0,0]
 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => 0
[1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
 => 3
[1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,1,0,0,0]
 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => 1
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
 => 3
[1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
 => 3
[1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,1,0,1,1,0,0,0,0]
 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => 1
[1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => 1
[1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,1,0,0,0]
 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => 1
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => 0
[1,1,0,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,1,0,0,1,0,0]
 => [3,1,1,2,1] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 3
[1,1,0,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,0,1,0,1,1,0,1,0,0,0]
 => [3,1,1,2,1] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 3
[1,1,0,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,1,0,0,1,0,1,0,0]
 => [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
 => ? = 3
[1,1,0,0,1,0,1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,0,1,1,0,1,0,0,1,0,0]
 => [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
 => ? = 3
[1,1,0,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,0,0,1,0,1,1,0,1,0,1,0,0,0]
 => [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
 => ? = 3
[1,1,0,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,0,0,1,0,1,1,1,1,0,0,0,0,0]
 => [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 1
[1,1,0,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,1,0,1,0,0]
 => [3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 3
[1,1,0,0,1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,0,0,1,1,1,0,0,0,0]
 => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 0
[1,1,0,0,1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,1,0,1,0,0,1,0,1,0,0]
 => [3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 3
[1,1,0,0,1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,1,0,1,0,0,1,0,0]
 => [3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 3
[1,1,0,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,0,1,1,0,1,0,1,0,1,0,0,0]
 => [3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 3
[1,1,0,0,1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,0,1,1,0,1,1,1,0,0,0,0,0]
 => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 0
[1,1,0,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,1,1,0,0,0]
 => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 0
[1,1,0,0,1,1,1,0,0,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,1,1,0,0,0,0]
 => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 0
[1,1,0,0,1,1,1,0,1,1,0,0,0,0]
 => [1,1,1,0,0,1,1,1,0,1,1,0,0,0,0,0]
 => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 0
[1,1,0,1,0,0,1,0,1,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,1,0,0,1,0,0]
 => [3,1,1,2,1] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 3
[1,1,0,1,0,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,1,0,1,1,0,1,0,0,0]
 => [3,1,1,2,1] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 3
[1,1,0,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,1,1,0,0,1,0,1,0,0]
 => [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
 => ? = 3
[1,1,0,1,0,0,1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,1,0,1,0,0,1,0,0]
 => [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
 => ? = 3
[1,1,0,1,0,0,1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,1,1,0,1,0,1,0,0,0]
 => [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
 => ? = 3
[1,1,0,1,0,0,1,1,1,1,0,0,0,0]
 => [1,1,1,0,1,0,0,1,1,1,1,0,0,0,0,0]
 => [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 1
[1,1,0,1,0,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,1,1,0,0,1,0,0]
 => [3,1,1,2,1] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 3
[1,1,0,1,0,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,1,1,0,1,0,0,0]
 => [3,1,1,2,1] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 3
[1,1,0,1,0,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,1,0,1,0,1,1,0,0,0,1,0,0]
 => [3,1,1,2,1] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 3
[1,1,0,1,0,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,1,0,0,1,0,0,0]
 => [3,1,1,2,1] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 3
[1,1,0,1,0,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,1,0,1,1,0,1,0,0,0,0]
 => [3,1,1,2,1] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 3
[1,1,0,1,0,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,1,0,0,0,1,0,1,0,0]
 => [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
 => ? = 3
[1,1,0,1,0,1,1,0,0,1,0,0,1,0]
 => [1,1,1,0,1,0,1,1,0,0,1,0,0,1,0,0]
 => [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
 => ? = 3
[1,1,0,1,0,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,1,0,0,1,0,1,0,0,0]
 => [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
 => ? = 3
[1,1,0,1,0,1,1,0,1,0,0,0,1,0]
 => [1,1,1,0,1,0,1,1,0,1,0,0,0,1,0,0]
 => [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
 => ? = 3
[1,1,0,1,0,1,1,0,1,0,0,1,0,0]
 => [1,1,1,0,1,0,1,1,0,1,0,0,1,0,0,0]
 => [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
 => ? = 3
[1,1,0,1,0,1,1,0,1,0,1,0,0,0]
 => [1,1,1,0,1,0,1,1,0,1,0,1,0,0,0,0]
 => [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
 => ? = 3
[1,1,0,1,0,1,1,1,1,0,0,0,0,0]
 => [1,1,1,0,1,0,1,1,1,1,0,0,0,0,0,0]
 => [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 1
[1,1,0,1,1,0,0,0,1,0,1,0,1,0]
 => [1,1,1,0,1,1,0,0,0,1,0,1,0,1,0,0]
 => [3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 3
[1,1,0,1,1,0,0,0,1,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,1,1,1,0,0,0,0]
 => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 0
[1,1,0,1,1,0,0,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,0,0,1,0,0,1,0,1,0,0]
 => [3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 3
[1,1,0,1,1,0,0,1,0,1,0,0,1,0]
 => [1,1,1,0,1,1,0,0,1,0,1,0,0,1,0,0]
 => [3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 3
[1,1,0,1,1,0,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,1,0,0,1,0,1,0,1,0,0,0]
 => [3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 3
[1,1,0,1,1,0,0,1,1,1,0,0,0,0]
 => [1,1,1,0,1,1,0,0,1,1,1,0,0,0,0,0]
 => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 0
[1,1,0,1,1,0,1,0,0,0,1,0,1,0]
 => [1,1,1,0,1,1,0,1,0,0,0,1,0,1,0,0]
 => [3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 3
[1,1,0,1,1,0,1,0,0,1,0,0,1,0]
 => [1,1,1,0,1,1,0,1,0,0,1,0,0,1,0,0]
 => [3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 3
[1,1,0,1,1,0,1,0,0,1,0,1,0,0]
 => [1,1,1,0,1,1,0,1,0,0,1,0,1,0,0,0]
 => [3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 3
[1,1,0,1,1,0,1,0,1,0,0,0,1,0]
 => [1,1,1,0,1,1,0,1,0,1,0,0,0,1,0,0]
 => [3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 3
[1,1,0,1,1,0,1,0,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,1,0,1,0,0,1,0,0,0]
 => [3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 3
[1,1,0,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,0,1,1,0,1,0,1,0,1,0,0,0,0]
 => [3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 3
[1,1,0,1,1,0,1,1,1,0,0,0,0,0]
 => [1,1,1,0,1,1,0,1,1,1,0,0,0,0,0,0]
 => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 0
[1,1,0,1,1,1,0,0,0,0,1,1,0,0]
 => [1,1,1,0,1,1,1,0,0,0,0,1,1,0,0,0]
 => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 0
[1,1,0,1,1,1,0,0,0,1,1,0,0,0]
 => [1,1,1,0,1,1,1,0,0,0,1,1,0,0,0,0]
 => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 0
[1,1,0,1,1,1,0,0,1,1,0,0,0,0]
 => [1,1,1,0,1,1,1,0,0,1,1,0,0,0,0,0]
 => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 0
[1,1,0,1,1,1,0,1,1,0,0,0,0,0]
 => [1,1,1,0,1,1,1,0,1,1,0,0,0,0,0,0]
 => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 0
Description
The number of rises of length 1 of a Dyck path.
Matching statistic: St000504
Mapping
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
Mp00171: Set partitions —intertwining number to dual major index⟶ Set partitions
St000504: Set partitions ⟶ ℤResult quality: 43% ●values known / values provided: 43%●distinct values known / distinct values provided: 80%
Values
[1,0]
 => [1,0]
 => {{1}}
 => {{1}}
 => ? = 0 + 1
[1,0,1,0]
 => [1,1,0,0]
 => {{1,2}}
 => {{1,2}}
 => 2 = 1 + 1
[1,1,0,0]
 => [1,0,1,0]
 => {{1},{2}}
 => {{1},{2}}
 => 1 = 0 + 1
[1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => {{1,2,3}}
 => {{1,2,3}}
 => 3 = 2 + 1
[1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => {{1,3},{2}}
 => {{1},{2,3}}
 => 1 = 0 + 1
[1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => {{1,2},{3}}
 => {{1,2},{3}}
 => 2 = 1 + 1
[1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => {{1},{2,3}}
 => {{1,3},{2}}
 => 2 = 1 + 1
[1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => {{1},{2},{3}}
 => {{1},{2},{3}}
 => 1 = 0 + 1
[1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => {{1,2,3,4}}
 => {{1,2,3,4}}
 => 4 = 3 + 1
[1,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => {{1,2,4},{3}}
 => {{1,2},{3,4}}
 => 2 = 1 + 1
[1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => {{1,4},{2,3}}
 => {{1,3},{2,4}}
 => 2 = 1 + 1
[1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => {{1,3,4},{2}}
 => {{1,4},{2,3}}
 => 2 = 1 + 1
[1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => {{1,4},{2},{3}}
 => {{1},{2},{3,4}}
 => 1 = 0 + 1
[1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => {{1,2,3},{4}}
 => {{1,2,3},{4}}
 => 3 = 2 + 1
[1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => {{1,3},{2},{4}}
 => {{1},{2,3},{4}}
 => 1 = 0 + 1
[1,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => {{1,2},{3,4}}
 => {{1,2,4},{3}}
 => 3 = 2 + 1
[1,1,0,1,0,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => {{1},{2,3,4}}
 => {{1,3,4},{2}}
 => 3 = 2 + 1
[1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0]
 => {{1},{2,4},{3}}
 => {{1},{2,4},{3}}
 => 1 = 0 + 1
[1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => {{1,2},{3},{4}}
 => {{1,2},{3},{4}}
 => 2 = 1 + 1
[1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => {{1},{2,3},{4}}
 => {{1,3},{2},{4}}
 => 2 = 1 + 1
[1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => {{1},{2},{3,4}}
 => {{1,4},{2},{3}}
 => 2 = 1 + 1
[1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => {{1},{2},{3},{4}}
 => {{1},{2},{3},{4}}
 => 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => {{1,2,3,4,5}}
 => {{1,2,3,4,5}}
 => 5 = 4 + 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => {{1,2,3,5},{4}}
 => {{1,2,3},{4,5}}
 => 3 = 2 + 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => {{1,2,5},{3,4}}
 => {{1,2,4},{3,5}}
 => 3 = 2 + 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => {{1,2,4,5},{3}}
 => {{1,2,5},{3,4}}
 => 3 = 2 + 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => {{1,2,5},{3},{4}}
 => {{1,2},{3},{4,5}}
 => 2 = 1 + 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => {{1,5},{2,3,4}}
 => {{1,3,4},{2,5}}
 => 3 = 2 + 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => {{1,5},{2,4},{3}}
 => {{1},{2,4},{3,5}}
 => 1 = 0 + 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}}
 => {{1,3,5},{2,4}}
 => 3 = 2 + 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => {{1,3,4,5},{2}}
 => {{1,4,5},{2,3}}
 => 3 = 2 + 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => {{1,3,5},{2},{4}}
 => {{1},{2,3,5},{4}}
 => 1 = 0 + 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => {{1,5},{2,3},{4}}
 => {{1,3},{2},{4,5}}
 => 2 = 1 + 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => {{1,5},{2},{3,4}}
 => {{1,4},{2},{3,5}}
 => 2 = 1 + 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => {{1,4,5},{2},{3}}
 => {{1,5},{2},{3,4}}
 => 2 = 1 + 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => {{1,5},{2},{3},{4}}
 => {{1},{2},{3},{4,5}}
 => 1 = 0 + 1
[1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => {{1,2,3,4},{5}}
 => {{1,2,3,4},{5}}
 => 4 = 3 + 1
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => {{1,2,4},{3},{5}}
 => {{1,2},{3,4},{5}}
 => 2 = 1 + 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => {{1,4},{2,3},{5}}
 => {{1,3},{2,4},{5}}
 => 2 = 1 + 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => {{1,3,4},{2},{5}}
 => {{1,4},{2,3},{5}}
 => 2 = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => {{1,4},{2},{3},{5}}
 => {{1},{2},{3,4},{5}}
 => 1 = 0 + 1
[1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => {{1,2,3},{4,5}}
 => {{1,2,3,5},{4}}
 => 4 = 3 + 1
[1,1,0,1,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => {{1,3},{2},{4,5}}
 => {{1,5},{2,3},{4}}
 => 2 = 1 + 1
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => {{1,2},{3,4,5}}
 => {{1,2,4,5},{3}}
 => 4 = 3 + 1
[1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => {{1},{2,3,4,5}}
 => {{1,3,4,5},{2}}
 => 4 = 3 + 1
[1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => {{1},{2,3,5},{4}}
 => {{1,3},{2,5},{4}}
 => 2 = 1 + 1
[1,1,0,1,1,0,0,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => {{1,2},{3,5},{4}}
 => {{1,2},{3,5},{4}}
 => 2 = 1 + 1
[1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => {{1},{2,5},{3,4}}
 => {{1,4},{2,5},{3}}
 => 2 = 1 + 1
[1,1,0,1,1,0,1,0,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => {{1},{2,4,5},{3}}
 => {{1,5},{2,4},{3}}
 => 2 = 1 + 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => {{1},{2,5},{3},{4}}
 => {{1},{2},{3,5},{4}}
 => 1 = 0 + 1
[1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => {{1,2,3},{4},{5}}
 => {{1,2,3},{4},{5}}
 => 3 = 2 + 1
[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
 => {{1,2,3,4,5,8},{6},{7}}
 => {{1,2,3,4,5},{6},{7,8}}
 => ? = 4 + 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
 => {{1,2,3,4,8},{5,7},{6}}
 => {{1,2,3,4},{5,7},{6,8}}
 => ? = 3 + 1
[1,0,1,0,1,0,1,0,1,1,0,1,1,0,0,0]
 => [1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0]
 => {{1,2,3,4,6,8},{5},{7}}
 => {{1,2,3,4},{5,6,8},{7}}
 => ? = 3 + 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0]
 => {{1,2,3,4,8},{5,6},{7}}
 => {{1,2,3,4,6},{5},{7,8}}
 => ? = 4 + 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0]
 => {{1,2,3,4,8},{5},{6,7}}
 => {{1,2,3,4,7},{5},{6,8}}
 => ? = 4 + 1
[1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0]
 => {{1,2,3,4,7,8},{5},{6}}
 => {{1,2,3,4,8},{5},{6,7}}
 => ? = 4 + 1
[1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
 => {{1,2,3,4,8},{5},{6},{7}}
 => {{1,2,3,4},{5},{6},{7,8}}
 => ? = 3 + 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0]
 => {{1,2,3,8},{4,5,7},{6}}
 => {{1,2,3,5},{4,7},{6,8}}
 => ? = 3 + 1
[1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,1,1,0,1,0,1,0,0,1,0,0,0,0]
 => {{1,2,3,8},{4,7},{5},{6}}
 => {{1,2,3},{4},{5,7},{6,8}}
 => ? = 2 + 1
[1,0,1,0,1,0,1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,1,0,1,1,1,0,1,0,0,0,0,0,0]
 => {{1,2,3,5,6,8},{4},{7}}
 => {{1,2,3,6},{4,5,8},{7}}
 => ? = 3 + 1
[1,0,1,0,1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,1,0,1,1,0,1,0,1,0,0,0,0,0]
 => {{1,2,3,5,8},{4},{6},{7}}
 => {{1,2,3},{4,5},{6,8},{7}}
 => ? = 2 + 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,1,0,1,0,0,0,0]
 => {{1,2,3,8},{4,5,6},{7}}
 => {{1,2,3,5,6},{4},{7,8}}
 => ? = 4 + 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,1,0,1,1,1,0,0,0,1,0,0,0,0]
 => {{1,2,3,8},{4},{5,6,7}}
 => {{1,2,3,6,7},{4},{5,8}}
 => ? = 4 + 1
[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]
 => {{1,2,3,7,8},{4,5},{6}}
 => ?
 => ? = 4 + 1
[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]
 => {{1,2,3,7,8},{4},{5,6}}
 => {{1,2,3,6,8},{4},{5,7}}
 => ? = 4 + 1
[1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,1,1,1,0,0,0,0,0,0]
 => {{1,2,3,6,7,8},{4},{5}}
 => {{1,2,3,7,8},{4},{5,6}}
 => ? = 4 + 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,0,1,0,1,0,1,0,0,0,0]
 => {{1,2,3,8},{4,5},{6},{7}}
 => {{1,2,3,5},{4},{6},{7,8}}
 => ? = 3 + 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,1,0,1,1,0,0,1,0,1,0,0,0,0]
 => {{1,2,3,8},{4},{5,6},{7}}
 => {{1,2,3,6},{4},{5},{7,8}}
 => ? = 3 + 1
[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]
 => {{1,2,3,7,8},{4},{5},{6}}
 => {{1,2,3,8},{4},{5},{6,7}}
 => ? = 3 + 1
[1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
 => {{1,2,3,8},{4},{5},{6},{7}}
 => {{1,2,3},{4},{5},{6},{7,8}}
 => ? = 2 + 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,1,0,1,0,0,0,0,1,0,0,0]
 => {{1,2,8},{3,4,5,7},{6}}
 => {{1,2,4,5},{3,7},{6,8}}
 => ? = 3 + 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,1,1,1,1,1,0,0,1,0,0,0,1,0,0,0]
 => {{1,2,8},{3,4,7},{5,6}}
 => {{1,2,4,6},{3,7},{5,8}}
 => ? = 3 + 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,1,0,1,1,0,0,0,0,1,0,0,0]
 => {{1,2,8},{3,4,6,7},{5}}
 => {{1,2,4,7},{3,6},{5,8}}
 => ? = 3 + 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,1,0,0,1,0,0,0]
 => {{1,2,8},{3,7},{4,5,6}}
 => {{1,2,5,6},{3,7},{4,8}}
 => ? = 3 + 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,0,1,0]
 => [1,1,1,1,1,0,0,1,1,0,0,0,1,0,0,0]
 => {{1,2,8},{3,6,7},{4,5}}
 => ?
 => ? = 3 + 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,0,1,1,1,0,0,0,0,1,0,0,0]
 => {{1,2,8},{3,5,6,7},{4}}
 => ?
 => ? = 3 + 1
[1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,1,0,1,0,1,0,0,1,0,0,0]
 => {{1,2,8},{3,7},{4},{5},{6}}
 => ?
 => ? = 1 + 1
[1,0,1,0,1,1,0,1,0,0,1,1,1,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,1,1,0,0,0,0]
 => {{1,2,7,8},{3,6},{4},{5}}
 => {{1,2,8},{3},{4,6},{5,7}}
 => ? = 2 + 1
[1,0,1,0,1,1,0,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,1,1,0,1,0,0,0,0,0,0]
 => {{1,2,4,5,6,8},{3},{7}}
 => {{1,2,5,6},{3,4,8},{7}}
 => ? = 3 + 1
[1,0,1,0,1,1,0,1,1,0,0,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,1,1,0,1,0,0,0,0]
 => {{1,2,6,8},{3,4,5},{7}}
 => ?
 => ? = 3 + 1
[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]
 => {{1,2,4,6,7,8},{3},{5}}
 => {{1,2,7,8},{3,4,6},{5}}
 => ? = 3 + 1
[1,0,1,0,1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,1,1,0,1,1,0,1,0,1,0,1,0,0,0,0]
 => {{1,2,4,8},{3},{5},{6},{7}}
 => {{1,2},{3,4},{5},{6,8},{7}}
 => ? = 1 + 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,1,0,1,0,0,0]
 => {{1,2,8},{3,4,5,6},{7}}
 => ?
 => ? = 4 + 1
[1,0,1,0,1,1,1,0,0,1,0,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,1,1,0,0,1,0,0,0]
 => {{1,2,8},{3,4,5},{6,7}}
 => {{1,2,4,5,7},{3},{6,8}}
 => ? = 4 + 1
[1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,1,1,1,0,0,0,0,1,0,0,0]
 => {{1,2,8},{3},{4,5,6,7}}
 => {{1,2,5,6,7},{3},{4,8}}
 => ? = 4 + 1
[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]
 => {{1,2,7,8},{3,4,5},{6}}
 => ?
 => ? = 4 + 1
[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]
 => {{1,2,7,8},{3},{4,5,6}}
 => {{1,2,5,6,8},{3},{4,7}}
 => ? = 4 + 1
[1,0,1,0,1,1,1,0,1,0,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,1,1,1,0,0,0,0,0]
 => {{1,2,6,7,8},{3},{4,5}}
 => ?
 => ? = 4 + 1
[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]
 => {{1,2,5,6,7,8},{3},{4}}
 => {{1,2,6,7,8},{3},{4,5}}
 => ? = 4 + 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,1,1,1,0,1,0,0,1,0,1,0,1,0,0,0]
 => {{1,2,8},{3,5},{4},{6},{7}}
 => ?
 => ? = 1 + 1
[1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0]
 => [1,1,1,0,1,1,1,0,0,0,1,0,1,0,0,0]
 => {{1,2,8},{3},{4,5,6},{7}}
 => {{1,2,5,6},{3},{4},{7,8}}
 => ? = 3 + 1
[1,0,1,0,1,1,1,1,0,0,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,1,0,0,1,0,1,0,0,0]
 => {{1,2,8},{3},{4,6},{5},{7}}
 => ?
 => ? = 1 + 1
[1,0,1,0,1,1,1,1,0,0,1,0,1,0,0,0]
 => [1,1,1,0,1,0,1,1,1,0,0,0,1,0,0,0]
 => {{1,2,8},{3},{4},{5,6,7}}
 => ?
 => ? = 3 + 1
[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]
 => {{1,2,6,7,8},{3},{4},{5}}
 => {{1,2,7,8},{3},{4},{5,6}}
 => ? = 3 + 1
[1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,1,1,0,0,1,0,1,0,1,0,1,0,0,0]
 => {{1,2,8},{3,4},{5},{6},{7}}
 => {{1,2,4},{3},{5},{6},{7,8}}
 => ? = 2 + 1
[1,0,1,0,1,1,1,1,1,0,0,0,1,0,0,0]
 => [1,1,1,0,1,0,1,1,0,0,1,0,1,0,0,0]
 => {{1,2,8},{3},{4},{5,6},{7}}
 => ?
 => ? = 2 + 1
[1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,0]
 => [1,1,1,0,1,0,1,0,1,1,0,0,1,0,0,0]
 => {{1,2,8},{3},{4},{5},{6,7}}
 => ?
 => ? = 2 + 1
[1,0,1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0]
 => {{1,2,7,8},{3},{4},{5},{6}}
 => {{1,2,8},{3},{4},{5},{6,7}}
 => ? = 2 + 1
[1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => {{1,2,8},{3},{4},{5},{6},{7}}
 => {{1,2},{3},{4},{5},{6},{7,8}}
 => ? = 1 + 1
Description
The cardinality of the first block of a set partition. The number of partitions of $\{1,\ldots,n\}$ into $k$ blocks in which the first block has cardinality $j+1$ is given by $\binom{n-1}{j}S(n-j-1,k-1)$, see [1, Theorem 1.1] and the references therein. Here, $S(n,k)$ are the ''Stirling numbers of the second kind'' counting all set partitions of $\{1,\ldots,n\}$ into $k$ blocks [2].
Matching statistic: St000502
(load all 5 compositions to match this statistic)
Mapping
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00028: Dyck paths —reverse⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000502: Set partitions ⟶ ℤResult quality: 43% ●values known / values provided: 43%●distinct values known / distinct values provided: 80%
Values
[1,0]
 => [1,0]
 => [1,0]
 => {{1}}
 => ? = 0
[1,0,1,0]
 => [1,1,0,0]
 => [1,1,0,0]
 => {{1,2}}
 => 1
[1,1,0,0]
 => [1,0,1,0]
 => [1,0,1,0]
 => {{1},{2}}
 => 0
[1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => {{1,2,3}}
 => 2
[1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => [1,1,0,1,0,0]
 => {{1,3},{2}}
 => 0
[1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => {{1},{2,3}}
 => 1
[1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => {{1,2},{3}}
 => 1
[1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => {{1},{2},{3}}
 => 0
[1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => {{1,2,3,4}}
 => 3
[1,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => {{1,2,4},{3}}
 => 1
[1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => {{1,3,4},{2}}
 => 1
[1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => {{1,4},{2,3}}
 => 1
[1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => {{1,4},{2},{3}}
 => 0
[1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => {{1},{2,3,4}}
 => 2
[1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => {{1},{2,4},{3}}
 => 0
[1,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => {{1,2},{3,4}}
 => 2
[1,1,0,1,0,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0]
 => {{1,2,3},{4}}
 => 2
[1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => {{1,3},{2},{4}}
 => 0
[1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => {{1},{2},{3,4}}
 => 1
[1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => {{1},{2,3},{4}}
 => 1
[1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => {{1,2},{3},{4}}
 => 1
[1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => {{1},{2},{3},{4}}
 => 0
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => {{1,2,3,4,5}}
 => 4
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => {{1,2,3,5},{4}}
 => 2
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => {{1,2,4,5},{3}}
 => 2
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => {{1,2,5},{3,4}}
 => 2
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => {{1,2,5},{3},{4}}
 => 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => {{1,3,4,5},{2}}
 => 2
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => {{1,3,5},{2},{4}}
 => 0
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => {{1,4,5},{2,3}}
 => 2
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => {{1,5},{2,3,4}}
 => 2
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => {{1,5},{2,4},{3}}
 => 0
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => {{1,4,5},{2},{3}}
 => 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => {{1,5},{2},{3,4}}
 => 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => {{1,5},{2,3},{4}}
 => 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => {{1,5},{2},{3},{4}}
 => 0
[1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => {{1},{2,3,4,5}}
 => 3
[1,1,0,0,1,0,1,1,0,0]
 => [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}}
 => 1
[1,1,0,0,1,1,0,0,1,0]
 => [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
[1,1,0,0,1,1,0,1,0,0]
 => [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}}
 => 1
[1,1,0,0,1,1,1,0,0,0]
 => [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}}
 => 0
[1,1,0,1,0,0,1,0,1,0]
 => [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}}
 => 3
[1,1,0,1,0,0,1,1,0,0]
 => [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}}
 => 1
[1,1,0,1,0,1,0,0,1,0]
 => [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}}
 => 3
[1,1,0,1,0,1,0,1,0,0]
 => [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}}
 => 3
[1,1,0,1,0,1,1,0,0,0]
 => [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}}
 => 1
[1,1,0,1,1,0,0,0,1,0]
 => [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
[1,1,0,1,1,0,0,1,0,0]
 => [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}}
 => 1
[1,1,0,1,1,0,1,0,0,0]
 => [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}}
 => 1
[1,1,0,1,1,1,0,0,0,0]
 => [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}}
 => 0
[1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => {{1},{2},{3,4,5}}
 => 2
[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
 => {{1,2,3,4,5,8},{6},{7}}
 => ? = 4
[1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
 => [1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0]
 => {{1,2,3,4,6,8},{5},{7}}
 => ? = 3
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
 => {{1,2,3,4,8},{5,6,7}}
 => ? = 5
[1,0,1,0,1,0,1,0,1,1,0,1,1,0,0,0]
 => [1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
 => {{1,2,3,4,8},{5,7},{6}}
 => ? = 3
[1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0]
 => {{1,2,3,4,7,8},{5},{6}}
 => ? = 4
[1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0]
 => [1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0]
 => {{1,2,3,4,8},{5},{6,7}}
 => ? = 4
[1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0]
 => {{1,2,3,4,8},{5,6},{7}}
 => ? = 4
[1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
 => {{1,2,3,4,8},{5},{6},{7}}
 => ? = 3
[1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0]
 => [1,1,1,1,0,1,1,1,0,1,0,0,0,0,0,0]
 => {{1,2,3,5,6,8},{4},{7}}
 => ? = 3
[1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,1,1,0,1,0,1,0,0,1,0,0,0,0]
 => [1,1,1,1,0,1,1,0,1,0,1,0,0,0,0,0]
 => {{1,2,3,5,8},{4},{6},{7}}
 => ? = 2
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
 => {{1,2,3,8},{4,5,6,7}}
 => ? = 5
[1,0,1,0,1,0,1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,1,0,1,1,1,0,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0]
 => {{1,2,3,8},{4,5,7},{6}}
 => ? = 3
[1,0,1,0,1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,1,0,1,1,0,1,0,1,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,1,0,0,1,0,0,0,0]
 => {{1,2,3,8},{4,7},{5},{6}}
 => ? = 2
[1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,1,0,1,0,0,0,0]
 => [1,1,1,1,0,1,0,1,1,1,0,0,0,0,0,0]
 => {{1,2,3,6,7,8},{4},{5}}
 => ? = 4
[1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,1,0,1,1,1,0,0,0,1,0,0,0,0]
 => [1,1,1,1,0,1,1,1,0,0,0,1,0,0,0,0]
 => {{1,2,3,8},{4},{5,6,7}}
 => ? = 4
[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]
 => [1,1,1,1,1,0,0,1,0,1,1,0,0,0,0,0]
 => {{1,2,3,7,8},{4,5},{6}}
 => ? = 4
[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]
 => [1,1,1,1,1,0,0,1,1,0,0,1,0,0,0,0]
 => {{1,2,3,8},{4,5},{6,7}}
 => ? = 4
[1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,1,0,1,0,0,0,0]
 => {{1,2,3,8},{4,5,6},{7}}
 => ? = 4
[1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,0,1,0,1,0,1,0,0,0,0]
 => [1,1,1,1,0,1,0,1,0,1,1,0,0,0,0,0]
 => {{1,2,3,7,8},{4},{5},{6}}
 => ? = 3
[1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,1,0,1,1,0,0,1,0,1,0,0,0,0]
 => [1,1,1,1,0,1,0,1,1,0,0,1,0,0,0,0]
 => {{1,2,3,8},{4},{5},{6,7}}
 => ? = 3
[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]
 => [1,1,1,1,1,0,0,1,0,1,0,1,0,0,0,0]
 => {{1,2,3,8},{4,5},{6},{7}}
 => ? = 3
[1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
 => [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
 => {{1,2,3,8},{4},{5},{6},{7}}
 => ? = 2
[1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,1,0,1,0,0,0,0,1,0,0,0]
 => [1,1,1,0,1,1,1,1,0,1,0,0,0,0,0,0]
 => {{1,2,4,5,6,8},{3},{7}}
 => ? = 3
[1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,1,1,1,1,1,0,0,1,0,0,0,1,0,0,0]
 => [1,1,1,0,1,1,1,0,1,1,0,0,0,0,0,0]
 => {{1,2,4,5,7,8},{3},{6}}
 => ? = 3
[1,0,1,0,1,1,0,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,1,0,1,1,0,0,0,0,1,0,0,0]
 => [1,1,1,0,1,1,1,1,0,0,1,0,0,0,0,0]
 => {{1,2,4,5,8},{3},{6,7}}
 => ? = 3
[1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,1,0,0,1,0,0,0]
 => [1,1,1,0,1,1,0,1,1,1,0,0,0,0,0,0]
 => {{1,2,4,6,7,8},{3},{5}}
 => ? = 3
[1,0,1,0,1,1,0,0,1,1,0,1,0,0,1,0]
 => [1,1,1,1,1,0,0,1,1,0,0,0,1,0,0,0]
 => [1,1,1,0,1,1,1,0,0,1,1,0,0,0,0,0]
 => {{1,2,4,7,8},{3},{5,6}}
 => ? = 3
[1,0,1,0,1,1,0,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,0,1,1,1,0,0,0,0,1,0,0,0]
 => [1,1,1,0,1,1,1,1,0,0,0,1,0,0,0,0]
 => {{1,2,4,8},{3},{5,6,7}}
 => ? = 3
[1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,1,0,1,0,1,0,0,1,0,0,0]
 => [1,1,1,0,1,1,0,1,0,1,0,1,0,0,0,0]
 => {{1,2,4,8},{3},{5},{6},{7}}
 => ? = 1
[1,0,1,0,1,1,0,1,0,0,1,1,1,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,1,1,0,0,0,0]
 => [1,1,1,1,0,0,1,1,0,1,0,1,0,0,0,0]
 => {{1,2,5,8},{3,4},{6},{7}}
 => ? = 2
[1,0,1,0,1,1,0,1,0,1,0,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,1,1,1,0,0,0,0,0]
 => {{1,2,6,7,8},{3,4,5}}
 => ? = 5
[1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
 => {{1,2,8},{3,4,5,6,7}}
 => ? = 5
[1,0,1,0,1,1,0,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,1,1,0,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,0,1,0,0,0,0,1,0,0,0]
 => {{1,2,8},{3,4,5,7},{6}}
 => ? = 3
[1,0,1,0,1,1,0,1,1,0,0,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,1,1,0,1,0,0,0,0]
 => [1,1,1,1,0,1,0,0,1,1,1,0,0,0,0,0]
 => {{1,2,6,7,8},{3,5},{4}}
 => ? = 3
[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]
 => [1,1,1,1,1,1,0,0,0,1,0,0,1,0,0,0]
 => {{1,2,8},{3,7},{4,5,6}}
 => ? = 3
[1,0,1,0,1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,1,1,0,1,1,0,1,0,1,0,1,0,0,0,0]
 => [1,1,1,1,0,1,0,1,0,1,0,0,1,0,0,0]
 => {{1,2,8},{3,7},{4},{5},{6}}
 => ? = 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,1,0,1,0,0,0]
 => [1,1,1,0,1,0,1,1,1,1,0,0,0,0,0,0]
 => {{1,2,5,6,7,8},{3},{4}}
 => ? = 4
[1,0,1,0,1,1,1,0,0,1,0,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,1,1,0,0,1,0,0,0]
 => [1,1,1,0,1,1,0,0,1,1,1,0,0,0,0,0]
 => {{1,2,6,7,8},{3},{4,5}}
 => ? = 4
[1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,1,1,1,0,0,0,0,1,0,0,0]
 => [1,1,1,0,1,1,1,1,0,0,0,0,1,0,0,0]
 => {{1,2,8},{3},{4,5,6,7}}
 => ? = 4
[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]
 => [1,1,1,1,0,0,1,0,1,1,1,0,0,0,0,0]
 => {{1,2,6,7,8},{3,4},{5}}
 => ? = 4
[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]
 => [1,1,1,1,0,0,1,1,1,0,0,0,1,0,0,0]
 => {{1,2,8},{3,4},{5,6,7}}
 => ? = 4
[1,0,1,0,1,1,1,0,1,0,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,1,1,0,0,1,0,0,0]
 => {{1,2,8},{3,4,5},{6,7}}
 => ? = 4
[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]
 => [1,1,1,1,1,1,0,0,0,0,1,0,1,0,0,0]
 => {{1,2,8},{3,4,5,6},{7}}
 => ? = 4
[1,0,1,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,1,1,1,0,1,0,0,1,0,1,0,1,0,0,0]
 => [1,1,1,0,1,0,1,0,1,1,0,1,0,0,0,0]
 => {{1,2,6,8},{3},{4},{5},{7}}
 => ? = 1
[1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0]
 => [1,1,1,0,1,1,1,0,0,0,1,0,1,0,0,0]
 => [1,1,1,0,1,0,1,1,1,0,0,0,1,0,0,0]
 => {{1,2,8},{3},{4},{5,6,7}}
 => ? = 3
[1,0,1,0,1,1,1,1,0,0,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,1,0,0,1,0,1,0,0,0]
 => [1,1,1,0,1,0,1,1,0,1,0,0,1,0,0,0]
 => {{1,2,8},{3},{4},{5,7},{6}}
 => ? = 1
[1,0,1,0,1,1,1,1,0,0,1,0,1,0,0,0]
 => [1,1,1,0,1,0,1,1,1,0,0,0,1,0,0,0]
 => [1,1,1,0,1,1,1,0,0,0,1,0,1,0,0,0]
 => {{1,2,8},{3},{4,5,6},{7}}
 => ? = 3
[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]
 => [1,1,1,1,1,0,0,0,1,0,1,0,1,0,0,0]
 => {{1,2,8},{3,4,5},{6},{7}}
 => ? = 3
[1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,1,1,0,0,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0]
 => {{1,2,7,8},{3},{4},{5},{6}}
 => ? = 2
Description
The number of successions of a set partitions. This is the number of indices $i$ such that $i$ and $i+1$ belonging to the same block.
Matching statistic: St001189
(load all 9 compositions to match this statistic)
Mapping
Mp00028: Dyck paths —reverse⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St001189: Dyck paths ⟶ ℤResult quality: 41% ●values known / values provided: 41%●distinct values known / distinct values provided: 70%
Values
[1,0]
 => [1,0]
 => [1,0]
 => 0
[1,0,1,0]
 => [1,0,1,0]
 => [1,1,0,0]
 => 1
[1,1,0,0]
 => [1,1,0,0]
 => [1,0,1,0]
 => 0
[1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 2
[1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => [1,1,0,1,0,0]
 => 0
[1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => 1
[1,1,0,1,0,0]
 => [1,1,0,1,0,0]
 => [1,1,0,0,1,0]
 => 1
[1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 0
[1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 3
[1,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,0]
 => 1
[1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 1
[1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 1
[1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => 0
[1,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => 2
[1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => 0
[1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 2
[1,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 2
[1,1,0,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 0
[1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => 1
[1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0]
 => 1
[1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => 1
[1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [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]
 => [1,1,1,1,1,0,0,0,0,0]
 => 4
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 2
[1,0,1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 2
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 2
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 2
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 0
[1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 2
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 2
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 0
[1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 0
[1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 3
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 0
[1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 3
[1,1,0,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 1
[1,1,0,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 3
[1,1,0,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 3
[1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => 1
[1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => 1
[1,1,0,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => 1
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 7
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => ? = 5
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
 => ? = 5
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
 => ? = 5
[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => ? = 4
[1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
 => ? = 5
[1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
 => ? = 3
[1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0]
 => ? = 5
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
 => ? = 5
[1,0,1,0,1,0,1,0,1,1,0,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => ? = 3
[1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0]
 => [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
 => ? = 4
[1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0,1,0,1,0,1,0]
 => [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => ? = 4
[1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0,1,0,1,0,1,0,1,0]
 => [1,1,0,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => ? = 4
[1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => ? = 3
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 5
[1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0]
 => ? = 3
[1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,1,1,1,1,0,1,0,1,0,0,0,0,0]
 => ? = 2
[1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,1,1,1,0,0,0,0,0,0]
 => ? = 5
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
 => ? = 5
[1,0,1,0,1,0,1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,0,1,0,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,1,1,1,1,0,0,0,1,0,0,0,0]
 => ? = 3
[1,0,1,0,1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,0,1,1,1,1,0,0,1,0,0,0,0]
 => ? = 2
[1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]
 => [1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
 => ? = 4
[1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0,1,0,1,0,1,0]
 => [1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
 => ? = 4
[1,0,1,0,1,0,1,1,1,0,1,0,0,0,1,0]
 => [1,0,1,1,1,0,1,0,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,1,1,1,0,1,1,0,0,0,0,0]
 => ? = 4
[1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,1,1,1,1,0,0,1,0,0,0,0]
 => ? = 4
[1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,0,1,1,1,1,0,1,0,0,0,0]
 => ? = 4
[1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,1,1,1,0,0,0,0,1,0,1,0,1,0]
 => [1,0,1,0,1,1,1,1,0,1,1,0,0,0,0,0]
 => ? = 3
[1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,0,0,1,0,1,0,1,0]
 => [1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,0]
 => ? = 3
[1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,0,1,1,1,1,0,1,0,0,0,0]
 => ? = 3
[1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
 => ? = 2
[1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 5
[1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,1,0,1,1,1,0,1,0,0,0,0,0,0]
 => ? = 3
[1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,1,0,1,1,0,1,1,0,0,0,0,0,0]
 => ? = 3
[1,0,1,0,1,1,0,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0]
 => ? = 3
[1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,1,1,0,1,0,1,1,1,0,0,0,0,0,0]
 => ? = 3
[1,0,1,0,1,1,0,0,1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,1,1,1,0,1,0,0,1,1,0,0,0,0,0]
 => ? = 3
[1,0,1,0,1,1,0,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0]
 => ? = 3
[1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,1,1,1,0,1,0,1,0,0,0,0]
 => ? = 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,1,0,0,1,1,1,1,0,0,0,0,0,0]
 => ? = 5
[1,0,1,0,1,1,0,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,1,0,0,1,0,1,0,0,0,0]
 => ? = 2
[1,0,1,0,1,1,0,1,0,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,1,0,1,0,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,1,1,1,0,0,0,0,0]
 => ? = 5
[1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
 => ? = 5
[1,0,1,0,1,1,0,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0,1,0,1,0]
 => [1,1,1,1,0,1,1,1,0,0,0,0,1,0,0,0]
 => ? = 3
[1,0,1,0,1,1,0,1,1,0,0,0,1,0,1,0]
 => [1,0,1,0,1,1,1,0,0,1,0,0,1,0,1,0]
 => [1,1,0,1,1,1,0,0,1,1,1,0,0,0,0,0]
 => ? = 3
[1,0,1,0,1,1,0,1,1,0,1,0,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,1,1,0,0,1,0,0,0]
 => ? = 3
[1,0,1,0,1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,1,1,1,0,0,1,0,0,0]
 => ? = 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
 => ? = 4
[1,0,1,0,1,1,1,0,0,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,1,1,0,0,0,1,0,1,0]
 => [1,0,1,1,1,1,0,0,1,1,1,0,0,0,0,0]
 => ? = 4
[1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0,1,0,1,0]
 => [1,0,1,1,1,1,1,1,0,0,0,0,1,0,0,0]
 => ? = 4
[1,0,1,0,1,1,1,0,1,0,0,0,1,0,1,0]
 => [1,0,1,0,1,1,1,0,1,0,0,0,1,0,1,0]
 => [1,1,0,0,1,1,1,0,1,1,1,0,0,0,0,0]
 => ? = 4
Description
The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St000441
(load all 4 compositions to match this statistic)
Mapping
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
St000441: Permutations ⟶ ℤResult quality: 39% ●values known / values provided: 39%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => [.,.]
 => [1] => 0
[1,0,1,0]
 => [2,1] => [[.,.],.]
 => [1,2] => 1
[1,1,0,0]
 => [1,2] => [.,[.,.]]
 => [2,1] => 0
[1,0,1,0,1,0]
 => [3,2,1] => [[[.,.],.],.]
 => [1,2,3] => 2
[1,0,1,1,0,0]
 => [2,3,1] => [[.,[.,.]],.]
 => [2,1,3] => 0
[1,1,0,0,1,0]
 => [3,1,2] => [[.,.],[.,.]]
 => [3,1,2] => 1
[1,1,0,1,0,0]
 => [2,1,3] => [[.,.],[.,.]]
 => [3,1,2] => 1
[1,1,1,0,0,0]
 => [1,2,3] => [.,[.,[.,.]]]
 => [3,2,1] => 0
[1,0,1,0,1,0,1,0]
 => [4,3,2,1] => [[[[.,.],.],.],.]
 => [1,2,3,4] => 3
[1,0,1,0,1,1,0,0]
 => [3,4,2,1] => [[[.,[.,.]],.],.]
 => [2,1,3,4] => 1
[1,0,1,1,0,0,1,0]
 => [4,2,3,1] => [[[.,.],[.,.]],.]
 => [3,1,2,4] => 1
[1,0,1,1,0,1,0,0]
 => [3,2,4,1] => [[[.,.],[.,.]],.]
 => [3,1,2,4] => 1
[1,0,1,1,1,0,0,0]
 => [2,3,4,1] => [[.,[.,[.,.]]],.]
 => [3,2,1,4] => 0
[1,1,0,0,1,0,1,0]
 => [4,3,1,2] => [[[.,.],.],[.,.]]
 => [4,1,2,3] => 2
[1,1,0,0,1,1,0,0]
 => [3,4,1,2] => [[.,[.,.]],[.,.]]
 => [4,2,1,3] => 0
[1,1,0,1,0,0,1,0]
 => [4,2,1,3] => [[[.,.],.],[.,.]]
 => [4,1,2,3] => 2
[1,1,0,1,0,1,0,0]
 => [3,2,1,4] => [[[.,.],.],[.,.]]
 => [4,1,2,3] => 2
[1,1,0,1,1,0,0,0]
 => [2,3,1,4] => [[.,[.,.]],[.,.]]
 => [4,2,1,3] => 0
[1,1,1,0,0,0,1,0]
 => [4,1,2,3] => [[.,.],[.,[.,.]]]
 => [4,3,1,2] => 1
[1,1,1,0,0,1,0,0]
 => [3,1,2,4] => [[.,.],[.,[.,.]]]
 => [4,3,1,2] => 1
[1,1,1,0,1,0,0,0]
 => [2,1,3,4] => [[.,.],[.,[.,.]]]
 => [4,3,1,2] => 1
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [.,[.,[.,[.,.]]]]
 => [4,3,2,1] => 0
[1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
 => [1,2,3,4,5] => 4
[1,0,1,0,1,0,1,1,0,0]
 => [4,5,3,2,1] => [[[[.,[.,.]],.],.],.]
 => [2,1,3,4,5] => 2
[1,0,1,0,1,1,0,0,1,0]
 => [5,3,4,2,1] => [[[[.,.],[.,.]],.],.]
 => [3,1,2,4,5] => 2
[1,0,1,0,1,1,0,1,0,0]
 => [4,3,5,2,1] => [[[[.,.],[.,.]],.],.]
 => [3,1,2,4,5] => 2
[1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => [[[.,[.,[.,.]]],.],.]
 => [3,2,1,4,5] => 1
[1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,3,1] => [[[[.,.],.],[.,.]],.]
 => [4,1,2,3,5] => 2
[1,0,1,1,0,0,1,1,0,0]
 => [4,5,2,3,1] => [[[.,[.,.]],[.,.]],.]
 => [4,2,1,3,5] => 0
[1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,4,1] => [[[[.,.],.],[.,.]],.]
 => [4,1,2,3,5] => 2
[1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,5,1] => [[[[.,.],.],[.,.]],.]
 => [4,1,2,3,5] => 2
[1,0,1,1,0,1,1,0,0,0]
 => [3,4,2,5,1] => [[[.,[.,.]],[.,.]],.]
 => [4,2,1,3,5] => 0
[1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => [[[.,.],[.,[.,.]]],.]
 => [4,3,1,2,5] => 1
[1,0,1,1,1,0,0,1,0,0]
 => [4,2,3,5,1] => [[[.,.],[.,[.,.]]],.]
 => [4,3,1,2,5] => 1
[1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => [[[.,.],[.,[.,.]]],.]
 => [4,3,1,2,5] => 1
[1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [[.,[.,[.,[.,.]]]],.]
 => [4,3,2,1,5] => 0
[1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,2] => [[[[.,.],.],.],[.,.]]
 => [5,1,2,3,4] => 3
[1,1,0,0,1,0,1,1,0,0]
 => [4,5,3,1,2] => [[[.,[.,.]],.],[.,.]]
 => [5,2,1,3,4] => 1
[1,1,0,0,1,1,0,0,1,0]
 => [5,3,4,1,2] => [[[.,.],[.,.]],[.,.]]
 => [5,3,1,2,4] => 1
[1,1,0,0,1,1,0,1,0,0]
 => [4,3,5,1,2] => [[[.,.],[.,.]],[.,.]]
 => [5,3,1,2,4] => 1
[1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => [[.,[.,[.,.]]],[.,.]]
 => [5,3,2,1,4] => 0
[1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,3] => [[[[.,.],.],.],[.,.]]
 => [5,1,2,3,4] => 3
[1,1,0,1,0,0,1,1,0,0]
 => [4,5,2,1,3] => [[[.,[.,.]],.],[.,.]]
 => [5,2,1,3,4] => 1
[1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,4] => [[[[.,.],.],.],[.,.]]
 => [5,1,2,3,4] => 3
[1,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1,5] => [[[[.,.],.],.],[.,.]]
 => [5,1,2,3,4] => 3
[1,1,0,1,0,1,1,0,0,0]
 => [3,4,2,1,5] => [[[.,[.,.]],.],[.,.]]
 => [5,2,1,3,4] => 1
[1,1,0,1,1,0,0,0,1,0]
 => [5,2,3,1,4] => [[[.,.],[.,.]],[.,.]]
 => [5,3,1,2,4] => 1
[1,1,0,1,1,0,0,1,0,0]
 => [4,2,3,1,5] => [[[.,.],[.,.]],[.,.]]
 => [5,3,1,2,4] => 1
[1,1,0,1,1,0,1,0,0,0]
 => [3,2,4,1,5] => [[[.,.],[.,.]],[.,.]]
 => [5,3,1,2,4] => 1
[1,1,0,1,1,1,0,0,0,0]
 => [2,3,4,1,5] => [[.,[.,[.,.]]],[.,.]]
 => [5,3,2,1,4] => 0
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [6,7,4,5,3,2,1] => [[[[[.,[.,.]],[.,.]],.],.],.]
 => [4,2,1,3,5,6,7] => ? = 2
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
 => [5,6,4,7,3,2,1] => [[[[[.,[.,.]],[.,.]],.],.],.]
 => [4,2,1,3,5,6,7] => ? = 2
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => [7,4,5,6,3,2,1] => [[[[[.,.],[.,[.,.]]],.],.],.]
 => [4,3,1,2,5,6,7] => ? = 3
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [6,4,5,7,3,2,1] => [[[[[.,.],[.,[.,.]]],.],.],.]
 => [4,3,1,2,5,6,7] => ? = 3
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => [5,4,6,7,3,2,1] => [[[[[.,.],[.,[.,.]]],.],.],.]
 => [4,3,1,2,5,6,7] => ? = 3
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [6,7,5,3,4,2,1] => [[[[[.,[.,.]],.],[.,.]],.],.]
 => [5,2,1,3,4,6,7] => ? = 2
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [7,5,6,3,4,2,1] => [[[[[.,.],[.,.]],[.,.]],.],.]
 => [5,3,1,2,4,6,7] => ? = 2
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
 => [6,5,7,3,4,2,1] => [[[[[.,.],[.,.]],[.,.]],.],.]
 => [5,3,1,2,4,6,7] => ? = 2
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => [5,6,7,3,4,2,1] => [[[[.,[.,[.,.]]],[.,.]],.],.]
 => [5,3,2,1,4,6,7] => ? = 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
 => [6,7,4,3,5,2,1] => [[[[[.,[.,.]],.],[.,.]],.],.]
 => [5,2,1,3,4,6,7] => ? = 2
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
 => [5,6,4,3,7,2,1] => [[[[[.,[.,.]],.],[.,.]],.],.]
 => [5,2,1,3,4,6,7] => ? = 2
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
 => [7,4,5,3,6,2,1] => [[[[[.,.],[.,.]],[.,.]],.],.]
 => [5,3,1,2,4,6,7] => ? = 2
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
 => [6,4,5,3,7,2,1] => [[[[[.,.],[.,.]],[.,.]],.],.]
 => [5,3,1,2,4,6,7] => ? = 2
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
 => [5,4,6,3,7,2,1] => [[[[[.,.],[.,.]],[.,.]],.],.]
 => [5,3,1,2,4,6,7] => ? = 2
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
 => [4,5,6,3,7,2,1] => [[[[.,[.,[.,.]]],[.,.]],.],.]
 => [5,3,2,1,4,6,7] => ? = 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => [7,6,3,4,5,2,1] => [[[[[.,.],.],[.,[.,.]]],.],.]
 => [5,4,1,2,3,6,7] => ? = 3
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => [6,7,3,4,5,2,1] => [[[[.,[.,.]],[.,[.,.]]],.],.]
 => [5,4,2,1,3,6,7] => ? = 1
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
 => [7,5,3,4,6,2,1] => [[[[[.,.],.],[.,[.,.]]],.],.]
 => [5,4,1,2,3,6,7] => ? = 3
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [6,5,3,4,7,2,1] => [[[[[.,.],.],[.,[.,.]]],.],.]
 => [5,4,1,2,3,6,7] => ? = 3
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
 => [5,6,3,4,7,2,1] => [[[[.,[.,.]],[.,[.,.]]],.],.]
 => [5,4,2,1,3,6,7] => ? = 1
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
 => [7,4,3,5,6,2,1] => [[[[[.,.],.],[.,[.,.]]],.],.]
 => [5,4,1,2,3,6,7] => ? = 3
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => [6,4,3,5,7,2,1] => [[[[[.,.],.],[.,[.,.]]],.],.]
 => [5,4,1,2,3,6,7] => ? = 3
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
 => [5,4,3,6,7,2,1] => [[[[[.,.],.],[.,[.,.]]],.],.]
 => [5,4,1,2,3,6,7] => ? = 3
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
 => [4,5,3,6,7,2,1] => [[[[.,[.,.]],[.,[.,.]]],.],.]
 => [5,4,2,1,3,6,7] => ? = 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => [7,3,4,5,6,2,1] => [[[[.,.],[.,[.,[.,.]]]],.],.]
 => [5,4,3,1,2,6,7] => ? = 2
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => [6,3,4,5,7,2,1] => [[[[.,.],[.,[.,[.,.]]]],.],.]
 => [5,4,3,1,2,6,7] => ? = 2
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
 => [5,3,4,6,7,2,1] => [[[[.,.],[.,[.,[.,.]]]],.],.]
 => [5,4,3,1,2,6,7] => ? = 2
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
 => [4,3,5,6,7,2,1] => [[[[.,.],[.,[.,[.,.]]]],.],.]
 => [5,4,3,1,2,6,7] => ? = 2
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [6,7,5,4,2,3,1] => [[[[[.,[.,.]],.],.],[.,.]],.]
 => [6,2,1,3,4,5,7] => ? = 2
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => [7,5,6,4,2,3,1] => [[[[[.,.],[.,.]],.],[.,.]],.]
 => [6,3,1,2,4,5,7] => ? = 2
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
 => [6,5,7,4,2,3,1] => [[[[[.,.],[.,.]],.],[.,.]],.]
 => [6,3,1,2,4,5,7] => ? = 2
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
 => [5,6,7,4,2,3,1] => [[[[.,[.,[.,.]]],.],[.,.]],.]
 => [6,3,2,1,4,5,7] => ? = 1
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [7,6,4,5,2,3,1] => [[[[[.,.],.],[.,.]],[.,.]],.]
 => [6,4,1,2,3,5,7] => ? = 2
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [6,7,4,5,2,3,1] => [[[[.,[.,.]],[.,.]],[.,.]],.]
 => [6,4,2,1,3,5,7] => ? = 0
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
 => [7,5,4,6,2,3,1] => [[[[[.,.],.],[.,.]],[.,.]],.]
 => [6,4,1,2,3,5,7] => ? = 2
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
 => [6,5,4,7,2,3,1] => [[[[[.,.],.],[.,.]],[.,.]],.]
 => [6,4,1,2,3,5,7] => ? = 2
[1,0,1,1,0,0,1,1,0,1,1,0,0,0]
 => [5,6,4,7,2,3,1] => [[[[.,[.,.]],[.,.]],[.,.]],.]
 => [6,4,2,1,3,5,7] => ? = 0
[1,0,1,1,0,0,1,1,1,0,0,0,1,0]
 => [7,4,5,6,2,3,1] => [[[[.,.],[.,[.,.]]],[.,.]],.]
 => [6,4,3,1,2,5,7] => ? = 1
[1,0,1,1,0,0,1,1,1,0,0,1,0,0]
 => [6,4,5,7,2,3,1] => [[[[.,.],[.,[.,.]]],[.,.]],.]
 => [6,4,3,1,2,5,7] => ? = 1
[1,0,1,1,0,0,1,1,1,0,1,0,0,0]
 => [5,4,6,7,2,3,1] => [[[[.,.],[.,[.,.]]],[.,.]],.]
 => [6,4,3,1,2,5,7] => ? = 1
[1,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => [4,5,6,7,2,3,1] => [[[.,[.,[.,[.,.]]]],[.,.]],.]
 => [6,4,3,2,1,5,7] => ? = 0
[1,0,1,1,0,1,0,0,1,0,1,1,0,0]
 => [6,7,5,3,2,4,1] => [[[[[.,[.,.]],.],.],[.,.]],.]
 => [6,2,1,3,4,5,7] => ? = 2
[1,0,1,1,0,1,0,0,1,1,0,0,1,0]
 => [7,5,6,3,2,4,1] => [[[[[.,.],[.,.]],.],[.,.]],.]
 => [6,3,1,2,4,5,7] => ? = 2
[1,0,1,1,0,1,0,0,1,1,0,1,0,0]
 => [6,5,7,3,2,4,1] => [[[[[.,.],[.,.]],.],[.,.]],.]
 => [6,3,1,2,4,5,7] => ? = 2
[1,0,1,1,0,1,0,0,1,1,1,0,0,0]
 => [5,6,7,3,2,4,1] => [[[[.,[.,[.,.]]],.],[.,.]],.]
 => [6,3,2,1,4,5,7] => ? = 1
[1,0,1,1,0,1,0,1,0,0,1,1,0,0]
 => [6,7,4,3,2,5,1] => [[[[[.,[.,.]],.],.],[.,.]],.]
 => [6,2,1,3,4,5,7] => ? = 2
[1,0,1,1,0,1,0,1,0,1,1,0,0,0]
 => [5,6,4,3,2,7,1] => [[[[[.,[.,.]],.],.],[.,.]],.]
 => [6,2,1,3,4,5,7] => ? = 2
[1,0,1,1,0,1,0,1,1,0,0,0,1,0]
 => [7,4,5,3,2,6,1] => [[[[[.,.],[.,.]],.],[.,.]],.]
 => [6,3,1,2,4,5,7] => ? = 2
[1,0,1,1,0,1,0,1,1,0,0,1,0,0]
 => [6,4,5,3,2,7,1] => [[[[[.,.],[.,.]],.],[.,.]],.]
 => [6,3,1,2,4,5,7] => ? = 2
[1,0,1,1,0,1,0,1,1,0,1,0,0,0]
 => [5,4,6,3,2,7,1] => [[[[[.,.],[.,.]],.],[.,.]],.]
 => [6,3,1,2,4,5,7] => ? = 2
Description
The number of successions of a permutation. A succession of a permutation $\pi$ is an index $i$ such that $\pi(i)+1 = \pi(i+1)$. Successions are also known as ''small ascents'' or ''1-rises''.
Matching statistic: St001091
Mapping
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00182: Skew partitions —outer shape⟶ Integer partitions
St001091: Integer partitions ⟶ ℤResult quality: 33% ●values known / values provided: 33%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => [[1],[]]
 => [1]
 => 0
[1,0,1,0]
 => [1,1] => [[1,1],[]]
 => [1,1]
 => 1
[1,1,0,0]
 => [2] => [[2],[]]
 => [2]
 => 0
[1,0,1,0,1,0]
 => [1,1,1] => [[1,1,1],[]]
 => [1,1,1]
 => 2
[1,0,1,1,0,0]
 => [1,2] => [[2,1],[]]
 => [2,1]
 => 0
[1,1,0,0,1,0]
 => [2,1] => [[2,2],[1]]
 => [2,2]
 => 1
[1,1,0,1,0,0]
 => [2,1] => [[2,2],[1]]
 => [2,2]
 => 1
[1,1,1,0,0,0]
 => [3] => [[3],[]]
 => [3]
 => 0
[1,0,1,0,1,0,1,0]
 => [1,1,1,1] => [[1,1,1,1],[]]
 => [1,1,1,1]
 => 3
[1,0,1,0,1,1,0,0]
 => [1,1,2] => [[2,1,1],[]]
 => [2,1,1]
 => 1
[1,0,1,1,0,0,1,0]
 => [1,2,1] => [[2,2,1],[1]]
 => [2,2,1]
 => 1
[1,0,1,1,0,1,0,0]
 => [1,2,1] => [[2,2,1],[1]]
 => [2,2,1]
 => 1
[1,0,1,1,1,0,0,0]
 => [1,3] => [[3,1],[]]
 => [3,1]
 => 0
[1,1,0,0,1,0,1,0]
 => [2,1,1] => [[2,2,2],[1,1]]
 => [2,2,2]
 => 2
[1,1,0,0,1,1,0,0]
 => [2,2] => [[3,2],[1]]
 => [3,2]
 => 0
[1,1,0,1,0,0,1,0]
 => [2,1,1] => [[2,2,2],[1,1]]
 => [2,2,2]
 => 2
[1,1,0,1,0,1,0,0]
 => [2,1,1] => [[2,2,2],[1,1]]
 => [2,2,2]
 => 2
[1,1,0,1,1,0,0,0]
 => [2,2] => [[3,2],[1]]
 => [3,2]
 => 0
[1,1,1,0,0,0,1,0]
 => [3,1] => [[3,3],[2]]
 => [3,3]
 => 1
[1,1,1,0,0,1,0,0]
 => [3,1] => [[3,3],[2]]
 => [3,3]
 => 1
[1,1,1,0,1,0,0,0]
 => [3,1] => [[3,3],[2]]
 => [3,3]
 => 1
[1,1,1,1,0,0,0,0]
 => [4] => [[4],[]]
 => [4]
 => 0
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1] => [[1,1,1,1,1],[]]
 => [1,1,1,1,1]
 => 4
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,2] => [[2,1,1,1],[]]
 => [2,1,1,1]
 => 2
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,2,1] => [[2,2,1,1],[1]]
 => [2,2,1,1]
 => 2
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,2,1] => [[2,2,1,1],[1]]
 => [2,2,1,1]
 => 2
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,3] => [[3,1,1],[]]
 => [3,1,1]
 => 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,2,1,1] => [[2,2,2,1],[1,1]]
 => [2,2,2,1]
 => 2
[1,0,1,1,0,0,1,1,0,0]
 => [1,2,2] => [[3,2,1],[1]]
 => [3,2,1]
 => 0
[1,0,1,1,0,1,0,0,1,0]
 => [1,2,1,1] => [[2,2,2,1],[1,1]]
 => [2,2,2,1]
 => 2
[1,0,1,1,0,1,0,1,0,0]
 => [1,2,1,1] => [[2,2,2,1],[1,1]]
 => [2,2,2,1]
 => 2
[1,0,1,1,0,1,1,0,0,0]
 => [1,2,2] => [[3,2,1],[1]]
 => [3,2,1]
 => 0
[1,0,1,1,1,0,0,0,1,0]
 => [1,3,1] => [[3,3,1],[2]]
 => [3,3,1]
 => 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,3,1] => [[3,3,1],[2]]
 => [3,3,1]
 => 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,3,1] => [[3,3,1],[2]]
 => [3,3,1]
 => 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,4] => [[4,1],[]]
 => [4,1]
 => 0
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
 => [2,2,2,2]
 => 3
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,2] => [[3,2,2],[1,1]]
 => [3,2,2]
 => 1
[1,1,0,0,1,1,0,0,1,0]
 => [2,2,1] => [[3,3,2],[2,1]]
 => [3,3,2]
 => 1
[1,1,0,0,1,1,0,1,0,0]
 => [2,2,1] => [[3,3,2],[2,1]]
 => [3,3,2]
 => 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,3] => [[4,2],[1]]
 => [4,2]
 => 0
[1,1,0,1,0,0,1,0,1,0]
 => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
 => [2,2,2,2]
 => 3
[1,1,0,1,0,0,1,1,0,0]
 => [2,1,2] => [[3,2,2],[1,1]]
 => [3,2,2]
 => 1
[1,1,0,1,0,1,0,0,1,0]
 => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
 => [2,2,2,2]
 => 3
[1,1,0,1,0,1,0,1,0,0]
 => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
 => [2,2,2,2]
 => 3
[1,1,0,1,0,1,1,0,0,0]
 => [2,1,2] => [[3,2,2],[1,1]]
 => [3,2,2]
 => 1
[1,1,0,1,1,0,0,0,1,0]
 => [2,2,1] => [[3,3,2],[2,1]]
 => [3,3,2]
 => 1
[1,1,0,1,1,0,0,1,0,0]
 => [2,2,1] => [[3,3,2],[2,1]]
 => [3,3,2]
 => 1
[1,1,0,1,1,0,1,0,0,0]
 => [2,2,1] => [[3,3,2],[2,1]]
 => [3,3,2]
 => 1
[1,1,0,1,1,1,0,0,0,0]
 => [2,3] => [[4,2],[1]]
 => [4,2]
 => 0
[1,0,1,1,1,0,0,0,1,0,1,0,1,0]
 => [1,3,1,1,1] => [[3,3,3,3,1],[2,2,2]]
 => [3,3,3,3,1]
 => ? = 3
[1,0,1,1,1,0,0,1,0,0,1,0,1,0]
 => [1,3,1,1,1] => [[3,3,3,3,1],[2,2,2]]
 => [3,3,3,3,1]
 => ? = 3
[1,0,1,1,1,0,0,1,0,1,0,0,1,0]
 => [1,3,1,1,1] => [[3,3,3,3,1],[2,2,2]]
 => [3,3,3,3,1]
 => ? = 3
[1,0,1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,3,1,1,1] => [[3,3,3,3,1],[2,2,2]]
 => [3,3,3,3,1]
 => ? = 3
[1,0,1,1,1,0,1,0,0,0,1,0,1,0]
 => [1,3,1,1,1] => [[3,3,3,3,1],[2,2,2]]
 => [3,3,3,3,1]
 => ? = 3
[1,0,1,1,1,0,1,0,0,1,0,0,1,0]
 => [1,3,1,1,1] => [[3,3,3,3,1],[2,2,2]]
 => [3,3,3,3,1]
 => ? = 3
[1,0,1,1,1,0,1,0,0,1,0,1,0,0]
 => [1,3,1,1,1] => [[3,3,3,3,1],[2,2,2]]
 => [3,3,3,3,1]
 => ? = 3
[1,0,1,1,1,0,1,0,1,0,0,0,1,0]
 => [1,3,1,1,1] => [[3,3,3,3,1],[2,2,2]]
 => [3,3,3,3,1]
 => ? = 3
[1,0,1,1,1,0,1,0,1,0,0,1,0,0]
 => [1,3,1,1,1] => [[3,3,3,3,1],[2,2,2]]
 => [3,3,3,3,1]
 => ? = 3
[1,0,1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,3,1,1,1] => [[3,3,3,3,1],[2,2,2]]
 => [3,3,3,3,1]
 => ? = 3
[1,0,1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,4,1,1] => [[4,4,4,1],[3,3]]
 => [4,4,4,1]
 => ? = 2
[1,0,1,1,1,1,0,0,0,1,0,0,1,0]
 => [1,4,1,1] => [[4,4,4,1],[3,3]]
 => [4,4,4,1]
 => ? = 2
[1,0,1,1,1,1,0,0,0,1,0,1,0,0]
 => [1,4,1,1] => [[4,4,4,1],[3,3]]
 => [4,4,4,1]
 => ? = 2
[1,0,1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,4,1,1] => [[4,4,4,1],[3,3]]
 => [4,4,4,1]
 => ? = 2
[1,0,1,1,1,1,0,0,1,0,0,1,0,0]
 => [1,4,1,1] => [[4,4,4,1],[3,3]]
 => [4,4,4,1]
 => ? = 2
[1,0,1,1,1,1,0,0,1,0,1,0,0,0]
 => [1,4,1,1] => [[4,4,4,1],[3,3]]
 => [4,4,4,1]
 => ? = 2
[1,0,1,1,1,1,0,1,0,0,0,0,1,0]
 => [1,4,1,1] => [[4,4,4,1],[3,3]]
 => [4,4,4,1]
 => ? = 2
[1,0,1,1,1,1,0,1,0,0,0,1,0,0]
 => [1,4,1,1] => [[4,4,4,1],[3,3]]
 => [4,4,4,1]
 => ? = 2
[1,0,1,1,1,1,0,1,0,0,1,0,0,0]
 => [1,4,1,1] => [[4,4,4,1],[3,3]]
 => [4,4,4,1]
 => ? = 2
[1,0,1,1,1,1,0,1,0,1,0,0,0,0]
 => [1,4,1,1] => [[4,4,4,1],[3,3]]
 => [4,4,4,1]
 => ? = 2
[1,1,0,0,1,0,1,1,0,0,1,0,1,0]
 => [2,1,2,1,1] => [[3,3,3,2,2],[2,2,1,1]]
 => [3,3,3,2,2]
 => ? = 3
[1,1,0,0,1,0,1,1,0,1,0,0,1,0]
 => [2,1,2,1,1] => [[3,3,3,2,2],[2,2,1,1]]
 => [3,3,3,2,2]
 => ? = 3
[1,1,0,0,1,0,1,1,0,1,0,1,0,0]
 => [2,1,2,1,1] => [[3,3,3,2,2],[2,2,1,1]]
 => [3,3,3,2,2]
 => ? = 3
[1,1,0,0,1,1,0,0,1,0,1,0,1,0]
 => [2,2,1,1,1] => [[3,3,3,3,2],[2,2,2,1]]
 => [3,3,3,3,2]
 => ? = 3
[1,1,0,0,1,1,0,1,0,0,1,0,1,0]
 => [2,2,1,1,1] => [[3,3,3,3,2],[2,2,2,1]]
 => [3,3,3,3,2]
 => ? = 3
[1,1,0,0,1,1,0,1,0,1,0,0,1,0]
 => [2,2,1,1,1] => [[3,3,3,3,2],[2,2,2,1]]
 => [3,3,3,3,2]
 => ? = 3
[1,1,0,0,1,1,0,1,0,1,0,1,0,0]
 => [2,2,1,1,1] => [[3,3,3,3,2],[2,2,2,1]]
 => [3,3,3,3,2]
 => ? = 3
[1,1,0,0,1,1,1,0,0,0,1,0,1,0]
 => [2,3,1,1] => [[4,4,4,2],[3,3,1]]
 => [4,4,4,2]
 => ? = 2
[1,1,0,0,1,1,1,0,0,1,0,0,1,0]
 => [2,3,1,1] => [[4,4,4,2],[3,3,1]]
 => [4,4,4,2]
 => ? = 2
[1,1,0,0,1,1,1,0,0,1,0,1,0,0]
 => [2,3,1,1] => [[4,4,4,2],[3,3,1]]
 => [4,4,4,2]
 => ? = 2
[1,1,0,0,1,1,1,0,1,0,0,0,1,0]
 => [2,3,1,1] => [[4,4,4,2],[3,3,1]]
 => [4,4,4,2]
 => ? = 2
[1,1,0,0,1,1,1,0,1,0,0,1,0,0]
 => [2,3,1,1] => [[4,4,4,2],[3,3,1]]
 => [4,4,4,2]
 => ? = 2
[1,1,0,0,1,1,1,0,1,0,1,0,0,0]
 => [2,3,1,1] => [[4,4,4,2],[3,3,1]]
 => [4,4,4,2]
 => ? = 2
[1,1,0,1,0,0,1,1,0,0,1,0,1,0]
 => [2,1,2,1,1] => [[3,3,3,2,2],[2,2,1,1]]
 => [3,3,3,2,2]
 => ? = 3
[1,1,0,1,0,0,1,1,0,1,0,0,1,0]
 => [2,1,2,1,1] => [[3,3,3,2,2],[2,2,1,1]]
 => [3,3,3,2,2]
 => ? = 3
[1,1,0,1,0,0,1,1,0,1,0,1,0,0]
 => [2,1,2,1,1] => [[3,3,3,2,2],[2,2,1,1]]
 => [3,3,3,2,2]
 => ? = 3
[1,1,0,1,0,1,1,0,0,0,1,0,1,0]
 => [2,1,2,1,1] => [[3,3,3,2,2],[2,2,1,1]]
 => [3,3,3,2,2]
 => ? = 3
[1,1,0,1,0,1,1,0,0,1,0,0,1,0]
 => [2,1,2,1,1] => [[3,3,3,2,2],[2,2,1,1]]
 => [3,3,3,2,2]
 => ? = 3
[1,1,0,1,0,1,1,0,0,1,0,1,0,0]
 => [2,1,2,1,1] => [[3,3,3,2,2],[2,2,1,1]]
 => [3,3,3,2,2]
 => ? = 3
[1,1,0,1,0,1,1,0,1,0,0,0,1,0]
 => [2,1,2,1,1] => [[3,3,3,2,2],[2,2,1,1]]
 => [3,3,3,2,2]
 => ? = 3
[1,1,0,1,0,1,1,0,1,0,0,1,0,0]
 => [2,1,2,1,1] => [[3,3,3,2,2],[2,2,1,1]]
 => [3,3,3,2,2]
 => ? = 3
[1,1,0,1,0,1,1,0,1,0,1,0,0,0]
 => [2,1,2,1,1] => [[3,3,3,2,2],[2,2,1,1]]
 => [3,3,3,2,2]
 => ? = 3
[1,1,0,1,1,0,0,0,1,0,1,0,1,0]
 => [2,2,1,1,1] => [[3,3,3,3,2],[2,2,2,1]]
 => [3,3,3,3,2]
 => ? = 3
[1,1,0,1,1,0,0,1,0,0,1,0,1,0]
 => [2,2,1,1,1] => [[3,3,3,3,2],[2,2,2,1]]
 => [3,3,3,3,2]
 => ? = 3
[1,1,0,1,1,0,0,1,0,1,0,0,1,0]
 => [2,2,1,1,1] => [[3,3,3,3,2],[2,2,2,1]]
 => [3,3,3,3,2]
 => ? = 3
[1,1,0,1,1,0,0,1,0,1,0,1,0,0]
 => [2,2,1,1,1] => [[3,3,3,3,2],[2,2,2,1]]
 => [3,3,3,3,2]
 => ? = 3
[1,1,0,1,1,0,1,0,0,0,1,0,1,0]
 => [2,2,1,1,1] => [[3,3,3,3,2],[2,2,2,1]]
 => [3,3,3,3,2]
 => ? = 3
[1,1,0,1,1,0,1,0,0,1,0,0,1,0]
 => [2,2,1,1,1] => [[3,3,3,3,2],[2,2,2,1]]
 => [3,3,3,3,2]
 => ? = 3
[1,1,0,1,1,0,1,0,0,1,0,1,0,0]
 => [2,2,1,1,1] => [[3,3,3,3,2],[2,2,2,1]]
 => [3,3,3,3,2]
 => ? = 3
[1,1,0,1,1,0,1,0,1,0,0,0,1,0]
 => [2,2,1,1,1] => [[3,3,3,3,2],[2,2,2,1]]
 => [3,3,3,3,2]
 => ? = 3
Description
The number of parts in an integer partition whose next smaller part has the same size. In other words, this is the number of distinct parts subtracted from the number of all parts.
The following 22 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St000214The number of adjacencies of a permutation. St000674The number of hills of a Dyck path. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001061The number of indices that are both descents and recoils of a permutation. St000237The number of small exceedances. St000247The number of singleton blocks of a set partition. St000248The number of anti-singletons of a set partition. St001126Number of simple module that are 1-regular in the corresponding Nakayama algebra. St001631The number of simple modules $S$ with $dim Ext^1(S,A)=1$ in the incidence algebra $A$ of the poset. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St000931The number of occurrences of the pattern UUU in a Dyck path. St001172The number of 1-rises at odd height of a Dyck path. St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. St000215The number of adjacencies of a permutation, zero appended. St000022The number of fixed points of a permutation. St000732The number of double deficiencies of a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001948The number of augmented double ascents of a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St000239The number of small weak excedances. St001621The number of atoms of a lattice.