Your data matches 3 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000708
Mapping
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00122: Dyck paths —Elizalde-Deutsch bijection⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000708: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
 => [1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => [1,1]
 => 1
[1,1,0,0]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => [2,1]
 => 2
[1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => [3,1,1]
 => 3
[1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0]
 => [2,1,1]
 => 2
[1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1]
 => 1
[1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => [2,2,1]
 => 4
[1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [3,2,1]
 => 6
[1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [3,3,1,1]
 => 9
[1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => [2,2,1,1]
 => 4
[1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [4,3,1,1]
 => 12
[1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => [4,2,1,1]
 => 8
[1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => [3,2,1,1]
 => 6
[1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => [3,1,1,1]
 => 3
[1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 1
[1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [4,1,1,1]
 => 4
[1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [4,2,2,1]
 => 16
[1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => [3,2,2,1]
 => 12
[1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => [2,1,1,1]
 => 2
[1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [2,2,2,1]
 => 8
[1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [3,3,2,1]
 => 18
[1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [4,3,2,1]
 => 24
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => [1,0,1,1,0,1,1,0,0,0,1,0]
 => [5,2,2,1,1]
 => 20
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,1,0,0]
 => [4,3,3,1,1]
 => 36
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,0,0]
 => [1,0,1,1,0,1,1,0,0,1,0,0]
 => [4,2,2,1,1]
 => 16
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,0,1,1,1,0,0,0,0]
 => [2,2,2,1,1]
 => 8
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,1,1,0,0,0]
 => [3,3,3,1,1]
 => 27
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,0,1,1,0,1,0,0,0]
 => [3,2,2,1,1]
 => 12
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => [1,0,1,1,0,1,0,0,1,1,0,0]
 => [4,4,2,1,1]
 => 32
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,0,1,0,1,1,0,0,0]
 => [3,3,2,1,1]
 => 18
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,1]
 => 30
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1,1]
 => 24
[1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0,1,0]
 => [5,3,1,1,1]
 => 15
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,1,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [5,1,1,1,1]
 => 5
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => [4,3,1,1,1]
 => 12
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,1,0,1,0,0,0]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [4,1,1,1,1]
 => 4
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [2,1,1,1,1]
 => 2
[1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [1,0,1,1,1,0,0,1,1,0,0,0]
 => [3,3,1,1,1]
 => 9
[1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,1,0,0,0]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [3,1,1,1,1]
 => 3
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [4,4,1,1,1]
 => 16
[1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,1,0,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,1,1,0,0,0]
 => [3,3,2,2,1]
 => 36
[1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [5,4,1,1,1]
 => 20
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,2,1]
 => 48
[1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [2,2,1,1,1]
 => 4
[1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1]
 => 1
[1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => [4,2,1,1,1]
 => 8
[1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,1,0,0,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => [4,2,2,2,1]
 => 32
[1,1,1,0,0,1,1,0,0,0]
 => [1,1,1,1,0,0,1,1,0,0,0,0]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [2,2,2,2,1]
 => 16
[1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => [1,0,1,1,1,0,1,0,0,0,1,0]
 => [5,2,1,1,1]
 => 10
[1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,1,0,1,0,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0,1,0]
 => [5,2,2,2,1]
 => 40
[1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => [3,2,1,1,1]
 => 6
Description
The product of the parts of an integer partition.
Matching statistic: St001808
(load all 4 compositions to match this statistic)
Mapping
Mp00122: Dyck paths —Elizalde-Deutsch bijection⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001808: Dyck paths ⟶ ℤResult quality: 92% ●values known / values provided: 92%●distinct values known / distinct values provided: 95%
Values
[1,0,1,0]
 => [1,1,0,0]
 => []
 => []
 => ? = 1
[1,1,0,0]
 => [1,0,1,0]
 => [1]
 => [1,0,1,0]
 => 2
[1,0,1,0,1,0]
 => [1,1,0,0,1,0]
 => [2]
 => [1,1,0,0,1,0]
 => 3
[1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => [1]
 => [1,0,1,0]
 => 2
[1,1,0,0,1,0]
 => [1,1,1,0,0,0]
 => []
 => []
 => ? = 1
[1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => [1,1]
 => [1,0,1,1,0,0]
 => 4
[1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => [2,1]
 => [1,0,1,0,1,0]
 => 6
[1,0,1,0,1,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 9
[1,0,1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [1,1]
 => [1,0,1,1,0,0]
 => 4
[1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 12
[1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 8
[1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => [2,1]
 => [1,0,1,0,1,0]
 => 6
[1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [2]
 => [1,1,0,0,1,0]
 => 3
[1,1,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0]
 => []
 => []
 => ? = 1
[1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 4
[1,1,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 16
[1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 12
[1,1,1,0,0,0,1,0]
 => [1,1,1,0,1,0,0,0]
 => [1]
 => [1,0,1,0]
 => 2
[1,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 8
[1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => 18
[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]
 => 24
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => 20
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => [3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => 36
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 16
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 8
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => 27
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 12
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => [3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => 32
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => 18
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => [4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => 30
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 24
[1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => 15
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 5
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 12
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 4
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1]
 => [1,0,1,0]
 => 2
[1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 9
[1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [2]
 => [1,1,0,0,1,0]
 => 3
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => 16
[1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => [2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => 36
[1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => 20
[1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => [3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => 48
[1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,1]
 => [1,0,1,1,0,0]
 => 4
[1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => []
 => ? = 1
[1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 8
[1,1,1,0,0,1,0,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 32
[1,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 16
[1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => 10
[1,1,1,0,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => 40
[1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [2,1]
 => [1,0,1,0,1,0]
 => 6
[1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => 24
[1,1,1,1,0,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => 54
[1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 32
[1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => 24
[1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => []
 => ? = 1
[1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => []
 => []
 => ? = 1
[1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => []
 => []
 => ? = 1
[1,1,1,1,1,0,0,0,0,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => []
 => []
 => ? = 1
[1,1,1,1,1,0,0,0,0,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
 => []
 => []
 => ? = 1
Description
The box weight or horizontal decoration of a Dyck path. Let a Dyck path $D = (d_1,d_2,\dots,d_n)$ with steps $d_i \in \{N=(0,1),E=(1,0)\}$ be given. For the $i$th step $d_i \in D$ we define the weight $$ \beta(d_i) = 1, \quad \text{ if } d_i=N, $$ and $$ \beta(d_i) = \sum_{k = 1}^{i} [\![ d_k = N]\!], \quad \text{ if } d_i=E, $$ where we use the Iverson bracket $[\![ A ]\!]$ that is equal to $1$ if $A$ is true, and $0$ otherwise. The '''box weight''' or '''horizontal deocration''' of $D$ is defined as $$ \prod_{i=1}^{n} \beta(d_i). $$ The name describes the fact that between each $E$ step and the line $y=-1$ exactly one unit box is marked.
Matching statistic: St001813
Mapping
Mp00122: Dyck paths —Elizalde-Deutsch bijection⟶ Dyck paths
Mp00242: Dyck paths —Hessenberg poset⟶ Posets
St001813: Posets ⟶ ℤResult quality: 84% ●values known / values provided: 84%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
 => [1,1,0,0]
 => ([],2)
 => 1
[1,1,0,0]
 => [1,0,1,0]
 => ([(0,1)],2)
 => 2
[1,0,1,0,1,0]
 => [1,1,0,0,1,0]
 => ([(0,1),(0,2)],3)
 => 3
[1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => ([(1,2)],3)
 => 2
[1,1,0,0,1,0]
 => [1,1,1,0,0,0]
 => ([],3)
 => 1
[1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => ([(0,2),(1,2)],3)
 => 4
[1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => ([(0,2),(2,1)],3)
 => 6
[1,0,1,0,1,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 9
[1,0,1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => ([(1,3),(2,3)],4)
 => 4
[1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => ([(0,3),(3,1),(3,2)],4)
 => 12
[1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => ([(0,2),(0,3),(3,1)],4)
 => 8
[1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => ([(0,3),(1,2),(1,3)],4)
 => 6
[1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => ([(1,2),(1,3)],4)
 => 3
[1,1,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0]
 => ([],4)
 => 1
[1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => ([(0,1),(0,2),(0,3)],4)
 => 4
[1,1,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 16
[1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0]
 => ([(0,3),(1,2),(2,3)],4)
 => 12
[1,1,1,0,0,0,1,0]
 => [1,1,1,0,1,0,0,0]
 => ([(2,3)],4)
 => 2
[1,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => ([(0,3),(1,3),(2,3)],4)
 => 8
[1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => ([(0,3),(1,3),(3,2)],4)
 => 18
[1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 24
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => 20
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
 => 36
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => ([(0,4),(1,2),(1,3),(3,4)],5)
 => 16
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => ([(1,4),(2,4),(3,4)],5)
 => 8
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 27
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => ([(0,4),(1,4),(2,3),(2,4)],5)
 => 12
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => 32
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => ([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 18
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => 30
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => ([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
 => 24
[1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => ([(0,3),(0,4),(4,1),(4,2)],5)
 => 15
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => ([(0,1),(0,2),(0,3),(0,4)],5)
 => 5
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => ([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => 12
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => ([(1,2),(1,3),(1,4)],5)
 => 4
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => ([(3,4)],5)
 => 2
[1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => 9
[1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => ([(2,3),(2,4)],5)
 => 3
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => 16
[1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => 36
[1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => ([(0,4),(4,1),(4,2),(4,3)],5)
 => 20
[1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => 48
[1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => ([(2,4),(3,4)],5)
 => 4
[1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => ([],5)
 => 1
[1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => ([(0,4),(1,2),(1,3),(1,4)],5)
 => 8
[1,1,1,0,0,1,0,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 32
[1,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 16
[1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => ([(0,2),(0,3),(0,4),(4,1)],5)
 => 10
[1,1,1,0,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 40
[1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => ([(1,4),(2,3),(2,4)],5)
 => 6
[1,1,1,0,0,0,1,1,1,0,0,1,0,0]
 => [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
 => ([(2,5),(2,6),(3,4),(3,5),(3,6)],7)
 => ? = 12
[1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
 => ([(4,7),(5,6),(5,7)],8)
 => ? = 6
[1,1,1,0,0,1,0,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
 => ([(3,7),(4,5),(4,6),(4,7)],8)
 => ? = 8
[1,1,1,0,1,0,0,0,1,1,1,0,1,0,0,0]
 => [1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0]
 => ([(4,6),(4,7),(5,6),(5,7)],8)
 => ? = 9
[1,1,1,0,1,0,0,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
 => ([(5,7),(6,7)],8)
 => ? = 4
[1,1,1,1,0,0,0,0,1,1,1,0,1,0,0,0]
 => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
 => ([(5,6),(5,7)],8)
 => ? = 3
[1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ([],8)
 => ? = 1
[1,1,1,1,0,1,0,0,0,0,1,1,1,0,0,0]
 => [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
 => ([(4,5),(4,6),(4,7)],8)
 => ? = 4
[1,1,1,1,1,0,0,0,0,0,1,1,0,1,0,0]
 => [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
 => ([(4,7),(5,7),(6,7)],8)
 => ? = 8
[1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
 => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => ([(6,7)],8)
 => ? = 2
[1,1,1,1,0,0,0,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
 => ([(7,8)],9)
 => ? = 2
[1,1,1,1,1,0,0,0,0,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => ([],9)
 => ? = 1
[1,1,1,1,1,1,0,0,0,0,0,0,1,1,1,0,0,0]
 => [1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]
 => ([(5,8),(6,7),(6,8)],9)
 => ? = 6
[1,1,1,1,1,0,0,0,0,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
 => ([],10)
 => ? = 1
[1,1,1,1,1,1,0,0,0,0,0,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
 => ([(8,9)],10)
 => ? = 2
[1,1,1,0,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0]
 => ([(5,8),(6,8),(7,8)],9)
 => ? = 8
[1,1,1,1,0,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
 => ([(6,7),(6,8)],9)
 => ? = 3
[1,1,1,1,0,0,0,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0]
 => ([(5,6),(5,7),(5,8)],9)
 => ? = 4
[1,1,1,1,1,0,0,0,0,0,1,1,1,0,1,0,0,0]
 => [1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
 => ([(6,8),(7,8)],9)
 => ? = 4
Description
The product of the sizes of the principal order filters in a poset.