Your data matches 7 different statistics following compositions of up to 3 maps.
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Matching statistic: St001914
Mapping
St001914: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => 1
[2]
 => 2
[1,1]
 => 2
[3]
 => 2
[2,1]
 => 1
[1,1,1]
 => 3
[4]
 => 4
[3,1]
 => 3
[2,2]
 => 3
[2,1,1]
 => 3
[1,1,1,1]
 => 5
[5]
 => 5
[4,1]
 => 4
[3,2]
 => 3
[3,1,1]
 => 3
[2,2,1]
 => 3
[2,1,1,1]
 => 5
[1,1,1,1,1]
 => 6
[6]
 => 4
[5,1]
 => 3
[4,2]
 => 2
[4,1,1]
 => 6
[3,3]
 => 5
[3,2,1]
 => 1
[3,1,1,1]
 => 3
[2,2,2]
 => 4
[2,2,1,1]
 => 7
[2,1,1,1,1]
 => 4
[1,1,1,1,1,1]
 => 5
[7]
 => 7
[6,1]
 => 6
[5,2]
 => 5
[5,1,1]
 => 6
[4,3]
 => 5
[4,2,1]
 => 4
[4,1,1,1]
 => 6
[3,3,1]
 => 4
[3,2,2]
 => 4
[3,2,1,1]
 => 4
[3,1,1,1,1]
 => 6
[2,2,2,1]
 => 7
[2,2,1,1,1]
 => 7
[2,1,1,1,1,1]
 => 7
[1,1,1,1,1,1,1]
 => 8
[8]
 => 8
[7,1]
 => 7
[6,2]
 => 6
[6,1,1]
 => 4
[5,3]
 => 3
[5,2,1]
 => 5
Description
The size of the orbit of an integer partition in Bulgarian solitaire. Bulgarian solitaire is the dynamical system where a move consists of removing the first column of the Ferrers diagram and inserting it as a row. This statistic returns the number of partitions that can be obtained from the given partition.
Matching statistic: St001232
(load all 9 compositions to match this statistic)
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00142: Dyck paths —promotion⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 57%
Values
[1]
 => [1,0,1,0]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1
[2]
 => [1,1,0,0,1,0]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 2
[1,1]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 2
[3]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => ? ∊ {1,2}
[2,1]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => ? ∊ {1,2}
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 3
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => ? = 3
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 4
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => 3
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => 5
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => ? ∊ {3,3,3,4,5,5,6}
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => ? ∊ {3,3,3,4,5,5,6}
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => ? ∊ {3,3,3,4,5,5,6}
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0]
 => ? ∊ {3,3,3,4,5,5,6}
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => ? ∊ {3,3,3,4,5,5,6}
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => ? ∊ {3,3,3,4,5,5,6}
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => ? ∊ {3,3,3,4,5,5,6}
[6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
 => ? ∊ {1,2,3,3,4,5,6}
[5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => ? ∊ {1,2,3,3,4,5,6}
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => ? ∊ {1,2,3,3,4,5,6}
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => ? ∊ {1,2,3,3,4,5,6}
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => ? ∊ {1,2,3,3,4,5,6}
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => ? ∊ {1,2,3,3,4,5,6}
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 4
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => 5
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => 4
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,0,1,1,1,0,1,0,0,0]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => 7
[1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,1,1,1,0,0,1,0,1,0,0,0]
 => ? ∊ {1,2,3,3,4,5,6}
[7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
 => ? ∊ {4,4,5,5,6,6,7,7,7,7,8}
[6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
 => ? ∊ {4,4,5,5,6,6,7,7,7,7,8}
[5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => ? ∊ {4,4,5,5,6,6,7,7,7,7,8}
[5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => [1,1,1,1,0,1,1,0,0,0,0,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => ? ∊ {4,4,5,5,6,6,7,7,7,7,8}
[4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => ? ∊ {4,4,5,5,6,6,7,7,7,7,8}
[4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 4
[4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => ? ∊ {4,4,5,5,6,6,7,7,7,7,8}
[3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 6
[3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 6
[3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 4
[3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0,1,0]
 => ? ∊ {4,4,5,5,6,6,7,7,7,7,8}
[2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => ? ∊ {4,4,5,5,6,6,7,7,7,7,8}
[2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,1,0,0,1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,1,0,1,0,0]
 => ? ∊ {4,4,5,5,6,6,7,7,7,7,8}
[2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,1,0,0,1,1,1,1,0,1,0,0,0,0]
 => [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
 => ? ∊ {4,4,5,5,6,6,7,7,7,7,8}
[1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,0,1,1,1,1,0,0,1,0,1,0,1,0,0,0]
 => ? ∊ {4,4,5,5,6,6,7,7,7,7,8}
[8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0]
 => ? ∊ {2,2,3,4,4,4,4,4,4,5,5,5,5,6,6,7,7,8,9}
[7,1]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
 => ? ∊ {2,2,3,4,4,4,4,4,4,5,5,5,5,6,6,7,7,8,9}
[6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => [1,1,1,0,1,1,0,1,0,1,0,0,0,0]
 => ? ∊ {2,2,3,4,4,4,4,4,4,5,5,5,5,6,6,7,7,8,9}
[6,1,1]
 => [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
 => [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,1,0,0,0]
 => ? ∊ {2,2,3,4,4,4,4,4,4,5,5,5,5,6,6,7,7,8,9}
[5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [1,1,1,1,1,0,0,0,1,0,0,0]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => ? ∊ {2,2,3,4,4,4,4,4,4,5,5,5,5,6,6,7,7,8,9}
[5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {2,2,3,4,4,4,4,4,4,5,5,5,5,6,6,7,7,8,9}
[5,1,1,1]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,1,1,1,0,1,0,0,1,0,0,0]
 => ? ∊ {2,2,3,4,4,4,4,4,4,5,5,5,5,6,6,7,7,8,9}
[4,4]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,1,0,1,0,1,1,0,0,0,0]
 => ? ∊ {2,2,3,4,4,4,4,4,4,5,5,5,5,6,6,7,7,8,9}
[4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => ? ∊ {2,2,3,4,4,4,4,4,4,5,5,5,5,6,6,7,7,8,9}
[4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => ? ∊ {2,2,3,4,4,4,4,4,4,5,5,5,5,6,6,7,7,8,9}
[4,2,1,1]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => ? ∊ {2,2,3,4,4,4,4,4,4,5,5,5,5,6,6,7,7,8,9}
[4,1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,1,1,0,0,0]
 => 7
[3,3,2]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => ? ∊ {2,2,3,4,4,4,4,4,4,5,5,5,5,6,6,7,7,8,9}
[3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => ? ∊ {2,2,3,4,4,4,4,4,4,5,5,5,5,6,6,7,7,8,9}
[3,2,2,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => ? ∊ {2,2,3,4,4,4,4,4,4,5,5,5,5,6,6,7,7,8,9}
[3,2,1,1,1]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,1,0,0,1,1,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0]
 => ? ∊ {2,2,3,4,4,4,4,4,4,5,5,5,5,6,6,7,7,8,9}
[3,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
 => [1,1,0,0,1,1,1,1,0,0,1,0,0,0]
 => [1,0,1,1,1,0,0,1,1,0,1,0,0,0]
 => ? ∊ {2,2,3,4,4,4,4,4,4,5,5,5,5,6,6,7,7,8,9}
[2,2,2,2]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => 8
[2,2,2,1,1]
 => [1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,1,0,0,1,0,1,1,1,0,0,0]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => 6
[2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
 => [1,1,0,0,1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,1,0,0,1,0,0]
 => ? ∊ {2,2,3,4,4,4,4,4,4,5,5,5,5,6,6,7,7,8,9}
[2,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => [1,1,0,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,0,1,1,1,1,1,0,0,1,0,1,0,0,0,0]
 => ? ∊ {2,2,3,4,4,4,4,4,4,5,5,5,5,6,6,7,7,8,9}
[1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,1,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,0,1,1,1,1,1,0,0,1,0,1,0,1,0,0,0,0]
 => ? ∊ {2,2,3,4,4,4,4,4,4,5,5,5,5,6,6,7,7,8,9}
[9]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0,0]
 => ? ∊ {4,4,4,4,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9,10,10,11}
[8,1]
 => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
 => [1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
 => ? ∊ {4,4,4,4,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9,10,10,11}
[7,2]
 => [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
 => [1,1,1,1,0,1,0,1,0,1,0,0,1,0,0,0]
 => ? ∊ {4,4,4,4,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9,10,10,11}
[3,3,1,1,1]
 => [1,0,1,1,1,0,0,1,1,0,0,0]
 => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => 5
[3,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
 => [1,1,0,0,1,1,1,0,1,0,1,0,0,0]
 => [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
 => 9
[4,3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,1,1,0,0,0]
 => 5
[4,2,2,2]
 => [1,1,0,0,1,1,1,0,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0,1,1,0,0]
 => 6
[4,2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,1,0,0]
 => 5
[3,3,3,1]
 => [1,1,0,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,0,0,1,1,0,0]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => 7
[3,3,2,2]
 => [1,1,0,0,1,1,0,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,1,0,0]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => 7
[3,3,2,1,1]
 => [1,0,1,1,0,1,0,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,0,0,0,0,1,0]
 => 5
[5,3,2,1]
 => [1,1,0,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 5
[5,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,1,0,0]
 => [1,1,0,0,1,1,1,0,1,0,0,0,1,0]
 => [1,0,1,1,1,1,0,0,1,1,0,0,0,0]
 => 10
[4,4,2,1]
 => [1,1,0,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [1,1,1,1,0,1,1,0,0,0,0,0]
 => 8
[4,3,3,1]
 => [1,1,0,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0,1,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => 9
[4,3,2,2]
 => [1,1,0,0,1,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => 8
[4,3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 5
[3,2,2,2,2]
 => [1,1,0,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,0,0,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,1,0,0,0]
 => 11
[3,2,2,2,1,1]
 => [1,0,1,1,0,1,1,1,0,1,0,0,0,0]
 => [1,1,0,0,1,0,1,1,1,0,1,0,0,0]
 => [1,0,1,1,1,1,1,0,0,0,1,0,0,0]
 => 8
[4,4,1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,1,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,1,1,0,0,1,1,0,0,0,1,0]
 => 8
[3,3,3,1,1,1]
 => [1,0,1,1,1,0,0,1,1,1,0,0,0,0]
 => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,0,1,1,1,0,0,1,0,0]
 => 7
[5,4,1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,1,0,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,1,1,1,0,0,0,0]
 => 9
[5,2,2,2,2]
 => [1,1,0,0,1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,1,0,0,0]
 => 10
[5,2,2,2,1,1]
 => [1,0,1,1,0,1,1,1,0,0,0,1,0,0]
 => [1,1,0,0,1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,1,0,0,0,1,1,0,0,0]
 => 8
[3,3,3,3,1]
 => [1,1,0,1,0,0,1,1,1,1,0,0,0,0]
 => [1,1,1,0,1,0,0,0,1,1,1,0,0,0]
 => [1,1,1,0,1,1,1,0,0,0,0,1,0,0]
 => 11
[3,3,3,2,2]
 => [1,1,0,0,1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,1,0,0]
 => 10
[3,3,3,2,1,1]
 => [1,0,1,1,0,1,0,1,1,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
 => 7
[5,3,3,1,1,1]
 => [1,0,1,1,1,0,0,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => 6
[4,4,3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,1,1,0,0,0]
 => [1,1,0,0,1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
 => 6
[4,4,2,2,2]
 => [1,1,0,0,1,1,1,0,0,1,1,0,0,0]
 => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0,1,1,0,0,1,0]
 => 7
[4,4,2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,1,1,0,0,0]
 => [1,1,0,0,1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
 => 6
[5,4,3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => 6
[5,4,2,2,2]
 => [1,1,0,0,1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,0,1,1,1,0,0,0]
 => 7
[5,4,2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => 6
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Matching statistic: St001879
(load all 6 compositions to match this statistic)
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00140: Dyck paths —logarithmic height to pruning number⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St001879: Posets ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 24%
Values
[1]
 => [1,0,1,0]
 => [.,[.,.]]
 => ([(0,1)],2)
 => ? = 1
[2]
 => [1,1,0,0,1,0]
 => [[.,[.,.]],.]
 => ([(0,2),(2,1)],3)
 => 2
[1,1]
 => [1,0,1,1,0,0]
 => [.,[[.,.],.]]
 => ([(0,2),(2,1)],3)
 => 2
[3]
 => [1,1,1,0,0,0,1,0]
 => [[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? ∊ {1,3}
[2,1]
 => [1,0,1,0,1,0]
 => [.,[.,[.,.]]]
 => ([(0,2),(2,1)],3)
 => 2
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(3,2)],4)
 => ? ∊ {1,3}
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [[[.,.],.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? ∊ {4,5}
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [[[.,[.,.]],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 3
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [[.,[[.,.],.]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 3
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [.,[[[.,.],.],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => 3
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [.,[[[.,.],.],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? ∊ {4,5}
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [[[.,.],[.,[.,.]]],[.,.]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => ? ∊ {4,5,5,6}
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [[.,[.,.]],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? ∊ {4,5,5,6}
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [[.,[.,[.,.]]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 3
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [.,[[.,[.,.]],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => 3
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [.,[.,[[.,.],.]]]
 => ([(0,3),(2,1),(3,2)],4)
 => 3
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [.,[[.,[.,.]],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? ∊ {4,5,5,6}
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [.,[[[.,.],[.,.]],[.,.]]]
 => ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
 => ? ∊ {4,5,5,6}
[6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [[[.,.],[.,[.,.]]],[[.,.],.]]
 => ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7)
 => ? ∊ {1,2,3,4,4,4,5,5,6,7}
[5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [[[[.,.],.],.],[.,[.,.]]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ? ∊ {1,2,3,4,4,4,5,5,6,7}
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [[.,.],[[.,[.,.]],.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? ∊ {1,2,3,4,4,4,5,5,6,7}
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [[[.,.],[.,[.,.]]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? ∊ {1,2,3,4,4,4,5,5,6,7}
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [[.,.],[.,[[.,.],.]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? ∊ {1,2,3,4,4,4,5,5,6,7}
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [.,[.,[.,[.,.]]]]
 => ([(0,3),(2,1),(3,2)],4)
 => 3
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [.,[[.,.],[[.,.],.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? ∊ {1,2,3,4,4,4,5,5,6,7}
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [[.,[[.,.],[.,.]]],.]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ? ∊ {1,2,3,4,4,4,5,5,6,7}
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [.,[[[.,.],[.,.]],.]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ? ∊ {1,2,3,4,4,4,5,5,6,7}
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [.,[[[[.,.],.],.],[.,.]]]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => ? ∊ {1,2,3,4,4,4,5,5,6,7}
[1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [.,[[[.,.],[.,.]],[[.,.],.]]]
 => ([(0,5),(1,5),(2,3),(3,6),(5,6),(6,4)],7)
 => ? ∊ {1,2,3,4,4,4,5,5,6,7}
[7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [[[.,.],[.,.]],[[.,.],[.,[.,.]]]]
 => ([(0,7),(1,5),(2,5),(3,4),(4,7),(5,6),(7,6)],8)
 => ? ∊ {5,5,6,6,6,6,7,7,7,7,8}
[6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [[[.,.],[.,[.,.]]],[.,[.,.]]]
 => ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7)
 => ? ∊ {5,5,6,6,6,6,7,7,7,7,8}
[5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [[[.,[.,.]],.],[.,[.,.]]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ? ∊ {5,5,6,6,6,6,7,7,7,7,8}
[5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => [[.,[[.,.],.]],[.,[.,.]]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ? ∊ {5,5,6,6,6,6,7,7,7,7,8}
[4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [[.,.],[.,[.,[.,.]]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? ∊ {5,5,6,6,6,6,7,7,7,7,8}
[4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [[[[.,[.,.]],.],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4
[4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => [.,[[.,.],[.,[.,.]]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? ∊ {5,5,6,6,6,6,7,7,7,7,8}
[3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [[[.,[[.,.],.]],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4
[3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [[.,[[[.,.],.],.]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4
[3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [.,[[[[.,.],.],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4
[3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [.,[[[.,[.,.]],.],[.,.]]]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => ? ∊ {5,5,6,6,6,6,7,7,7,7,8}
[2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [.,[.,[[.,.],[.,.]]]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ? ∊ {5,5,6,6,6,6,7,7,7,7,8}
[2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [.,[[.,[[.,.],.]],[.,.]]]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => ? ∊ {5,5,6,6,6,6,7,7,7,7,8}
[2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [.,[[[.,.],[.,.]],[.,[.,.]]]]
 => ([(0,5),(1,5),(2,3),(3,6),(5,6),(6,4)],7)
 => ? ∊ {5,5,6,6,6,6,7,7,7,7,8}
[1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [.,[[[.,.],[.,.]],[[.,.],[.,.]]]]
 => ([(0,6),(1,6),(2,5),(3,5),(5,7),(6,7),(7,4)],8)
 => ? ∊ {5,5,6,6,6,6,7,7,7,7,8}
[8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [[[[.,.],.],[.,.]],[[.,.],[.,[.,.]]]]
 => ([(0,8),(1,7),(2,4),(3,5),(4,7),(5,8),(7,6),(8,6)],9)
 => ? ∊ {2,2,3,5,5,5,5,6,6,6,7,7,7,8,8,9}
[7,1]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => [[[.,.],[.,[.,.]]],[[[.,.],.],.]]
 => ([(0,7),(1,3),(2,4),(3,7),(4,5),(5,6),(7,6)],8)
 => ? ∊ {2,2,3,5,5,5,5,6,6,6,7,7,7,8,8,9}
[6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [[[[.,.],.],[.,[.,.]]],[.,.]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(6,5)],7)
 => ? ∊ {2,2,3,5,5,5,5,6,6,6,7,7,7,8,8,9}
[6,1,1]
 => [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
 => [[[[.,.],[.,[.,.]]],.],[.,.]]
 => ([(0,6),(1,5),(2,3),(3,6),(4,5),(6,4)],7)
 => ? ∊ {2,2,3,5,5,5,5,6,6,6,7,7,7,8,8,9}
[5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [[[.,.],.],[[.,[.,.]],.]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ? ∊ {2,2,3,5,5,5,5,6,6,6,7,7,7,8,8,9}
[5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [[.,[.,[.,.]]],[.,[.,.]]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ? ∊ {2,2,3,5,5,5,5,6,6,6,7,7,7,8,8,9}
[5,1,1,1]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => [[[[.,.],.],[.,[.,.]]],.]
 => ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
 => ? ∊ {2,2,3,5,5,5,5,6,6,6,7,7,7,8,8,9}
[4,4]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [[[.,.],.],[.,[[.,.],.]]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ? ∊ {2,2,3,5,5,5,5,6,6,6,7,7,7,8,8,9}
[4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => [[[.,[.,[.,.]]],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4
[4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [[.,[[.,[.,.]],.]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4
[4,2,1,1]
 => [1,0,1,1,0,1,0,0,1,0]
 => [.,[[[.,[.,.]],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4
[4,1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [.,[[[.,.],.],[[.,.],.]]]
 => ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
 => ? ∊ {2,2,3,5,5,5,5,6,6,6,7,7,7,8,8,9}
[3,3,2]
 => [1,1,0,0,1,0,1,1,0,0]
 => [[.,[.,[[.,.],.]]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4
[3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,0]
 => [.,[[.,[[.,.],.]],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4
[3,2,2,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [.,[.,[[[.,.],.],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4
[3,2,1,1,1]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => [.,[[.,[.,[.,.]]],[.,.]]]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => ? ∊ {2,2,3,5,5,5,5,6,6,6,7,7,7,8,8,9}
[3,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
 => [.,[[[[.,.],.],[.,.]],[.,.]]]
 => ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7)
 => ? ∊ {2,2,3,5,5,5,5,6,6,6,7,7,7,8,8,9}
[2,2,2,2]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [[.,[[[.,.],.],[.,.]]],.]
 => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
 => ? ∊ {2,2,3,5,5,5,5,6,6,6,7,7,7,8,8,9}
[2,2,2,1,1]
 => [1,0,1,1,0,1,1,1,0,0,0,0]
 => [.,[[[[.,.],.],[.,.]],.]]
 => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
 => ? ∊ {2,2,3,5,5,5,5,6,6,6,7,7,7,8,8,9}
[2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
 => [.,[[[[.,.],[.,.]],.],[.,.]]]
 => ([(0,6),(1,5),(2,5),(4,6),(5,4),(6,3)],7)
 => ? ∊ {2,2,3,5,5,5,5,6,6,6,7,7,7,8,8,9}
[2,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => [.,[[[.,.],[.,.]],[[[.,.],.],.]]]
 => ([(0,6),(1,6),(2,4),(4,5),(5,7),(6,7),(7,3)],8)
 => ? ∊ {2,2,3,5,5,5,5,6,6,6,7,7,7,8,8,9}
[1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [.,[[[[.,.],.],[.,.]],[[.,.],[.,.]]]]
 => ([(0,8),(1,6),(2,6),(3,5),(5,8),(6,7),(7,4),(8,7)],9)
 => ? ∊ {2,2,3,5,5,5,5,6,6,6,7,7,7,8,8,9}
[9]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => [[[[.,.],[.,.]],[.,.]],[[.,.],[.,[.,.]]]]
 => ([(0,6),(1,6),(2,9),(3,8),(4,5),(5,9),(6,8),(8,7),(9,7)],10)
 => ? ∊ {5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9,9,10,10,11}
[8,1]
 => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
 => [[[.,[.,.]],[.,.]],[[.,.],[.,[.,.]]]]
 => ([(0,8),(1,7),(2,4),(3,5),(4,7),(5,8),(7,6),(8,6)],9)
 => ? ∊ {5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9,9,10,10,11}
[7,2]
 => [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
 => [[[.,.],[.,[.,.]]],[[.,[.,.]],.]]
 => ([(0,7),(1,3),(2,4),(3,7),(4,5),(5,6),(7,6)],8)
 => ? ∊ {5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9,9,10,10,11}
[7,1,1]
 => [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
 => [[[.,.],[.,[.,.]]],[.,[[.,.],.]]]
 => ([(0,7),(1,3),(2,4),(3,7),(4,5),(5,6),(7,6)],8)
 => ? ∊ {5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9,9,10,10,11}
[4,3,2]
 => [1,1,0,0,1,0,1,0,1,0]
 => [[.,[.,[.,[.,.]]]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4
[4,3,1,1]
 => [1,0,1,1,0,0,1,0,1,0]
 => [.,[[.,[.,[.,.]]],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4
[4,2,2,1]
 => [1,0,1,0,1,1,0,0,1,0]
 => [.,[.,[[.,[.,.]],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4
[3,3,2,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [.,[.,[.,[[.,.],.]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4
[4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => [.,[.,[.,[.,[.,.]]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4
[5,3,2,1]
 => [1,1,0,1,0,1,0,1,0,0,1,0]
 => [[[[[.,[.,.]],.],.],.],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5
[4,4,2,1]
 => [1,1,0,1,0,1,0,0,1,1,0,0]
 => [[[[.,[[.,.],.]],.],.],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5
[4,3,3,1]
 => [1,1,0,1,0,0,1,1,0,1,0,0]
 => [[[.,[[[.,.],.],.]],.],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5
[4,3,2,2]
 => [1,1,0,0,1,1,0,1,0,1,0,0]
 => [[.,[[[[.,.],.],.],.]],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5
[4,3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [.,[[[[[.,.],.],.],.],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5
[5,4,2,1]
 => [1,1,0,1,0,1,0,0,1,0,1,0]
 => [[[[.,[.,[.,.]]],.],.],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5
[5,3,3,1]
 => [1,1,0,1,0,0,1,1,0,0,1,0]
 => [[[.,[[.,[.,.]],.]],.],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5
[5,3,2,2]
 => [1,1,0,0,1,1,0,1,0,0,1,0]
 => [[.,[[[.,[.,.]],.],.]],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5
[5,3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0,1,0]
 => [.,[[[[.,[.,.]],.],.],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5
[4,4,3,1]
 => [1,1,0,1,0,0,1,0,1,1,0,0]
 => [[[.,[.,[[.,.],.]]],.],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5
[4,4,2,2]
 => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [[.,[[.,[[.,.],.]],.]],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5
[4,4,2,1,1]
 => [1,0,1,1,0,1,0,0,1,1,0,0]
 => [.,[[[.,[[.,.],.]],.],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5
[4,3,3,2]
 => [1,1,0,0,1,0,1,1,0,1,0,0]
 => [[.,[.,[[[.,.],.],.]]],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5
[4,3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,1,0,0]
 => [.,[[.,[[[.,.],.],.]],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5
[4,3,2,2,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => [.,[.,[[[[.,.],.],.],.]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5
[5,4,3,1]
 => [1,1,0,1,0,0,1,0,1,0,1,0]
 => [[[.,[.,[.,[.,.]]]],.],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5
[5,4,2,2]
 => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [[.,[[.,[.,[.,.]]],.]],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5
[5,4,2,1,1]
 => [1,0,1,1,0,1,0,0,1,0,1,0]
 => [.,[[[.,[.,[.,.]]],.],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5
[5,3,3,2]
 => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [[.,[.,[[.,[.,.]],.]]],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5
[5,3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => [.,[[.,[[.,[.,.]],.]],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5
[5,3,2,2,1]
 => [1,0,1,0,1,1,0,1,0,0,1,0]
 => [.,[.,[[[.,[.,.]],.],.]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5
[4,4,3,2]
 => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [[.,[.,[.,[[.,.],.]]]],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5
[4,4,3,1,1]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [.,[[.,[.,[[.,.],.]]],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5
[4,4,2,2,1]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [.,[.,[[.,[[.,.],.]],.]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5
[4,3,3,2,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [.,[.,[.,[[[.,.],.],.]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5
Description
The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice.
Matching statistic: St001880
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00140: Dyck paths —logarithmic height to pruning number⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St001880: Posets ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 24%
Values
[1]
 => [1,0,1,0]
 => [.,[.,.]]
 => ([(0,1)],2)
 => ? = 1 + 1
[2]
 => [1,1,0,0,1,0]
 => [[.,[.,.]],.]
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[1,1]
 => [1,0,1,1,0,0]
 => [.,[[.,.],.]]
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[3]
 => [1,1,1,0,0,0,1,0]
 => [[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? ∊ {1,3} + 1
[2,1]
 => [1,0,1,0,1,0]
 => [.,[.,[.,.]]]
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(3,2)],4)
 => ? ∊ {1,3} + 1
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [[[.,.],.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? ∊ {4,5} + 1
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [[[.,[.,.]],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 3 + 1
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [[.,[[.,.],.]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 3 + 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [.,[[[.,.],.],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 3 + 1
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [.,[[[.,.],.],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? ∊ {4,5} + 1
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [[[.,.],[.,[.,.]]],[.,.]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => ? ∊ {4,5,5,6} + 1
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [[.,[.,.]],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? ∊ {4,5,5,6} + 1
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [[.,[.,[.,.]]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 3 + 1
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [.,[[.,[.,.]],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 3 + 1
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [.,[.,[[.,.],.]]]
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 3 + 1
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [.,[[.,[.,.]],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? ∊ {4,5,5,6} + 1
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [.,[[[.,.],[.,.]],[.,.]]]
 => ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
 => ? ∊ {4,5,5,6} + 1
[6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [[[.,.],[.,[.,.]]],[[.,.],.]]
 => ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7)
 => ? ∊ {1,2,3,4,4,4,5,5,6,7} + 1
[5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [[[[.,.],.],.],[.,[.,.]]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ? ∊ {1,2,3,4,4,4,5,5,6,7} + 1
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [[.,.],[[.,[.,.]],.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? ∊ {1,2,3,4,4,4,5,5,6,7} + 1
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [[[.,.],[.,[.,.]]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? ∊ {1,2,3,4,4,4,5,5,6,7} + 1
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [[.,.],[.,[[.,.],.]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? ∊ {1,2,3,4,4,4,5,5,6,7} + 1
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [.,[.,[.,[.,.]]]]
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 3 + 1
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [.,[[.,.],[[.,.],.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? ∊ {1,2,3,4,4,4,5,5,6,7} + 1
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [[.,[[.,.],[.,.]]],.]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ? ∊ {1,2,3,4,4,4,5,5,6,7} + 1
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [.,[[[.,.],[.,.]],.]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ? ∊ {1,2,3,4,4,4,5,5,6,7} + 1
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [.,[[[[.,.],.],.],[.,.]]]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => ? ∊ {1,2,3,4,4,4,5,5,6,7} + 1
[1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [.,[[[.,.],[.,.]],[[.,.],.]]]
 => ([(0,5),(1,5),(2,3),(3,6),(5,6),(6,4)],7)
 => ? ∊ {1,2,3,4,4,4,5,5,6,7} + 1
[7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [[[.,.],[.,.]],[[.,.],[.,[.,.]]]]
 => ([(0,7),(1,5),(2,5),(3,4),(4,7),(5,6),(7,6)],8)
 => ? ∊ {5,5,6,6,6,6,7,7,7,7,8} + 1
[6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [[[.,.],[.,[.,.]]],[.,[.,.]]]
 => ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7)
 => ? ∊ {5,5,6,6,6,6,7,7,7,7,8} + 1
[5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [[[.,[.,.]],.],[.,[.,.]]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ? ∊ {5,5,6,6,6,6,7,7,7,7,8} + 1
[5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => [[.,[[.,.],.]],[.,[.,.]]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ? ∊ {5,5,6,6,6,6,7,7,7,7,8} + 1
[4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [[.,.],[.,[.,[.,.]]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? ∊ {5,5,6,6,6,6,7,7,7,7,8} + 1
[4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [[[[.,[.,.]],.],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 4 + 1
[4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => [.,[[.,.],[.,[.,.]]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? ∊ {5,5,6,6,6,6,7,7,7,7,8} + 1
[3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [[[.,[[.,.],.]],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 4 + 1
[3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [[.,[[[.,.],.],.]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 4 + 1
[3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [.,[[[[.,.],.],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 4 + 1
[3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [.,[[[.,[.,.]],.],[.,.]]]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => ? ∊ {5,5,6,6,6,6,7,7,7,7,8} + 1
[2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [.,[.,[[.,.],[.,.]]]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ? ∊ {5,5,6,6,6,6,7,7,7,7,8} + 1
[2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [.,[[.,[[.,.],.]],[.,.]]]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => ? ∊ {5,5,6,6,6,6,7,7,7,7,8} + 1
[2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [.,[[[.,.],[.,.]],[.,[.,.]]]]
 => ([(0,5),(1,5),(2,3),(3,6),(5,6),(6,4)],7)
 => ? ∊ {5,5,6,6,6,6,7,7,7,7,8} + 1
[1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [.,[[[.,.],[.,.]],[[.,.],[.,.]]]]
 => ([(0,6),(1,6),(2,5),(3,5),(5,7),(6,7),(7,4)],8)
 => ? ∊ {5,5,6,6,6,6,7,7,7,7,8} + 1
[8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [[[[.,.],.],[.,.]],[[.,.],[.,[.,.]]]]
 => ([(0,8),(1,7),(2,4),(3,5),(4,7),(5,8),(7,6),(8,6)],9)
 => ? ∊ {2,2,3,5,5,5,5,6,6,6,7,7,7,8,8,9} + 1
[7,1]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => [[[.,.],[.,[.,.]]],[[[.,.],.],.]]
 => ([(0,7),(1,3),(2,4),(3,7),(4,5),(5,6),(7,6)],8)
 => ? ∊ {2,2,3,5,5,5,5,6,6,6,7,7,7,8,8,9} + 1
[6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [[[[.,.],.],[.,[.,.]]],[.,.]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(6,5)],7)
 => ? ∊ {2,2,3,5,5,5,5,6,6,6,7,7,7,8,8,9} + 1
[6,1,1]
 => [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
 => [[[[.,.],[.,[.,.]]],.],[.,.]]
 => ([(0,6),(1,5),(2,3),(3,6),(4,5),(6,4)],7)
 => ? ∊ {2,2,3,5,5,5,5,6,6,6,7,7,7,8,8,9} + 1
[5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [[[.,.],.],[[.,[.,.]],.]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ? ∊ {2,2,3,5,5,5,5,6,6,6,7,7,7,8,8,9} + 1
[5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [[.,[.,[.,.]]],[.,[.,.]]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ? ∊ {2,2,3,5,5,5,5,6,6,6,7,7,7,8,8,9} + 1
[5,1,1,1]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => [[[[.,.],.],[.,[.,.]]],.]
 => ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
 => ? ∊ {2,2,3,5,5,5,5,6,6,6,7,7,7,8,8,9} + 1
[4,4]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [[[.,.],.],[.,[[.,.],.]]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ? ∊ {2,2,3,5,5,5,5,6,6,6,7,7,7,8,8,9} + 1
[4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => [[[.,[.,[.,.]]],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 4 + 1
[4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [[.,[[.,[.,.]],.]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 4 + 1
[4,2,1,1]
 => [1,0,1,1,0,1,0,0,1,0]
 => [.,[[[.,[.,.]],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 4 + 1
[4,1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [.,[[[.,.],.],[[.,.],.]]]
 => ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
 => ? ∊ {2,2,3,5,5,5,5,6,6,6,7,7,7,8,8,9} + 1
[3,3,2]
 => [1,1,0,0,1,0,1,1,0,0]
 => [[.,[.,[[.,.],.]]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 4 + 1
[3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,0]
 => [.,[[.,[[.,.],.]],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 4 + 1
[3,2,2,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [.,[.,[[[.,.],.],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 4 + 1
[3,2,1,1,1]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => [.,[[.,[.,[.,.]]],[.,.]]]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => ? ∊ {2,2,3,5,5,5,5,6,6,6,7,7,7,8,8,9} + 1
[3,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
 => [.,[[[[.,.],.],[.,.]],[.,.]]]
 => ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7)
 => ? ∊ {2,2,3,5,5,5,5,6,6,6,7,7,7,8,8,9} + 1
[2,2,2,2]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [[.,[[[.,.],.],[.,.]]],.]
 => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
 => ? ∊ {2,2,3,5,5,5,5,6,6,6,7,7,7,8,8,9} + 1
[2,2,2,1,1]
 => [1,0,1,1,0,1,1,1,0,0,0,0]
 => [.,[[[[.,.],.],[.,.]],.]]
 => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
 => ? ∊ {2,2,3,5,5,5,5,6,6,6,7,7,7,8,8,9} + 1
[2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
 => [.,[[[[.,.],[.,.]],.],[.,.]]]
 => ([(0,6),(1,5),(2,5),(4,6),(5,4),(6,3)],7)
 => ? ∊ {2,2,3,5,5,5,5,6,6,6,7,7,7,8,8,9} + 1
[2,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => [.,[[[.,.],[.,.]],[[[.,.],.],.]]]
 => ([(0,6),(1,6),(2,4),(4,5),(5,7),(6,7),(7,3)],8)
 => ? ∊ {2,2,3,5,5,5,5,6,6,6,7,7,7,8,8,9} + 1
[1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [.,[[[[.,.],.],[.,.]],[[.,.],[.,.]]]]
 => ([(0,8),(1,6),(2,6),(3,5),(5,8),(6,7),(7,4),(8,7)],9)
 => ? ∊ {2,2,3,5,5,5,5,6,6,6,7,7,7,8,8,9} + 1
[9]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => [[[[.,.],[.,.]],[.,.]],[[.,.],[.,[.,.]]]]
 => ([(0,6),(1,6),(2,9),(3,8),(4,5),(5,9),(6,8),(8,7),(9,7)],10)
 => ? ∊ {5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9,9,10,10,11} + 1
[8,1]
 => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
 => [[[.,[.,.]],[.,.]],[[.,.],[.,[.,.]]]]
 => ([(0,8),(1,7),(2,4),(3,5),(4,7),(5,8),(7,6),(8,6)],9)
 => ? ∊ {5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9,9,10,10,11} + 1
[7,2]
 => [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
 => [[[.,.],[.,[.,.]]],[[.,[.,.]],.]]
 => ([(0,7),(1,3),(2,4),(3,7),(4,5),(5,6),(7,6)],8)
 => ? ∊ {5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9,9,10,10,11} + 1
[7,1,1]
 => [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
 => [[[.,.],[.,[.,.]]],[.,[[.,.],.]]]
 => ([(0,7),(1,3),(2,4),(3,7),(4,5),(5,6),(7,6)],8)
 => ? ∊ {5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9,9,10,10,11} + 1
[4,3,2]
 => [1,1,0,0,1,0,1,0,1,0]
 => [[.,[.,[.,[.,.]]]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 4 + 1
[4,3,1,1]
 => [1,0,1,1,0,0,1,0,1,0]
 => [.,[[.,[.,[.,.]]],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 4 + 1
[4,2,2,1]
 => [1,0,1,0,1,1,0,0,1,0]
 => [.,[.,[[.,[.,.]],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 4 + 1
[3,3,2,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [.,[.,[.,[[.,.],.]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 4 + 1
[4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => [.,[.,[.,[.,[.,.]]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 4 + 1
[5,3,2,1]
 => [1,1,0,1,0,1,0,1,0,0,1,0]
 => [[[[[.,[.,.]],.],.],.],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 5 + 1
[4,4,2,1]
 => [1,1,0,1,0,1,0,0,1,1,0,0]
 => [[[[.,[[.,.],.]],.],.],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 5 + 1
[4,3,3,1]
 => [1,1,0,1,0,0,1,1,0,1,0,0]
 => [[[.,[[[.,.],.],.]],.],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 5 + 1
[4,3,2,2]
 => [1,1,0,0,1,1,0,1,0,1,0,0]
 => [[.,[[[[.,.],.],.],.]],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 5 + 1
[4,3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [.,[[[[[.,.],.],.],.],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 5 + 1
[5,4,2,1]
 => [1,1,0,1,0,1,0,0,1,0,1,0]
 => [[[[.,[.,[.,.]]],.],.],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 5 + 1
[5,3,3,1]
 => [1,1,0,1,0,0,1,1,0,0,1,0]
 => [[[.,[[.,[.,.]],.]],.],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 5 + 1
[5,3,2,2]
 => [1,1,0,0,1,1,0,1,0,0,1,0]
 => [[.,[[[.,[.,.]],.],.]],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 5 + 1
[5,3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0,1,0]
 => [.,[[[[.,[.,.]],.],.],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 5 + 1
[4,4,3,1]
 => [1,1,0,1,0,0,1,0,1,1,0,0]
 => [[[.,[.,[[.,.],.]]],.],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 5 + 1
[4,4,2,2]
 => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [[.,[[.,[[.,.],.]],.]],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 5 + 1
[4,4,2,1,1]
 => [1,0,1,1,0,1,0,0,1,1,0,0]
 => [.,[[[.,[[.,.],.]],.],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 5 + 1
[4,3,3,2]
 => [1,1,0,0,1,0,1,1,0,1,0,0]
 => [[.,[.,[[[.,.],.],.]]],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 5 + 1
[4,3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,1,0,0]
 => [.,[[.,[[[.,.],.],.]],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 5 + 1
[4,3,2,2,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => [.,[.,[[[[.,.],.],.],.]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 5 + 1
[5,4,3,1]
 => [1,1,0,1,0,0,1,0,1,0,1,0]
 => [[[.,[.,[.,[.,.]]]],.],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 5 + 1
[5,4,2,2]
 => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [[.,[[.,[.,[.,.]]],.]],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 5 + 1
[5,4,2,1,1]
 => [1,0,1,1,0,1,0,0,1,0,1,0]
 => [.,[[[.,[.,[.,.]]],.],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 5 + 1
[5,3,3,2]
 => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [[.,[.,[[.,[.,.]],.]]],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 5 + 1
[5,3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => [.,[[.,[[.,[.,.]],.]],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 5 + 1
[5,3,2,2,1]
 => [1,0,1,0,1,1,0,1,0,0,1,0]
 => [.,[.,[[[.,[.,.]],.],.]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 5 + 1
[4,4,3,2]
 => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [[.,[.,[.,[[.,.],.]]]],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 5 + 1
[4,4,3,1,1]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [.,[[.,[.,[[.,.],.]]],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 5 + 1
[4,4,2,2,1]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [.,[.,[[.,[[.,.],.]],.]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 5 + 1
[4,3,3,2,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [.,[.,[.,[[[.,.],.],.]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 5 + 1
Description
The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice.
Matching statistic: St001637
(load all 2 compositions to match this statistic)
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St001637: Posets ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 19%
Values
[1]
 => [1,0,1,0]
 => [1,0,1,0]
 => ([(0,1)],2)
 => 1
[2]
 => [1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => ([(0,2),(2,1)],3)
 => 2
[1,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => ([(0,2),(2,1)],3)
 => 2
[3]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 3
[2,1]
 => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => ([(0,2),(2,1)],3)
 => 2
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 1
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
 => ? ∊ {4,5}
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 3
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 3
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 3
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? ∊ {4,5}
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => ([(0,4),(0,5),(1,10),(2,7),(3,8),(4,3),(4,6),(5,1),(5,6),(6,8),(6,10),(8,9),(9,7),(10,2),(10,9)],11)
 => ? ∊ {4,5,5,6}
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ? ∊ {4,5,5,6}
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 3
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 3
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 3
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => ? ∊ {4,5,5,6}
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
 => ? ∊ {4,5,5,6}
[6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => ([(0,5),(0,6),(1,10),(2,11),(3,4),(3,14),(4,8),(5,2),(5,13),(6,3),(6,13),(8,9),(9,7),(10,7),(11,1),(11,12),(12,9),(12,10),(13,11),(13,14),(14,8),(14,12)],15)
 => ? ∊ {1,2,3,4,4,4,5,5,6,7}
[5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
 => ? ∊ {1,2,3,4,4,4,5,5,6,7}
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => ? ∊ {1,2,3,4,4,4,5,5,6,7}
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ? ∊ {1,2,3,4,4,4,5,5,6,7}
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => ? ∊ {1,2,3,4,4,4,5,5,6,7}
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 3
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
 => ? ∊ {1,2,3,4,4,4,5,5,6,7}
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
 => ? ∊ {1,2,3,4,4,4,5,5,6,7}
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => ? ∊ {1,2,3,4,4,4,5,5,6,7}
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
 => ? ∊ {1,2,3,4,4,4,5,5,6,7}
[1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => ([(0,5),(0,6),(1,4),(1,14),(2,11),(3,10),(4,3),(4,12),(5,1),(5,13),(6,2),(6,13),(8,9),(9,7),(10,7),(11,8),(12,9),(12,10),(13,11),(13,14),(14,8),(14,12)],15)
 => ? ∊ {1,2,3,4,4,4,5,5,6,7}
[7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
 => ([(0,6),(0,7),(1,13),(2,4),(2,17),(3,5),(3,18),(4,9),(5,12),(6,2),(6,15),(7,3),(7,15),(9,10),(10,11),(11,8),(12,1),(12,16),(13,8),(14,10),(14,16),(15,17),(15,18),(16,11),(16,13),(17,9),(17,14),(18,12),(18,14)],19)
 => ? ∊ {5,5,6,6,6,6,7,7,7,7,8}
[6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
 => ([(0,6),(1,11),(2,8),(3,9),(4,3),(4,7),(5,1),(5,7),(6,4),(6,5),(7,9),(7,11),(9,10),(10,8),(11,2),(11,10)],12)
 => ? ∊ {5,5,6,6,6,6,7,7,7,7,8}
[5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => ([(0,4),(0,5),(1,6),(3,7),(4,8),(5,1),(5,8),(6,7),(7,2),(8,3),(8,6)],9)
 => ? ∊ {5,5,6,6,6,6,7,7,7,7,8}
[5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
 => ? ∊ {5,5,6,6,6,6,7,7,7,7,8}
[4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => ? ∊ {5,5,6,6,6,6,7,7,7,7,8}
[4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4
[4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
 => ? ∊ {5,5,6,6,6,6,7,7,7,7,8}
[3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4
[3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4
[3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4
[3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
 => ? ∊ {5,5,6,6,6,6,7,7,7,7,8}
[2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? ∊ {5,5,6,6,6,6,7,7,7,7,8}
[2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
 => ? ∊ {5,5,6,6,6,6,7,7,7,7,8}
[2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => ([(0,6),(1,9),(2,8),(3,5),(3,7),(4,1),(4,7),(5,2),(5,10),(6,3),(6,4),(7,9),(7,10),(8,12),(9,11),(10,8),(10,11),(11,12)],13)
 => ? ∊ {5,5,6,6,6,6,7,7,7,7,8}
[1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => ([(0,6),(0,7),(1,4),(1,16),(2,5),(2,15),(3,13),(4,12),(5,3),(5,19),(6,1),(6,17),(7,2),(7,17),(9,11),(10,8),(11,8),(12,9),(13,10),(14,9),(14,18),(15,14),(15,19),(16,12),(16,14),(17,15),(17,16),(18,10),(18,11),(19,13),(19,18)],20)
 => ? ∊ {5,5,6,6,6,6,7,7,7,7,8}
[8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
 => ([(0,7),(0,8),(1,15),(2,4),(2,22),(3,5),(3,23),(4,6),(4,21),(5,14),(6,10),(7,2),(7,20),(8,3),(8,20),(10,11),(11,12),(12,9),(13,9),(14,1),(14,19),(15,13),(16,11),(16,17),(17,12),(17,13),(18,16),(18,19),(19,15),(19,17),(20,22),(20,23),(21,10),(21,16),(22,18),(22,21),(23,14),(23,18)],24)
 => ? ∊ {2,2,3,5,5,5,5,6,6,6,7,7,7,8,8,9}
[7,1]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => ([(0,1),(1,3),(1,4),(2,12),(3,7),(3,15),(4,6),(4,15),(5,9),(6,11),(7,5),(7,13),(9,10),(10,8),(11,2),(11,14),(12,8),(13,9),(13,14),(14,10),(14,12),(15,11),(15,13)],16)
 => ? ∊ {2,2,3,5,5,5,5,6,6,6,7,7,7,8,8,9}
[6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => ([(0,5),(0,6),(2,11),(3,10),(4,9),(5,3),(5,7),(6,4),(6,7),(7,9),(7,10),(8,11),(9,8),(10,2),(10,8),(11,1)],12)
 => ? ∊ {2,2,3,5,5,5,5,6,6,6,7,7,7,8,8,9}
[6,1,1]
 => [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
 => [1,1,0,1,1,1,1,0,0,1,0,0,0,0]
 => ([(0,6),(1,11),(2,8),(3,9),(4,3),(4,7),(5,1),(5,7),(6,4),(6,5),(7,9),(7,11),(9,10),(10,8),(11,2),(11,10)],12)
 => ? ∊ {2,2,3,5,5,5,5,6,6,6,7,7,7,8,8,9}
[5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0]
 => ([(0,4),(0,5),(1,6),(3,7),(4,8),(5,1),(5,8),(6,7),(7,2),(8,3),(8,6)],9)
 => ? ∊ {2,2,3,5,5,5,5,6,6,6,7,7,7,8,8,9}
[5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
 => ? ∊ {2,2,3,5,5,5,5,6,6,6,7,7,7,8,8,9}
[5,1,1,1]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,1,0,0,0]
 => ([(0,4),(0,5),(1,7),(2,9),(3,6),(4,8),(5,2),(5,8),(6,7),(8,3),(8,9),(9,1),(9,6)],10)
 => ? ∊ {2,2,3,5,5,5,5,6,6,6,7,7,7,8,8,9}
[4,4]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => ([(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,1),(5,7),(7,8),(8,2),(8,3)],9)
 => ? ∊ {2,2,3,5,5,5,5,6,6,6,7,7,7,8,8,9}
[4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4
[4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4
[4,2,1,1]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4
[4,1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
 => ? ∊ {2,2,3,5,5,5,5,6,6,6,7,7,7,8,8,9}
[3,3,2]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4
[3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4
[3,2,2,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4
[3,2,1,1,1]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
 => ? ∊ {2,2,3,5,5,5,5,6,6,6,7,7,7,8,8,9}
[3,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => ([(0,5),(0,6),(2,9),(3,8),(4,2),(4,10),(5,3),(5,7),(6,4),(6,7),(7,8),(7,10),(8,11),(9,12),(10,9),(10,11),(11,12),(12,1)],13)
 => ? ∊ {2,2,3,5,5,5,5,6,6,6,7,7,7,8,8,9}
[2,2,2,2]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,1,1,0,0,0,0]
 => ([(0,4),(0,5),(1,7),(2,9),(3,6),(4,8),(5,2),(5,8),(6,7),(8,3),(8,9),(9,1),(9,6)],10)
 => ? ∊ {2,2,3,5,5,5,5,6,6,6,7,7,7,8,8,9}
[2,2,2,1,1]
 => [1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,1),(5,6),(6,4),(6,8),(8,9),(9,7)],10)
 => ? ∊ {2,2,3,5,5,5,5,6,6,6,7,7,7,8,8,9}
[2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
 => [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
 => ([(0,6),(1,9),(2,8),(3,5),(3,7),(4,1),(4,7),(5,2),(5,10),(6,3),(6,4),(7,9),(7,10),(8,12),(9,11),(10,8),(10,11),(11,12)],13)
 => ? ∊ {2,2,3,5,5,5,5,6,6,6,7,7,7,8,8,9}
[2,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => ([(0,1),(1,2),(1,3),(2,5),(2,13),(3,7),(3,13),(4,12),(5,11),(6,4),(6,15),(7,6),(7,14),(9,10),(10,8),(11,9),(12,8),(13,11),(13,14),(14,9),(14,15),(15,10),(15,12)],16)
 => ? ∊ {2,2,3,5,5,5,5,6,6,6,7,7,7,8,8,9}
[1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
 => ([(0,7),(0,8),(1,6),(1,19),(2,4),(2,18),(3,14),(4,15),(5,3),(5,23),(6,5),(6,22),(7,1),(7,20),(8,2),(8,20),(10,11),(11,12),(12,9),(13,9),(14,13),(15,10),(16,12),(16,13),(17,11),(17,16),(18,15),(18,21),(19,21),(19,22),(20,18),(20,19),(21,10),(21,17),(22,17),(22,23),(23,14),(23,16)],24)
 => ? ∊ {2,2,3,5,5,5,5,6,6,6,7,7,7,8,8,9}
[9]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0]
 => ([(0,8),(0,9),(1,12),(2,7),(2,23),(3,6),(3,22),(4,17),(5,16),(6,5),(6,28),(7,4),(7,27),(8,2),(8,24),(9,3),(9,24),(11,13),(12,15),(13,14),(14,10),(15,10),(16,1),(16,26),(17,11),(18,20),(18,26),(19,18),(19,25),(20,13),(20,21),(21,14),(21,15),(22,19),(22,28),(23,19),(23,27),(24,22),(24,23),(25,11),(25,20),(26,12),(26,21),(27,17),(27,25),(28,16),(28,18)],29)
 => ? ∊ {5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9,9,10,10,11}
[8,1]
 => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
 => [1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
 => ([(0,1),(1,3),(1,4),(2,12),(3,7),(3,16),(4,8),(4,16),(5,13),(6,14),(7,5),(7,18),(8,6),(8,19),(10,11),(11,9),(12,9),(13,10),(14,2),(14,17),(15,10),(15,17),(16,18),(16,19),(17,11),(17,12),(18,13),(18,15),(19,14),(19,15)],20)
 => ? ∊ {5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9,9,10,10,11}
[7,2]
 => [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
 => ([(0,6),(0,7),(2,11),(3,12),(4,5),(4,15),(5,9),(6,3),(6,14),(7,4),(7,14),(8,1),(9,10),(10,8),(11,8),(12,2),(12,13),(13,10),(13,11),(14,12),(14,15),(15,9),(15,13)],16)
 => ? ∊ {5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9,9,10,10,11}
[7,1,1]
 => [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
 => [1,1,0,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => ([(0,1),(1,3),(1,4),(2,12),(3,7),(3,15),(4,6),(4,15),(5,9),(6,11),(7,5),(7,13),(9,10),(10,8),(11,2),(11,14),(12,8),(13,9),(13,14),(14,10),(14,12),(15,11),(15,13)],16)
 => ? ∊ {5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9,9,10,10,11}
[6,3]
 => [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
 => [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
 => ([(0,5),(0,6),(2,11),(3,10),(4,9),(5,3),(5,7),(6,4),(6,7),(7,9),(7,10),(8,11),(9,8),(10,2),(10,8),(11,1)],12)
 => ? ∊ {5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9,9,10,10,11}
[6,2,1]
 => [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
 => [1,0,1,0,1,1,1,1,0,0,1,0,0,0]
 => ([(0,5),(1,8),(2,9),(3,7),(4,3),(4,9),(5,6),(6,2),(6,4),(7,8),(9,1),(9,7)],10)
 => ? ∊ {5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9,9,10,10,11}
[4,3,2]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4
[4,3,1,1]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4
[4,2,2,1]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4
[3,3,2,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4
[4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4
Description
The number of (upper) dissectors of a poset.
Matching statistic: St001668
(load all 2 compositions to match this statistic)
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St001668: Posets ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 19%
Values
[1]
 => [1,0,1,0]
 => [1,0,1,0]
 => ([(0,1)],2)
 => 1
[2]
 => [1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => ([(0,2),(2,1)],3)
 => 2
[1,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => ([(0,2),(2,1)],3)
 => 2
[3]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 3
[2,1]
 => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => ([(0,2),(2,1)],3)
 => 2
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 1
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
 => ? ∊ {4,5}
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 3
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 3
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 3
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? ∊ {4,5}
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => ([(0,4),(0,5),(1,10),(2,7),(3,8),(4,3),(4,6),(5,1),(5,6),(6,8),(6,10),(8,9),(9,7),(10,2),(10,9)],11)
 => ? ∊ {4,5,5,6}
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ? ∊ {4,5,5,6}
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 3
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 3
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 3
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => ? ∊ {4,5,5,6}
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
 => ? ∊ {4,5,5,6}
[6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => ([(0,5),(0,6),(1,10),(2,11),(3,4),(3,14),(4,8),(5,2),(5,13),(6,3),(6,13),(8,9),(9,7),(10,7),(11,1),(11,12),(12,9),(12,10),(13,11),(13,14),(14,8),(14,12)],15)
 => ? ∊ {1,2,3,4,4,4,5,5,6,7}
[5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
 => ? ∊ {1,2,3,4,4,4,5,5,6,7}
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => ? ∊ {1,2,3,4,4,4,5,5,6,7}
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ? ∊ {1,2,3,4,4,4,5,5,6,7}
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => ? ∊ {1,2,3,4,4,4,5,5,6,7}
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 3
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
 => ? ∊ {1,2,3,4,4,4,5,5,6,7}
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
 => ? ∊ {1,2,3,4,4,4,5,5,6,7}
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => ? ∊ {1,2,3,4,4,4,5,5,6,7}
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
 => ? ∊ {1,2,3,4,4,4,5,5,6,7}
[1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => ([(0,5),(0,6),(1,4),(1,14),(2,11),(3,10),(4,3),(4,12),(5,1),(5,13),(6,2),(6,13),(8,9),(9,7),(10,7),(11,8),(12,9),(12,10),(13,11),(13,14),(14,8),(14,12)],15)
 => ? ∊ {1,2,3,4,4,4,5,5,6,7}
[7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
 => ([(0,6),(0,7),(1,13),(2,4),(2,17),(3,5),(3,18),(4,9),(5,12),(6,2),(6,15),(7,3),(7,15),(9,10),(10,11),(11,8),(12,1),(12,16),(13,8),(14,10),(14,16),(15,17),(15,18),(16,11),(16,13),(17,9),(17,14),(18,12),(18,14)],19)
 => ? ∊ {5,5,6,6,6,6,7,7,7,7,8}
[6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
 => ([(0,6),(1,11),(2,8),(3,9),(4,3),(4,7),(5,1),(5,7),(6,4),(6,5),(7,9),(7,11),(9,10),(10,8),(11,2),(11,10)],12)
 => ? ∊ {5,5,6,6,6,6,7,7,7,7,8}
[5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => ([(0,4),(0,5),(1,6),(3,7),(4,8),(5,1),(5,8),(6,7),(7,2),(8,3),(8,6)],9)
 => ? ∊ {5,5,6,6,6,6,7,7,7,7,8}
[5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
 => ? ∊ {5,5,6,6,6,6,7,7,7,7,8}
[4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => ? ∊ {5,5,6,6,6,6,7,7,7,7,8}
[4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4
[4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
 => ? ∊ {5,5,6,6,6,6,7,7,7,7,8}
[3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4
[3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4
[3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4
[3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
 => ? ∊ {5,5,6,6,6,6,7,7,7,7,8}
[2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? ∊ {5,5,6,6,6,6,7,7,7,7,8}
[2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
 => ? ∊ {5,5,6,6,6,6,7,7,7,7,8}
[2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => ([(0,6),(1,9),(2,8),(3,5),(3,7),(4,1),(4,7),(5,2),(5,10),(6,3),(6,4),(7,9),(7,10),(8,12),(9,11),(10,8),(10,11),(11,12)],13)
 => ? ∊ {5,5,6,6,6,6,7,7,7,7,8}
[1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => ([(0,6),(0,7),(1,4),(1,16),(2,5),(2,15),(3,13),(4,12),(5,3),(5,19),(6,1),(6,17),(7,2),(7,17),(9,11),(10,8),(11,8),(12,9),(13,10),(14,9),(14,18),(15,14),(15,19),(16,12),(16,14),(17,15),(17,16),(18,10),(18,11),(19,13),(19,18)],20)
 => ? ∊ {5,5,6,6,6,6,7,7,7,7,8}
[8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
 => ([(0,7),(0,8),(1,15),(2,4),(2,22),(3,5),(3,23),(4,6),(4,21),(5,14),(6,10),(7,2),(7,20),(8,3),(8,20),(10,11),(11,12),(12,9),(13,9),(14,1),(14,19),(15,13),(16,11),(16,17),(17,12),(17,13),(18,16),(18,19),(19,15),(19,17),(20,22),(20,23),(21,10),(21,16),(22,18),(22,21),(23,14),(23,18)],24)
 => ? ∊ {2,2,3,5,5,5,5,6,6,6,7,7,7,8,8,9}
[7,1]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => ([(0,1),(1,3),(1,4),(2,12),(3,7),(3,15),(4,6),(4,15),(5,9),(6,11),(7,5),(7,13),(9,10),(10,8),(11,2),(11,14),(12,8),(13,9),(13,14),(14,10),(14,12),(15,11),(15,13)],16)
 => ? ∊ {2,2,3,5,5,5,5,6,6,6,7,7,7,8,8,9}
[6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => ([(0,5),(0,6),(2,11),(3,10),(4,9),(5,3),(5,7),(6,4),(6,7),(7,9),(7,10),(8,11),(9,8),(10,2),(10,8),(11,1)],12)
 => ? ∊ {2,2,3,5,5,5,5,6,6,6,7,7,7,8,8,9}
[6,1,1]
 => [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
 => [1,1,0,1,1,1,1,0,0,1,0,0,0,0]
 => ([(0,6),(1,11),(2,8),(3,9),(4,3),(4,7),(5,1),(5,7),(6,4),(6,5),(7,9),(7,11),(9,10),(10,8),(11,2),(11,10)],12)
 => ? ∊ {2,2,3,5,5,5,5,6,6,6,7,7,7,8,8,9}
[5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0]
 => ([(0,4),(0,5),(1,6),(3,7),(4,8),(5,1),(5,8),(6,7),(7,2),(8,3),(8,6)],9)
 => ? ∊ {2,2,3,5,5,5,5,6,6,6,7,7,7,8,8,9}
[5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
 => ? ∊ {2,2,3,5,5,5,5,6,6,6,7,7,7,8,8,9}
[5,1,1,1]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,1,0,0,0]
 => ([(0,4),(0,5),(1,7),(2,9),(3,6),(4,8),(5,2),(5,8),(6,7),(8,3),(8,9),(9,1),(9,6)],10)
 => ? ∊ {2,2,3,5,5,5,5,6,6,6,7,7,7,8,8,9}
[4,4]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => ([(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,1),(5,7),(7,8),(8,2),(8,3)],9)
 => ? ∊ {2,2,3,5,5,5,5,6,6,6,7,7,7,8,8,9}
[4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4
[4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4
[4,2,1,1]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4
[4,1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
 => ? ∊ {2,2,3,5,5,5,5,6,6,6,7,7,7,8,8,9}
[3,3,2]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4
[3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4
[3,2,2,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4
[3,2,1,1,1]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
 => ? ∊ {2,2,3,5,5,5,5,6,6,6,7,7,7,8,8,9}
[3,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => ([(0,5),(0,6),(2,9),(3,8),(4,2),(4,10),(5,3),(5,7),(6,4),(6,7),(7,8),(7,10),(8,11),(9,12),(10,9),(10,11),(11,12),(12,1)],13)
 => ? ∊ {2,2,3,5,5,5,5,6,6,6,7,7,7,8,8,9}
[2,2,2,2]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,1,1,0,0,0,0]
 => ([(0,4),(0,5),(1,7),(2,9),(3,6),(4,8),(5,2),(5,8),(6,7),(8,3),(8,9),(9,1),(9,6)],10)
 => ? ∊ {2,2,3,5,5,5,5,6,6,6,7,7,7,8,8,9}
[2,2,2,1,1]
 => [1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,1),(5,6),(6,4),(6,8),(8,9),(9,7)],10)
 => ? ∊ {2,2,3,5,5,5,5,6,6,6,7,7,7,8,8,9}
[2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
 => [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
 => ([(0,6),(1,9),(2,8),(3,5),(3,7),(4,1),(4,7),(5,2),(5,10),(6,3),(6,4),(7,9),(7,10),(8,12),(9,11),(10,8),(10,11),(11,12)],13)
 => ? ∊ {2,2,3,5,5,5,5,6,6,6,7,7,7,8,8,9}
[2,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => ([(0,1),(1,2),(1,3),(2,5),(2,13),(3,7),(3,13),(4,12),(5,11),(6,4),(6,15),(7,6),(7,14),(9,10),(10,8),(11,9),(12,8),(13,11),(13,14),(14,9),(14,15),(15,10),(15,12)],16)
 => ? ∊ {2,2,3,5,5,5,5,6,6,6,7,7,7,8,8,9}
[1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
 => ([(0,7),(0,8),(1,6),(1,19),(2,4),(2,18),(3,14),(4,15),(5,3),(5,23),(6,5),(6,22),(7,1),(7,20),(8,2),(8,20),(10,11),(11,12),(12,9),(13,9),(14,13),(15,10),(16,12),(16,13),(17,11),(17,16),(18,15),(18,21),(19,21),(19,22),(20,18),(20,19),(21,10),(21,17),(22,17),(22,23),(23,14),(23,16)],24)
 => ? ∊ {2,2,3,5,5,5,5,6,6,6,7,7,7,8,8,9}
[9]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0]
 => ([(0,8),(0,9),(1,12),(2,7),(2,23),(3,6),(3,22),(4,17),(5,16),(6,5),(6,28),(7,4),(7,27),(8,2),(8,24),(9,3),(9,24),(11,13),(12,15),(13,14),(14,10),(15,10),(16,1),(16,26),(17,11),(18,20),(18,26),(19,18),(19,25),(20,13),(20,21),(21,14),(21,15),(22,19),(22,28),(23,19),(23,27),(24,22),(24,23),(25,11),(25,20),(26,12),(26,21),(27,17),(27,25),(28,16),(28,18)],29)
 => ? ∊ {5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9,9,10,10,11}
[8,1]
 => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
 => [1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
 => ([(0,1),(1,3),(1,4),(2,12),(3,7),(3,16),(4,8),(4,16),(5,13),(6,14),(7,5),(7,18),(8,6),(8,19),(10,11),(11,9),(12,9),(13,10),(14,2),(14,17),(15,10),(15,17),(16,18),(16,19),(17,11),(17,12),(18,13),(18,15),(19,14),(19,15)],20)
 => ? ∊ {5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9,9,10,10,11}
[7,2]
 => [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
 => ([(0,6),(0,7),(2,11),(3,12),(4,5),(4,15),(5,9),(6,3),(6,14),(7,4),(7,14),(8,1),(9,10),(10,8),(11,8),(12,2),(12,13),(13,10),(13,11),(14,12),(14,15),(15,9),(15,13)],16)
 => ? ∊ {5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9,9,10,10,11}
[7,1,1]
 => [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
 => [1,1,0,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => ([(0,1),(1,3),(1,4),(2,12),(3,7),(3,15),(4,6),(4,15),(5,9),(6,11),(7,5),(7,13),(9,10),(10,8),(11,2),(11,14),(12,8),(13,9),(13,14),(14,10),(14,12),(15,11),(15,13)],16)
 => ? ∊ {5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9,9,10,10,11}
[6,3]
 => [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
 => [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
 => ([(0,5),(0,6),(2,11),(3,10),(4,9),(5,3),(5,7),(6,4),(6,7),(7,9),(7,10),(8,11),(9,8),(10,2),(10,8),(11,1)],12)
 => ? ∊ {5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9,9,10,10,11}
[6,2,1]
 => [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
 => [1,0,1,0,1,1,1,1,0,0,1,0,0,0]
 => ([(0,5),(1,8),(2,9),(3,7),(4,3),(4,9),(5,6),(6,2),(6,4),(7,8),(9,1),(9,7)],10)
 => ? ∊ {5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9,9,10,10,11}
[4,3,2]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4
[4,3,1,1]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4
[4,2,2,1]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4
[3,3,2,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4
[4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4
Description
The number of points of the poset minus the width of the poset.
Matching statistic: St000422
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00140: Dyck paths —logarithmic height to pruning number⟶ Binary trees
Mp00011: Binary trees —to graph⟶ Graphs
St000422: Graphs ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 14%
Values
[1]
 => [1,0,1,0]
 => [.,[.,.]]
 => ([(0,1)],2)
 => 2 = 1 + 1
[2]
 => [1,1,0,0,1,0]
 => [[.,[.,.]],.]
 => ([(0,2),(1,2)],3)
 => ? ∊ {2,2} + 1
[1,1]
 => [1,0,1,1,0,0]
 => [.,[[.,.],.]]
 => ([(0,2),(1,2)],3)
 => ? ∊ {2,2} + 1
[3]
 => [1,1,1,0,0,0,1,0]
 => [[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? ∊ {1,2,3} + 1
[2,1]
 => [1,0,1,0,1,0]
 => [.,[.,[.,.]]]
 => ([(0,2),(1,2)],3)
 => ? ∊ {1,2,3} + 1
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(2,3)],4)
 => ? ∊ {1,2,3} + 1
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [[[.,.],.],[.,[.,.]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? ∊ {3,3,3,4,5} + 1
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [[[.,[.,.]],.],.]
 => ([(0,3),(1,2),(2,3)],4)
 => ? ∊ {3,3,3,4,5} + 1
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [[.,[[.,.],.]],.]
 => ([(0,3),(1,2),(2,3)],4)
 => ? ∊ {3,3,3,4,5} + 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [.,[[[.,.],.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? ∊ {3,3,3,4,5} + 1
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [.,[[[.,.],.],[.,.]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? ∊ {3,3,3,4,5} + 1
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [[[.,.],[.,[.,.]]],[.,.]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ? ∊ {3,3,3,4,5,6} + 1
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [[.,[.,.]],[.,[.,.]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? ∊ {3,3,3,4,5,6} + 1
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [[.,[.,[.,.]]],.]
 => ([(0,3),(1,2),(2,3)],4)
 => ? ∊ {3,3,3,4,5,6} + 1
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [.,[[.,[.,.]],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? ∊ {3,3,3,4,5,6} + 1
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [.,[.,[[.,.],.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? ∊ {3,3,3,4,5,6} + 1
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [.,[[.,[.,.]],[.,.]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? ∊ {3,3,3,4,5,6} + 1
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [.,[[[.,.],[.,.]],[.,.]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6 = 5 + 1
[6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [[[.,.],[.,[.,.]]],[[.,.],.]]
 => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
 => ? ∊ {1,2,3,3,4,4,4,5,5,6,7} + 1
[5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [[[[.,.],.],.],[.,[.,.]]]
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => ? ∊ {1,2,3,3,4,4,4,5,5,6,7} + 1
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [[.,.],[[.,[.,.]],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? ∊ {1,2,3,3,4,4,4,5,5,6,7} + 1
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [[[.,.],[.,[.,.]]],.]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? ∊ {1,2,3,3,4,4,4,5,5,6,7} + 1
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [[.,.],[.,[[.,.],.]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? ∊ {1,2,3,3,4,4,4,5,5,6,7} + 1
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [.,[.,[.,[.,.]]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? ∊ {1,2,3,3,4,4,4,5,5,6,7} + 1
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [.,[[.,.],[[.,.],.]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? ∊ {1,2,3,3,4,4,4,5,5,6,7} + 1
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [[.,[[.,.],[.,.]]],.]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? ∊ {1,2,3,3,4,4,4,5,5,6,7} + 1
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [.,[[[.,.],[.,.]],.]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? ∊ {1,2,3,3,4,4,4,5,5,6,7} + 1
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [.,[[[[.,.],.],.],[.,.]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ? ∊ {1,2,3,3,4,4,4,5,5,6,7} + 1
[1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [.,[[[.,.],[.,.]],[[.,.],.]]]
 => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
 => ? ∊ {1,2,3,3,4,4,4,5,5,6,7} + 1
[7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [[[.,.],[.,.]],[[.,.],[.,[.,.]]]]
 => ([(0,7),(1,6),(2,6),(3,4),(4,7),(5,6),(5,7)],8)
 => ? ∊ {4,4,4,4,5,5,6,6,6,6,7,7,7,7,8} + 1
[6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [[[.,.],[.,[.,.]]],[.,[.,.]]]
 => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
 => ? ∊ {4,4,4,4,5,5,6,6,6,6,7,7,7,7,8} + 1
[5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [[[.,[.,.]],.],[.,[.,.]]]
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => ? ∊ {4,4,4,4,5,5,6,6,6,6,7,7,7,7,8} + 1
[5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => [[.,[[.,.],.]],[.,[.,.]]]
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => ? ∊ {4,4,4,4,5,5,6,6,6,6,7,7,7,7,8} + 1
[4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [[.,.],[.,[.,[.,.]]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? ∊ {4,4,4,4,5,5,6,6,6,6,7,7,7,7,8} + 1
[4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [[[[.,[.,.]],.],.],.]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? ∊ {4,4,4,4,5,5,6,6,6,6,7,7,7,7,8} + 1
[4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => [.,[[.,.],[.,[.,.]]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? ∊ {4,4,4,4,5,5,6,6,6,6,7,7,7,7,8} + 1
[3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [[[.,[[.,.],.]],.],.]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? ∊ {4,4,4,4,5,5,6,6,6,6,7,7,7,7,8} + 1
[3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [[.,[[[.,.],.],.]],.]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? ∊ {4,4,4,4,5,5,6,6,6,6,7,7,7,7,8} + 1
[3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [.,[[[[.,.],.],.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? ∊ {4,4,4,4,5,5,6,6,6,6,7,7,7,7,8} + 1
[3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [.,[[[.,[.,.]],.],[.,.]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ? ∊ {4,4,4,4,5,5,6,6,6,6,7,7,7,7,8} + 1
[2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [.,[.,[[.,.],[.,.]]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? ∊ {4,4,4,4,5,5,6,6,6,6,7,7,7,7,8} + 1
[2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [.,[[.,[[.,.],.]],[.,.]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ? ∊ {4,4,4,4,5,5,6,6,6,6,7,7,7,7,8} + 1
[2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [.,[[[.,.],[.,.]],[.,[.,.]]]]
 => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
 => ? ∊ {4,4,4,4,5,5,6,6,6,6,7,7,7,7,8} + 1
[1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [.,[[[.,.],[.,.]],[[.,.],[.,.]]]]
 => ([(0,6),(1,6),(2,5),(3,5),(4,7),(5,7),(6,7)],8)
 => ? ∊ {4,4,4,4,5,5,6,6,6,6,7,7,7,7,8} + 1
[8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [[[[.,.],.],[.,.]],[[.,.],[.,[.,.]]]]
 => ([(0,8),(1,7),(2,4),(3,5),(4,7),(5,8),(6,7),(6,8)],9)
 => ? ∊ {2,2,3,4,4,4,4,4,4,5,5,5,5,6,6,6,7,7,8,8,9} + 1
[7,1]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => [[[.,.],[.,[.,.]]],[[[.,.],.],.]]
 => ([(0,6),(1,5),(2,7),(3,4),(3,5),(4,7),(6,7)],8)
 => ? ∊ {2,2,3,4,4,4,4,4,4,5,5,5,5,6,6,6,7,7,8,8,9} + 1
[6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [[[[.,.],.],[.,[.,.]]],[.,.]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 8 = 7 + 1
[6,1,1]
 => [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
 => [[[[.,.],[.,[.,.]]],.],[.,.]]
 => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
 => ? ∊ {2,2,3,4,4,4,4,4,4,5,5,5,5,6,6,6,7,7,8,8,9} + 1
[5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [[[.,.],.],[[.,[.,.]],.]]
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => ? ∊ {2,2,3,4,4,4,4,4,4,5,5,5,5,6,6,6,7,7,8,8,9} + 1
[5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [[.,[.,[.,.]]],[.,[.,.]]]
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => ? ∊ {2,2,3,4,4,4,4,4,4,5,5,5,5,6,6,6,7,7,8,8,9} + 1
[5,1,1,1]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => [[[[.,.],.],[.,[.,.]]],.]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ? ∊ {2,2,3,4,4,4,4,4,4,5,5,5,5,6,6,6,7,7,8,8,9} + 1
[4,4]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [[[.,.],.],[.,[[.,.],.]]]
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => ? ∊ {2,2,3,4,4,4,4,4,4,5,5,5,5,6,6,6,7,7,8,8,9} + 1
[4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => [[[.,[.,[.,.]]],.],.]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? ∊ {2,2,3,4,4,4,4,4,4,5,5,5,5,6,6,6,7,7,8,8,9} + 1
[6,3]
 => [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
 => [[[.,[.,.]],[.,[.,.]]],[.,.]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 8 = 7 + 1
[3,3,1,1,1]
 => [1,0,1,1,1,0,0,1,1,0,0,0]
 => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6 = 5 + 1
[6,2,2,2]
 => [1,1,1,0,0,1,1,1,0,0,0,0,1,0]
 => [[.,.],[[[.,.],.],[.,[.,.]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 8 = 7 + 1
[6,3,2,2]
 => [1,1,1,0,0,1,1,0,1,0,0,0,1,0]
 => [[.,.],[[.,[.,.]],[.,[.,.]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 8 = 7 + 1
[6,2,2,2,1]
 => [1,1,0,1,0,1,1,1,0,0,0,0,1,0]
 => [[[[[.,.],.],[.,[.,.]]],.],.]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 8 = 7 + 1
[5,2,2,2,2]
 => [1,1,0,0,1,1,1,1,0,0,0,1,0,0]
 => [[.,[[[.,.],.],[[.,.],.]]],.]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 8 = 7 + 1
[5,2,2,2,1,1]
 => [1,0,1,1,0,1,1,1,0,0,0,1,0,0]
 => [.,[[[[.,.],.],[[.,.],.]],.]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 8 = 7 + 1
[6,3,2,2,1]
 => [1,1,0,1,0,1,1,0,1,0,0,0,1,0]
 => [[[[.,[.,.]],[.,[.,.]]],.],.]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 8 = 7 + 1
[6,2,2,2,2]
 => [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
 => [[.,[[[.,.],.],[.,[.,.]]]],.]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 8 = 7 + 1
[6,2,2,2,1,1]
 => [1,0,1,1,0,1,1,1,0,0,0,0,1,0]
 => [.,[[[[.,.],.],[.,[.,.]]],.]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 8 = 7 + 1
[5,3,2,2,2]
 => [1,1,0,0,1,1,1,0,1,0,0,1,0,0]
 => [[.,[[.,[.,.]],[[.,.],.]]],.]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 8 = 7 + 1
[5,3,2,2,1,1]
 => [1,0,1,1,0,1,1,0,1,0,0,1,0,0]
 => [.,[[[.,[.,.]],[[.,.],.]],.]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 8 = 7 + 1
[5,2,2,2,2,1]
 => [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => [.,[.,[[[.,.],.],[[.,.],.]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 8 = 7 + 1
[6,3,2,2,2]
 => [1,1,0,0,1,1,1,0,1,0,0,0,1,0]
 => [[.,[[.,[.,.]],[.,[.,.]]]],.]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 8 = 7 + 1
[6,3,2,2,1,1]
 => [1,0,1,1,0,1,1,0,1,0,0,0,1,0]
 => [.,[[[.,[.,.]],[.,[.,.]]],.]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 8 = 7 + 1
[6,2,2,2,2,1]
 => [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => [.,[.,[[[.,.],.],[.,[.,.]]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 8 = 7 + 1
[5,3,2,2,2,1]
 => [1,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => [.,[.,[[.,[.,.]],[[.,.],.]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 8 = 7 + 1
[6,3,2,2,2,1]
 => [1,0,1,0,1,1,1,0,1,0,0,0,1,0]
 => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 8 = 7 + 1
Description
The energy of a graph, if it is integral. The energy of a graph is the sum of the absolute values of its eigenvalues. This statistic is only defined for graphs with integral energy. It is known, that the energy is never an odd integer [2]. In fact, it is never the square root of an odd integer [3]. The energy of a graph is the sum of the energies of the connected components of a graph. The energy of the complete graph $K_n$ equals $2n-2$. For this reason, we do not define the energy of the empty graph.