Your data matches 7 different statistics following compositions of up to 3 maps.
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Matching statistic: St001002
Mapping
St001002: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => 3
[1,0,1,0]
 => 3
[1,1,0,0]
 => 6
[1,0,1,0,1,0]
 => 3
[1,0,1,1,0,0]
 => 5
[1,1,0,0,1,0]
 => 5
[1,1,0,1,0,0]
 => 5
[1,1,1,0,0,0]
 => 10
[1,0,1,0,1,0,1,0]
 => 4
[1,0,1,0,1,1,0,0]
 => 5
[1,0,1,1,0,0,1,0]
 => 5
[1,0,1,1,0,1,0,0]
 => 4
[1,0,1,1,1,0,0,0]
 => 8
[1,1,0,0,1,0,1,0]
 => 5
[1,1,0,0,1,1,0,0]
 => 7
[1,1,0,1,0,0,1,0]
 => 4
[1,1,0,1,0,1,0,0]
 => 4
[1,1,0,1,1,0,0,0]
 => 7
[1,1,1,0,0,0,1,0]
 => 8
[1,1,1,0,0,1,0,0]
 => 7
[1,1,1,0,1,0,0,0]
 => 8
[1,1,1,1,0,0,0,0]
 => 15
[1,0,1,0,1,0,1,0,1,0]
 => 5
[1,0,1,0,1,0,1,1,0,0]
 => 6
[1,0,1,0,1,1,0,0,1,0]
 => 5
[1,0,1,0,1,1,0,1,0,0]
 => 5
[1,0,1,0,1,1,1,0,0,0]
 => 8
[1,0,1,1,0,0,1,0,1,0]
 => 5
[1,0,1,1,0,0,1,1,0,0]
 => 7
[1,0,1,1,0,1,0,0,1,0]
 => 4
[1,0,1,1,0,1,0,1,0,0]
 => 5
[1,0,1,1,0,1,1,0,0,0]
 => 6
[1,0,1,1,1,0,0,0,1,0]
 => 7
[1,0,1,1,1,0,0,1,0,0]
 => 7
[1,0,1,1,1,0,1,0,0,0]
 => 6
[1,0,1,1,1,1,0,0,0,0]
 => 12
[1,1,0,0,1,0,1,0,1,0]
 => 6
[1,1,0,0,1,0,1,1,0,0]
 => 7
[1,1,0,0,1,1,0,0,1,0]
 => 7
[1,1,0,0,1,1,0,1,0,0]
 => 6
[1,1,0,0,1,1,1,0,0,0]
 => 10
[1,1,0,1,0,0,1,0,1,0]
 => 5
[1,1,0,1,0,0,1,1,0,0]
 => 6
[1,1,0,1,0,1,0,0,1,0]
 => 5
[1,1,0,1,0,1,0,1,0,0]
 => 4
[1,1,0,1,0,1,1,0,0,0]
 => 6
[1,1,0,1,1,0,0,0,1,0]
 => 7
[1,1,0,1,1,0,0,1,0,0]
 => 5
[1,1,0,1,1,0,1,0,0,0]
 => 5
[1,1,0,1,1,1,0,0,0,0]
 => 10
Description
Number of indecomposable modules with projective and injective dimension at most 1 in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St000259
Mapping
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000259: Graphs ⟶ ℤResult quality: 18% ●values known / values provided: 22%●distinct values known / distinct values provided: 18%
Values
[1,0]
 => [1] => [1] => ([],1)
 => 0 = 3 - 3
[1,0,1,0]
 => [2,1] => [2] => ([],2)
 => ? ∊ {3,6} - 3
[1,1,0,0]
 => [1,2] => [2] => ([],2)
 => ? ∊ {3,6} - 3
[1,0,1,0,1,0]
 => [3,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 2 = 5 - 3
[1,0,1,1,0,0]
 => [2,3,1] => [3] => ([],3)
 => ? ∊ {3,5,5,10} - 3
[1,1,0,0,1,0]
 => [3,1,2] => [3] => ([],3)
 => ? ∊ {3,5,5,10} - 3
[1,1,0,1,0,0]
 => [2,1,3] => [3] => ([],3)
 => ? ∊ {3,5,5,10} - 3
[1,1,1,0,0,0]
 => [1,2,3] => [3] => ([],3)
 => ? ∊ {3,5,5,10} - 3
[1,0,1,0,1,0,1,0]
 => [4,3,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 5 - 3
[1,0,1,0,1,1,0,0]
 => [3,4,2,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 5 - 3
[1,0,1,1,0,0,1,0]
 => [4,2,3,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 5 - 3
[1,0,1,1,0,1,0,0]
 => [3,2,4,1] => [2,2] => ([(1,3),(2,3)],4)
 => ? ∊ {4,4,4,4,7,7,7,8,8,8,15} - 3
[1,0,1,1,1,0,0,0]
 => [2,3,4,1] => [4] => ([],4)
 => ? ∊ {4,4,4,4,7,7,7,8,8,8,15} - 3
[1,1,0,0,1,0,1,0]
 => [4,3,1,2] => [1,3] => ([(2,3)],4)
 => ? ∊ {4,4,4,4,7,7,7,8,8,8,15} - 3
[1,1,0,0,1,1,0,0]
 => [3,4,1,2] => [4] => ([],4)
 => ? ∊ {4,4,4,4,7,7,7,8,8,8,15} - 3
[1,1,0,1,0,0,1,0]
 => [4,2,1,3] => [2,2] => ([(1,3),(2,3)],4)
 => ? ∊ {4,4,4,4,7,7,7,8,8,8,15} - 3
[1,1,0,1,0,1,0,0]
 => [3,2,1,4] => [2,2] => ([(1,3),(2,3)],4)
 => ? ∊ {4,4,4,4,7,7,7,8,8,8,15} - 3
[1,1,0,1,1,0,0,0]
 => [2,3,1,4] => [4] => ([],4)
 => ? ∊ {4,4,4,4,7,7,7,8,8,8,15} - 3
[1,1,1,0,0,0,1,0]
 => [4,1,2,3] => [4] => ([],4)
 => ? ∊ {4,4,4,4,7,7,7,8,8,8,15} - 3
[1,1,1,0,0,1,0,0]
 => [3,1,2,4] => [4] => ([],4)
 => ? ∊ {4,4,4,4,7,7,7,8,8,8,15} - 3
[1,1,1,0,1,0,0,0]
 => [2,1,3,4] => [4] => ([],4)
 => ? ∊ {4,4,4,4,7,7,7,8,8,8,15} - 3
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [4] => ([],4)
 => ? ∊ {4,4,4,4,7,7,7,8,8,8,15} - 3
[1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 5 - 3
[1,0,1,0,1,0,1,1,0,0]
 => [4,5,3,2,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 5 - 3
[1,0,1,0,1,1,0,0,1,0]
 => [5,3,4,2,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 5 - 3
[1,0,1,0,1,1,0,1,0,0]
 => [4,3,5,2,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 5 - 3
[1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 5 - 3
[1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,3,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 5 - 3
[1,0,1,1,0,0,1,1,0,0]
 => [4,5,2,3,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 5 - 3
[1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,4,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 5 - 3
[1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,5,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {4,4,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,9,10,10,10,10,10,10,12,12,12,21} - 3
[1,0,1,1,0,1,1,0,0,0]
 => [3,4,2,5,1] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {4,4,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,9,10,10,10,10,10,10,12,12,12,21} - 3
[1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 5 - 3
[1,0,1,1,1,0,0,1,0,0]
 => [4,2,3,5,1] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {4,4,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,9,10,10,10,10,10,10,12,12,12,21} - 3
[1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => [2,3] => ([(2,4),(3,4)],5)
 => ? ∊ {4,4,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,9,10,10,10,10,10,10,12,12,12,21} - 3
[1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [5] => ([],5)
 => ? ∊ {4,4,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,9,10,10,10,10,10,10,12,12,12,21} - 3
[1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {4,4,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,9,10,10,10,10,10,10,12,12,12,21} - 3
[1,1,0,0,1,0,1,1,0,0]
 => [4,5,3,1,2] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {4,4,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,9,10,10,10,10,10,10,12,12,12,21} - 3
[1,1,0,0,1,1,0,0,1,0]
 => [5,3,4,1,2] => [1,4] => ([(3,4)],5)
 => ? ∊ {4,4,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,9,10,10,10,10,10,10,12,12,12,21} - 3
[1,1,0,0,1,1,0,1,0,0]
 => [4,3,5,1,2] => [1,4] => ([(3,4)],5)
 => ? ∊ {4,4,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,9,10,10,10,10,10,10,12,12,12,21} - 3
[1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => [5] => ([],5)
 => ? ∊ {4,4,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,9,10,10,10,10,10,10,12,12,12,21} - 3
[1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,3] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {4,4,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,9,10,10,10,10,10,10,12,12,12,21} - 3
[1,1,0,1,0,0,1,1,0,0]
 => [4,5,2,1,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {4,4,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,9,10,10,10,10,10,10,12,12,12,21} - 3
[1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,4] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {4,4,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,9,10,10,10,10,10,10,12,12,12,21} - 3
[1,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1,5] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {4,4,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,9,10,10,10,10,10,10,12,12,12,21} - 3
[1,1,0,1,0,1,1,0,0,0]
 => [3,4,2,1,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {4,4,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,9,10,10,10,10,10,10,12,12,12,21} - 3
[1,1,0,1,1,0,0,0,1,0]
 => [5,2,3,1,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {4,4,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,9,10,10,10,10,10,10,12,12,12,21} - 3
[1,1,0,1,1,0,0,1,0,0]
 => [4,2,3,1,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {4,4,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,9,10,10,10,10,10,10,12,12,12,21} - 3
[1,1,0,1,1,0,1,0,0,0]
 => [3,2,4,1,5] => [2,3] => ([(2,4),(3,4)],5)
 => ? ∊ {4,4,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,9,10,10,10,10,10,10,12,12,12,21} - 3
[1,1,0,1,1,1,0,0,0,0]
 => [2,3,4,1,5] => [5] => ([],5)
 => ? ∊ {4,4,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,9,10,10,10,10,10,10,12,12,12,21} - 3
[1,1,1,0,0,0,1,0,1,0]
 => [5,4,1,2,3] => [1,4] => ([(3,4)],5)
 => ? ∊ {4,4,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,9,10,10,10,10,10,10,12,12,12,21} - 3
[1,1,1,0,0,0,1,1,0,0]
 => [4,5,1,2,3] => [5] => ([],5)
 => ? ∊ {4,4,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,9,10,10,10,10,10,10,12,12,12,21} - 3
[1,1,1,0,0,1,0,0,1,0]
 => [5,3,1,2,4] => [1,4] => ([(3,4)],5)
 => ? ∊ {4,4,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,9,10,10,10,10,10,10,12,12,12,21} - 3
[1,1,1,0,0,1,0,1,0,0]
 => [4,3,1,2,5] => [1,4] => ([(3,4)],5)
 => ? ∊ {4,4,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,9,10,10,10,10,10,10,12,12,12,21} - 3
[1,1,1,0,0,1,1,0,0,0]
 => [3,4,1,2,5] => [5] => ([],5)
 => ? ∊ {4,4,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,9,10,10,10,10,10,10,12,12,12,21} - 3
[1,1,1,0,1,0,0,0,1,0]
 => [5,2,1,3,4] => [2,3] => ([(2,4),(3,4)],5)
 => ? ∊ {4,4,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,9,10,10,10,10,10,10,12,12,12,21} - 3
[1,1,1,0,1,0,0,1,0,0]
 => [4,2,1,3,5] => [2,3] => ([(2,4),(3,4)],5)
 => ? ∊ {4,4,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,9,10,10,10,10,10,10,12,12,12,21} - 3
[1,1,1,0,1,0,1,0,0,0]
 => [3,2,1,4,5] => [2,3] => ([(2,4),(3,4)],5)
 => ? ∊ {4,4,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,9,10,10,10,10,10,10,12,12,12,21} - 3
[1,1,1,0,1,1,0,0,0,0]
 => [2,3,1,4,5] => [5] => ([],5)
 => ? ∊ {4,4,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,9,10,10,10,10,10,10,12,12,12,21} - 3
[1,1,1,1,0,0,0,0,1,0]
 => [5,1,2,3,4] => [5] => ([],5)
 => ? ∊ {4,4,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,9,10,10,10,10,10,10,12,12,12,21} - 3
[1,1,1,1,0,0,0,1,0,0]
 => [4,1,2,3,5] => [5] => ([],5)
 => ? ∊ {4,4,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,9,10,10,10,10,10,10,12,12,12,21} - 3
[1,1,1,1,0,0,1,0,0,0]
 => [3,1,2,4,5] => [5] => ([],5)
 => ? ∊ {4,4,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,9,10,10,10,10,10,10,12,12,12,21} - 3
[1,1,1,1,0,1,0,0,0,0]
 => [2,1,3,4,5] => [5] => ([],5)
 => ? ∊ {4,4,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,9,10,10,10,10,10,10,12,12,12,21} - 3
[1,1,1,1,1,0,0,0,0,0]
 => [1,2,3,4,5] => [5] => ([],5)
 => ? ∊ {4,4,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,9,10,10,10,10,10,10,12,12,12,21} - 3
Description
The diameter of a connected graph. This is the greatest distance between any pair of vertices.
Matching statistic: St001232
(load all 2 compositions to match this statistic)
Mapping
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 27%
Values
[1,0]
 => [1,1,0,0]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1 = 3 - 2
[1,0,1,0]
 => [1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 1 = 3 - 2
[1,1,0,0]
 => [1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => ? = 6 - 2
[1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 3 = 5 - 2
[1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => ? ∊ {3,5,10} - 2
[1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 3 = 5 - 2
[1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => ? ∊ {3,5,10} - 2
[1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => ? ∊ {3,5,10} - 2
[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]
 => [1,1,0,0,1,1,0,0,1,0]
 => 3 = 5 - 2
[1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => ? ∊ {4,4,4,4,5,7,7,7,8,8,8,15} - 2
[1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 3 = 5 - 2
[1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => ? ∊ {4,4,4,4,5,7,7,7,8,8,8,15} - 2
[1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => ? ∊ {4,4,4,4,5,7,7,7,8,8,8,15} - 2
[1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => ? ∊ {4,4,4,4,5,7,7,7,8,8,8,15} - 2
[1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => ? ∊ {4,4,4,4,5,7,7,7,8,8,8,15} - 2
[1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => ? ∊ {4,4,4,4,5,7,7,7,8,8,8,15} - 2
[1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => ? ∊ {4,4,4,4,5,7,7,7,8,8,8,15} - 2
[1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => ? ∊ {4,4,4,4,5,7,7,7,8,8,8,15} - 2
[1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => ? ∊ {4,4,4,4,5,7,7,7,8,8,8,15} - 2
[1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => ? ∊ {4,4,4,4,5,7,7,7,8,8,8,15} - 2
[1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => ? ∊ {4,4,4,4,5,7,7,7,8,8,8,15} - 2
[1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => ? ∊ {4,4,4,4,5,7,7,7,8,8,8,15} - 2
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [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 = 7 - 2
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,0,1,1,0,0,1,0,1,0]
 => ? ∊ {4,4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,8,8,9,10,10,10,10,10,10,12,12,12,21} - 2
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => [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 = 7 - 2
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,0,0]
 => [1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,0,1,1,0,0,1,0,1,0]
 => ? ∊ {4,4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,8,8,9,10,10,10,10,10,10,12,12,12,21} - 2
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0]
 => ? ∊ {4,4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,8,8,9,10,10,10,10,10,10,12,12,12,21} - 2
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0,1,0]
 => ? ∊ {4,4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,8,8,9,10,10,10,10,10,10,12,12,12,21} - 2
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,0,1,1,0,0,1,0,1,0]
 => ? ∊ {4,4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,8,8,9,10,10,10,10,10,10,12,12,12,21} - 2
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0,1,0]
 => ? ∊ {4,4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,8,8,9,10,10,10,10,10,10,12,12,12,21} - 2
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => [1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,0,1,1,0,0,1,0,1,0]
 => ? ∊ {4,4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,8,8,9,10,10,10,10,10,10,12,12,12,21} - 2
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0]
 => ? ∊ {4,4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,8,8,9,10,10,10,10,10,10,12,12,12,21} - 2
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0,1,0]
 => ? ∊ {4,4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,8,8,9,10,10,10,10,10,10,12,12,12,21} - 2
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => [1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,0,1,1,0,0,1,0,1,0]
 => ? ∊ {4,4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,8,8,9,10,10,10,10,10,10,12,12,12,21} - 2
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0]
 => ? ∊ {4,4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,8,8,9,10,10,10,10,10,10,12,12,12,21} - 2
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0]
 => ? ∊ {4,4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,8,8,9,10,10,10,10,10,10,12,12,12,21} - 2
[1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => 5 = 7 - 2
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,1,0,0,0]
 => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => ? ∊ {4,4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,8,8,9,10,10,10,10,10,10,12,12,12,21} - 2
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => 5 = 7 - 2
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,1,0,1,0,0,0]
 => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => ? ∊ {4,4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,8,8,9,10,10,10,10,10,10,12,12,12,21} - 2
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0]
 => ? ∊ {4,4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,8,8,9,10,10,10,10,10,10,12,12,12,21} - 2
[1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0,1,0]
 => ? ∊ {4,4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,8,8,9,10,10,10,10,10,10,12,12,12,21} - 2
[1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,1,0,0,0]
 => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => ? ∊ {4,4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,8,8,9,10,10,10,10,10,10,12,12,12,21} - 2
[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]
 => [1,1,0,0,1,0,1,1,0,0,1,0]
 => ? ∊ {4,4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,8,8,9,10,10,10,10,10,10,12,12,12,21} - 2
[1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => ? ∊ {4,4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,8,8,9,10,10,10,10,10,10,12,12,12,21} - 2
[1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,1,0,1,1,0,0,0,0]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0]
 => ? ∊ {4,4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,8,8,9,10,10,10,10,10,10,12,12,12,21} - 2
[1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0,1,0]
 => ? ∊ {4,4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,8,8,9,10,10,10,10,10,10,12,12,12,21} - 2
[1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,1,0,0,0]
 => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => ? ∊ {4,4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,8,8,9,10,10,10,10,10,10,12,12,12,21} - 2
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0]
 => ? ∊ {4,4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,8,8,9,10,10,10,10,10,10,12,12,12,21} - 2
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0]
 => ? ∊ {4,4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,8,8,9,10,10,10,10,10,10,12,12,12,21} - 2
[1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0]
 => ? ∊ {4,4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,8,8,9,10,10,10,10,10,10,12,12,12,21} - 2
[1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => ? ∊ {4,4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,8,8,9,10,10,10,10,10,10,12,12,12,21} - 2
[1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0]
 => ? ∊ {4,4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,8,8,9,10,10,10,10,10,10,12,12,12,21} - 2
[1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,1,0,0,1,0,1,0,0,0]
 => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => ? ∊ {4,4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,8,8,9,10,10,10,10,10,10,12,12,12,21} - 2
[1,1,1,0,0,1,1,0,0,0]
 => [1,1,1,1,0,0,1,1,0,0,0,0]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0]
 => ? ∊ {4,4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,8,8,9,10,10,10,10,10,10,12,12,12,21} - 2
[1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0]
 => ? ∊ {4,4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,8,8,9,10,10,10,10,10,10,12,12,12,21} - 2
[1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,1,0,1,0,0,1,0,0,0]
 => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => ? ∊ {4,4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,8,8,9,10,10,10,10,10,10,12,12,12,21} - 2
[1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0]
 => ? ∊ {4,4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,8,8,9,10,10,10,10,10,10,12,12,12,21} - 2
[1,1,1,0,1,1,0,0,0,0]
 => [1,1,1,1,0,1,1,0,0,0,0,0]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0]
 => ? ∊ {4,4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,8,8,9,10,10,10,10,10,10,12,12,12,21} - 2
[1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0]
 => ? ∊ {4,4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,8,8,9,10,10,10,10,10,10,12,12,12,21} - 2
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Matching statistic: St001279
Mapping
Mp00103: Dyck paths —peeling map⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St001279: Integer partitions ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 27%
Values
[1,0]
 => [1,0]
 => [1,0]
 => []
 => ? = 3
[1,0,1,0]
 => [1,0,1,0]
 => [1,1,0,0]
 => []
 => ? ∊ {3,6}
[1,1,0,0]
 => [1,0,1,0]
 => [1,1,0,0]
 => []
 => ? ∊ {3,6}
[1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => []
 => ? ∊ {3,5,5,5,10}
[1,0,1,1,0,0]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => []
 => ? ∊ {3,5,5,5,10}
[1,1,0,0,1,0]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => []
 => ? ∊ {3,5,5,5,10}
[1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => []
 => ? ∊ {3,5,5,5,10}
[1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => []
 => ? ∊ {3,5,5,5,10}
[1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? ∊ {4,4,4,5,5,5,7,7,7,8,8,8,15}
[1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? ∊ {4,4,4,5,5,5,7,7,7,8,8,8,15}
[1,0,1,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? ∊ {4,4,4,5,5,5,7,7,7,8,8,8,15}
[1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? ∊ {4,4,4,5,5,5,7,7,7,8,8,8,15}
[1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? ∊ {4,4,4,5,5,5,7,7,7,8,8,8,15}
[1,1,0,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? ∊ {4,4,4,5,5,5,7,7,7,8,8,8,15}
[1,1,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? ∊ {4,4,4,5,5,5,7,7,7,8,8,8,15}
[1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? ∊ {4,4,4,5,5,5,7,7,7,8,8,8,15}
[1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? ∊ {4,4,4,5,5,5,7,7,7,8,8,8,15}
[1,1,0,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? ∊ {4,4,4,5,5,5,7,7,7,8,8,8,15}
[1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? ∊ {4,4,4,5,5,5,7,7,7,8,8,8,15}
[1,1,1,0,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? ∊ {4,4,4,5,5,5,7,7,7,8,8,8,15}
[1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? ∊ {4,4,4,5,5,5,7,7,7,8,8,8,15}
[1,1,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => [2,2]
 => 4
[1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? ∊ {4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,8,9,10,10,10,10,10,10,12,12,12,21}
[1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? ∊ {4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,8,9,10,10,10,10,10,10,12,12,12,21}
[1,0,1,0,1,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? ∊ {4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,8,9,10,10,10,10,10,10,12,12,12,21}
[1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? ∊ {4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,8,9,10,10,10,10,10,10,12,12,12,21}
[1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? ∊ {4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,8,9,10,10,10,10,10,10,12,12,12,21}
[1,0,1,1,0,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? ∊ {4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,8,9,10,10,10,10,10,10,12,12,12,21}
[1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? ∊ {4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,8,9,10,10,10,10,10,10,12,12,12,21}
[1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? ∊ {4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,8,9,10,10,10,10,10,10,12,12,12,21}
[1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? ∊ {4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,8,9,10,10,10,10,10,10,12,12,12,21}
[1,0,1,1,0,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? ∊ {4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,8,9,10,10,10,10,10,10,12,12,12,21}
[1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? ∊ {4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,8,9,10,10,10,10,10,10,12,12,12,21}
[1,0,1,1,1,0,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? ∊ {4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,8,9,10,10,10,10,10,10,12,12,12,21}
[1,0,1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? ∊ {4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,8,9,10,10,10,10,10,10,12,12,12,21}
[1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [3,3]
 => 6
[1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? ∊ {4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,8,9,10,10,10,10,10,10,12,12,12,21}
[1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? ∊ {4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,8,9,10,10,10,10,10,10,12,12,12,21}
[1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? ∊ {4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,8,9,10,10,10,10,10,10,12,12,12,21}
[1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? ∊ {4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,8,9,10,10,10,10,10,10,12,12,12,21}
[1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? ∊ {4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,8,9,10,10,10,10,10,10,12,12,12,21}
[1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? ∊ {4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,8,9,10,10,10,10,10,10,12,12,12,21}
[1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? ∊ {4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,8,9,10,10,10,10,10,10,12,12,12,21}
[1,1,0,1,0,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? ∊ {4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,8,9,10,10,10,10,10,10,12,12,12,21}
[1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? ∊ {4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,8,9,10,10,10,10,10,10,12,12,12,21}
[1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? ∊ {4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,8,9,10,10,10,10,10,10,12,12,12,21}
[1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? ∊ {4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,8,9,10,10,10,10,10,10,12,12,12,21}
[1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? ∊ {4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,8,9,10,10,10,10,10,10,12,12,12,21}
[1,1,0,1,1,0,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? ∊ {4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,8,9,10,10,10,10,10,10,12,12,12,21}
[1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [3,3]
 => 6
[1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? ∊ {4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,8,9,10,10,10,10,10,10,12,12,12,21}
[1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? ∊ {4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,8,9,10,10,10,10,10,10,12,12,12,21}
[1,1,1,0,0,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? ∊ {4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,8,9,10,10,10,10,10,10,12,12,12,21}
[1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [3,3]
 => 6
[1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,2,2]
 => 6
[1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,2,2]
 => 6
[1,1,1,1,0,0,1,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,2,2]
 => 6
[1,1,1,1,0,1,0,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [2,2]
 => 4
[1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [3,3,2]
 => 8
Description
The sum of the parts of an integer partition that are at least two.
Matching statistic: St001880
(load all 9 compositions to match this statistic)
Mapping
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001880: Posets ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 27%
Values
[1,0]
 => [1] => ([],1)
 => ? = 3
[1,0,1,0]
 => [2,1] => ([],2)
 => ? ∊ {3,6}
[1,1,0,0]
 => [1,2] => ([(0,1)],2)
 => ? ∊ {3,6}
[1,0,1,0,1,0]
 => [2,3,1] => ([(1,2)],3)
 => ? ∊ {5,5,5,10}
[1,0,1,1,0,0]
 => [2,1,3] => ([(0,2),(1,2)],3)
 => ? ∊ {5,5,5,10}
[1,1,0,0,1,0]
 => [1,3,2] => ([(0,1),(0,2)],3)
 => ? ∊ {5,5,5,10}
[1,1,0,1,0,0]
 => [3,1,2] => ([(1,2)],3)
 => ? ∊ {5,5,5,10}
[1,1,1,0,0,0]
 => [1,2,3] => ([(0,2),(2,1)],3)
 => 3
[1,0,1,0,1,0,1,0]
 => [2,3,4,1] => ([(1,2),(2,3)],4)
 => ? ∊ {4,4,5,5,5,7,7,7,8,8,8,15}
[1,0,1,0,1,1,0,0]
 => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
 => ? ∊ {4,4,5,5,5,7,7,7,8,8,8,15}
[1,0,1,1,0,0,1,0]
 => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ? ∊ {4,4,5,5,5,7,7,7,8,8,8,15}
[1,0,1,1,0,1,0,0]
 => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
 => ? ∊ {4,4,5,5,5,7,7,7,8,8,8,15}
[1,0,1,1,1,0,0,0]
 => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
 => ? ∊ {4,4,5,5,5,7,7,7,8,8,8,15}
[1,1,0,0,1,0,1,0]
 => [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
 => ? ∊ {4,4,5,5,5,7,7,7,8,8,8,15}
[1,1,0,0,1,1,0,0]
 => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 4
[1,1,0,1,0,0,1,0]
 => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
 => ? ∊ {4,4,5,5,5,7,7,7,8,8,8,15}
[1,1,0,1,0,1,0,0]
 => [3,4,1,2] => ([(0,3),(1,2)],4)
 => ? ∊ {4,4,5,5,5,7,7,7,8,8,8,15}
[1,1,0,1,1,0,0,0]
 => [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
 => ? ∊ {4,4,5,5,5,7,7,7,8,8,8,15}
[1,1,1,0,0,0,1,0]
 => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
 => ? ∊ {4,4,5,5,5,7,7,7,8,8,8,15}
[1,1,1,0,0,1,0,0]
 => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
 => ? ∊ {4,4,5,5,5,7,7,7,8,8,8,15}
[1,1,1,0,1,0,0,0]
 => [4,1,2,3] => ([(1,2),(2,3)],4)
 => ? ∊ {4,4,5,5,5,7,7,7,8,8,8,15}
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,0,1,0,1,0,1,0,1,0]
 => [2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
 => ? ∊ {5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,9,10,10,10,10,10,10,12,12,12,21}
[1,0,1,0,1,0,1,1,0,0]
 => [2,3,4,1,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? ∊ {5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,9,10,10,10,10,10,10,12,12,12,21}
[1,0,1,0,1,1,0,0,1,0]
 => [2,3,1,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
 => ? ∊ {5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,9,10,10,10,10,10,10,12,12,12,21}
[1,0,1,0,1,1,0,1,0,0]
 => [2,3,5,1,4] => ([(0,4),(1,2),(2,3),(2,4)],5)
 => ? ∊ {5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,9,10,10,10,10,10,10,12,12,12,21}
[1,0,1,0,1,1,1,0,0,0]
 => [2,3,1,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? ∊ {5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,9,10,10,10,10,10,10,12,12,12,21}
[1,0,1,1,0,0,1,0,1,0]
 => [2,1,4,5,3] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => ? ∊ {5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,9,10,10,10,10,10,10,12,12,12,21}
[1,0,1,1,0,0,1,1,0,0]
 => [2,1,4,3,5] => ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => ? ∊ {5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,9,10,10,10,10,10,10,12,12,12,21}
[1,0,1,1,0,1,0,0,1,0]
 => [2,4,1,5,3] => ([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
 => ? ∊ {5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,9,10,10,10,10,10,10,12,12,12,21}
[1,0,1,1,0,1,0,1,0,0]
 => [2,4,5,1,3] => ([(0,4),(1,2),(1,4),(2,3)],5)
 => ? ∊ {5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,9,10,10,10,10,10,10,12,12,12,21}
[1,0,1,1,0,1,1,0,0,0]
 => [2,4,1,3,5] => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => ? ∊ {5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,9,10,10,10,10,10,10,12,12,12,21}
[1,0,1,1,1,0,0,0,1,0]
 => [2,1,3,5,4] => ([(0,4),(1,4),(4,2),(4,3)],5)
 => ? ∊ {5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,9,10,10,10,10,10,10,12,12,12,21}
[1,0,1,1,1,0,0,1,0,0]
 => [2,1,5,3,4] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => ? ∊ {5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,9,10,10,10,10,10,10,12,12,12,21}
[1,0,1,1,1,0,1,0,0,0]
 => [2,5,1,3,4] => ([(0,4),(1,2),(1,4),(4,3)],5)
 => ? ∊ {5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,9,10,10,10,10,10,10,12,12,12,21}
[1,0,1,1,1,1,0,0,0,0]
 => [2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ? ∊ {5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,9,10,10,10,10,10,10,12,12,12,21}
[1,1,0,0,1,0,1,0,1,0]
 => [1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
 => ? ∊ {5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,9,10,10,10,10,10,10,12,12,12,21}
[1,1,0,0,1,0,1,1,0,0]
 => [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => 4
[1,1,0,0,1,1,0,0,1,0]
 => [1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ? ∊ {5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,9,10,10,10,10,10,10,12,12,12,21}
[1,1,0,0,1,1,0,1,0,0]
 => [1,3,5,2,4] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ? ∊ {5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,9,10,10,10,10,10,10,12,12,12,21}
[1,1,0,0,1,1,1,0,0,0]
 => [1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 5
[1,1,0,1,0,0,1,0,1,0]
 => [3,1,4,5,2] => ([(0,4),(1,2),(1,4),(4,3)],5)
 => ? ∊ {5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,9,10,10,10,10,10,10,12,12,12,21}
[1,1,0,1,0,0,1,1,0,0]
 => [3,1,4,2,5] => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => ? ∊ {5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,9,10,10,10,10,10,10,12,12,12,21}
[1,1,0,1,0,1,0,0,1,0]
 => [3,4,1,5,2] => ([(0,3),(1,2),(1,4),(3,4)],5)
 => ? ∊ {5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,9,10,10,10,10,10,10,12,12,12,21}
[1,1,0,1,0,1,0,1,0,0]
 => [3,4,5,1,2] => ([(0,3),(1,4),(4,2)],5)
 => ? ∊ {5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,9,10,10,10,10,10,10,12,12,12,21}
[1,1,0,1,0,1,1,0,0,0]
 => [3,4,1,2,5] => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? ∊ {5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,9,10,10,10,10,10,10,12,12,12,21}
[1,1,0,1,1,0,0,0,1,0]
 => [3,1,2,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
 => ? ∊ {5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,9,10,10,10,10,10,10,12,12,12,21}
[1,1,0,1,1,0,0,1,0,0]
 => [3,1,5,2,4] => ([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
 => ? ∊ {5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,9,10,10,10,10,10,10,12,12,12,21}
[1,1,0,1,1,0,1,0,0,0]
 => [3,5,1,2,4] => ([(0,3),(1,2),(1,4),(3,4)],5)
 => ? ∊ {5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,9,10,10,10,10,10,10,12,12,12,21}
[1,1,0,1,1,1,0,0,0,0]
 => [3,1,2,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? ∊ {5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,9,10,10,10,10,10,10,12,12,12,21}
[1,1,1,0,0,0,1,0,1,0]
 => [1,2,4,5,3] => ([(0,4),(3,2),(4,1),(4,3)],5)
 => ? ∊ {5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,9,10,10,10,10,10,10,12,12,12,21}
[1,1,1,0,0,0,1,1,0,0]
 => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 5
[1,1,1,0,0,1,0,0,1,0]
 => [1,4,2,5,3] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ? ∊ {5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,9,10,10,10,10,10,10,12,12,12,21}
[1,1,1,0,0,1,0,1,0,0]
 => [1,4,5,2,3] => ([(0,3),(0,4),(3,2),(4,1)],5)
 => ? ∊ {5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,9,10,10,10,10,10,10,12,12,12,21}
[1,1,1,0,0,1,1,0,0,0]
 => [1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => 4
[1,1,1,0,1,0,0,0,1,0]
 => [4,1,2,5,3] => ([(0,4),(1,2),(2,3),(2,4)],5)
 => ? ∊ {5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,9,10,10,10,10,10,10,12,12,12,21}
[1,1,1,0,1,0,0,1,0,0]
 => [4,1,5,2,3] => ([(0,4),(1,2),(1,4),(2,3)],5)
 => ? ∊ {5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,9,10,10,10,10,10,10,12,12,12,21}
[1,1,1,1,1,0,0,0,0,0]
 => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5
Description
The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice.
Matching statistic: St000422
(load all 5 compositions to match this statistic)
Mapping
Mp00026: Dyck paths —to ordered tree⟶ Ordered trees
Mp00046: Ordered trees —to graph⟶ Graphs
St000422: Graphs ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 27%
Values
[1,0]
 => [[]]
 => ([(0,1)],2)
 => 2 = 3 - 1
[1,0,1,0]
 => [[],[]]
 => ([(0,2),(1,2)],3)
 => ? ∊ {3,6} - 1
[1,1,0,0]
 => [[[]]]
 => ([(0,2),(1,2)],3)
 => ? ∊ {3,6} - 1
[1,0,1,0,1,0]
 => [[],[],[]]
 => ([(0,3),(1,3),(2,3)],4)
 => ? ∊ {3,5,5,5,10} - 1
[1,0,1,1,0,0]
 => [[],[[]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? ∊ {3,5,5,5,10} - 1
[1,1,0,0,1,0]
 => [[[]],[]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? ∊ {3,5,5,5,10} - 1
[1,1,0,1,0,0]
 => [[[],[]]]
 => ([(0,3),(1,3),(2,3)],4)
 => ? ∊ {3,5,5,5,10} - 1
[1,1,1,0,0,0]
 => [[[[]]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? ∊ {3,5,5,5,10} - 1
[1,0,1,0,1,0,1,0]
 => [[],[],[],[]]
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4 = 5 - 1
[1,0,1,0,1,1,0,0]
 => [[],[],[[]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? ∊ {4,4,4,4,5,7,7,7,8,8,8,15} - 1
[1,0,1,1,0,0,1,0]
 => [[],[[]],[]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? ∊ {4,4,4,4,5,7,7,7,8,8,8,15} - 1
[1,0,1,1,0,1,0,0]
 => [[],[[],[]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? ∊ {4,4,4,4,5,7,7,7,8,8,8,15} - 1
[1,0,1,1,1,0,0,0]
 => [[],[[[]]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? ∊ {4,4,4,4,5,7,7,7,8,8,8,15} - 1
[1,1,0,0,1,0,1,0]
 => [[[]],[],[]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? ∊ {4,4,4,4,5,7,7,7,8,8,8,15} - 1
[1,1,0,0,1,1,0,0]
 => [[[]],[[]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? ∊ {4,4,4,4,5,7,7,7,8,8,8,15} - 1
[1,1,0,1,0,0,1,0]
 => [[[],[]],[]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? ∊ {4,4,4,4,5,7,7,7,8,8,8,15} - 1
[1,1,0,1,0,1,0,0]
 => [[[],[],[]]]
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4 = 5 - 1
[1,1,0,1,1,0,0,0]
 => [[[],[[]]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? ∊ {4,4,4,4,5,7,7,7,8,8,8,15} - 1
[1,1,1,0,0,0,1,0]
 => [[[[]]],[]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? ∊ {4,4,4,4,5,7,7,7,8,8,8,15} - 1
[1,1,1,0,0,1,0,0]
 => [[[[]],[]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? ∊ {4,4,4,4,5,7,7,7,8,8,8,15} - 1
[1,1,1,0,1,0,0,0]
 => [[[[],[]]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? ∊ {4,4,4,4,5,7,7,7,8,8,8,15} - 1
[1,1,1,1,0,0,0,0]
 => [[[[[]]]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? ∊ {4,4,4,4,5,7,7,7,8,8,8,15} - 1
[1,0,1,0,1,0,1,0,1,0]
 => [[],[],[],[],[]]
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ? ∊ {4,4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,8,8,9,10,10,10,10,10,10,12,12,12,21} - 1
[1,0,1,0,1,0,1,1,0,0]
 => [[],[],[],[[]]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ? ∊ {4,4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,8,8,9,10,10,10,10,10,10,12,12,12,21} - 1
[1,0,1,0,1,1,0,0,1,0]
 => [[],[],[[]],[]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ? ∊ {4,4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,8,8,9,10,10,10,10,10,10,12,12,12,21} - 1
[1,0,1,0,1,1,0,1,0,0]
 => [[],[],[[],[]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6 = 7 - 1
[1,0,1,0,1,1,1,0,0,0]
 => [[],[],[[[]]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ? ∊ {4,4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,8,8,9,10,10,10,10,10,10,12,12,12,21} - 1
[1,0,1,1,0,0,1,0,1,0]
 => [[],[[]],[],[]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ? ∊ {4,4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,8,8,9,10,10,10,10,10,10,12,12,12,21} - 1
[1,0,1,1,0,0,1,1,0,0]
 => [[],[[]],[[]]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ? ∊ {4,4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,8,8,9,10,10,10,10,10,10,12,12,12,21} - 1
[1,0,1,1,0,1,0,0,1,0]
 => [[],[[],[]],[]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6 = 7 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [[],[[],[],[]]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ? ∊ {4,4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,8,8,9,10,10,10,10,10,10,12,12,12,21} - 1
[1,0,1,1,0,1,1,0,0,0]
 => [[],[[],[[]]]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ? ∊ {4,4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,8,8,9,10,10,10,10,10,10,12,12,12,21} - 1
[1,0,1,1,1,0,0,0,1,0]
 => [[],[[[]]],[]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ? ∊ {4,4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,8,8,9,10,10,10,10,10,10,12,12,12,21} - 1
[1,0,1,1,1,0,0,1,0,0]
 => [[],[[[]],[]]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ? ∊ {4,4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,8,8,9,10,10,10,10,10,10,12,12,12,21} - 1
[1,0,1,1,1,0,1,0,0,0]
 => [[],[[[],[]]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ? ∊ {4,4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,8,8,9,10,10,10,10,10,10,12,12,12,21} - 1
[1,0,1,1,1,1,0,0,0,0]
 => [[],[[[[]]]]]
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => ? ∊ {4,4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,8,8,9,10,10,10,10,10,10,12,12,12,21} - 1
[1,1,0,0,1,0,1,0,1,0]
 => [[[]],[],[],[]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ? ∊ {4,4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,8,8,9,10,10,10,10,10,10,12,12,12,21} - 1
[1,1,0,0,1,0,1,1,0,0]
 => [[[]],[],[[]]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ? ∊ {4,4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,8,8,9,10,10,10,10,10,10,12,12,12,21} - 1
[1,1,0,0,1,1,0,0,1,0]
 => [[[]],[[]],[]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ? ∊ {4,4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,8,8,9,10,10,10,10,10,10,12,12,12,21} - 1
[1,1,0,0,1,1,0,1,0,0]
 => [[[]],[[],[]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ? ∊ {4,4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,8,8,9,10,10,10,10,10,10,12,12,12,21} - 1
[1,1,0,0,1,1,1,0,0,0]
 => [[[]],[[[]]]]
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => ? ∊ {4,4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,8,8,9,10,10,10,10,10,10,12,12,12,21} - 1
[1,1,0,1,0,0,1,0,1,0]
 => [[[],[]],[],[]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6 = 7 - 1
[1,1,0,1,0,0,1,1,0,0]
 => [[[],[]],[[]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ? ∊ {4,4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,8,8,9,10,10,10,10,10,10,12,12,12,21} - 1
[1,1,0,1,0,1,0,0,1,0]
 => [[[],[],[]],[]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ? ∊ {4,4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,8,8,9,10,10,10,10,10,10,12,12,12,21} - 1
[1,1,0,1,0,1,0,1,0,0]
 => [[[],[],[],[]]]
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ? ∊ {4,4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,8,8,9,10,10,10,10,10,10,12,12,12,21} - 1
[1,1,0,1,0,1,1,0,0,0]
 => [[[],[],[[]]]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ? ∊ {4,4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,8,8,9,10,10,10,10,10,10,12,12,12,21} - 1
[1,1,0,1,1,0,0,0,1,0]
 => [[[],[[]]],[]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ? ∊ {4,4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,8,8,9,10,10,10,10,10,10,12,12,12,21} - 1
[1,1,0,1,1,0,0,1,0,0]
 => [[[],[[]],[]]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ? ∊ {4,4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,8,8,9,10,10,10,10,10,10,12,12,12,21} - 1
[1,1,0,1,1,0,1,0,0,0]
 => [[[],[[],[]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6 = 7 - 1
[1,1,0,1,1,1,0,0,0,0]
 => [[[],[[[]]]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ? ∊ {4,4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,8,8,9,10,10,10,10,10,10,12,12,12,21} - 1
[1,1,1,0,0,0,1,0,1,0]
 => [[[[]]],[],[]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ? ∊ {4,4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,8,8,9,10,10,10,10,10,10,12,12,12,21} - 1
[1,1,1,0,0,0,1,1,0,0]
 => [[[[]]],[[]]]
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => ? ∊ {4,4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,8,8,9,10,10,10,10,10,10,12,12,12,21} - 1
[1,1,1,0,0,1,0,0,1,0]
 => [[[[]],[]],[]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ? ∊ {4,4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,8,8,9,10,10,10,10,10,10,12,12,12,21} - 1
[1,1,1,0,0,1,0,1,0,0]
 => [[[[]],[],[]]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ? ∊ {4,4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,8,8,9,10,10,10,10,10,10,12,12,12,21} - 1
[1,1,1,0,0,1,1,0,0,0]
 => [[[[]],[[]]]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ? ∊ {4,4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,8,8,9,10,10,10,10,10,10,12,12,12,21} - 1
[1,1,1,0,1,0,0,0,1,0]
 => [[[[],[]]],[]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ? ∊ {4,4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,8,8,9,10,10,10,10,10,10,12,12,12,21} - 1
[1,1,1,0,1,0,0,1,0,0]
 => [[[[],[]],[]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6 = 7 - 1
[1,1,1,0,1,0,1,0,0,0]
 => [[[[],[],[]]]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ? ∊ {4,4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,8,8,9,10,10,10,10,10,10,12,12,12,21} - 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.
Matching statistic: St000454
Mapping
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 27%
Values
[1,0]
 => [2,1] => ([(0,1)],2)
 => 1 = 3 - 2
[1,0,1,0]
 => [3,1,2] => ([(0,2),(1,2)],3)
 => ? ∊ {3,6} - 2
[1,1,0,0]
 => [2,3,1] => ([(0,2),(1,2)],3)
 => ? ∊ {3,6} - 2
[1,0,1,0,1,0]
 => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => ? ∊ {3,5,5,5,10} - 2
[1,0,1,1,0,0]
 => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
 => ? ∊ {3,5,5,5,10} - 2
[1,1,0,0,1,0]
 => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => ? ∊ {3,5,5,5,10} - 2
[1,1,0,1,0,0]
 => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? ∊ {3,5,5,5,10} - 2
[1,1,1,0,0,0]
 => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => ? ∊ {3,5,5,5,10} - 2
[1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 4 - 2
[1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? ∊ {4,4,5,5,7,7,7,8,8,8,15} - 2
[1,0,1,1,0,0,1,0]
 => [3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? ∊ {4,4,5,5,7,7,7,8,8,8,15} - 2
[1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {4,4,5,5,7,7,7,8,8,8,15} - 2
[1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? ∊ {4,4,5,5,7,7,7,8,8,8,15} - 2
[1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? ∊ {4,4,5,5,7,7,7,8,8,8,15} - 2
[1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? ∊ {4,4,5,5,7,7,7,8,8,8,15} - 2
[1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {4,4,5,5,7,7,7,8,8,8,15} - 2
[1,1,0,1,0,1,0,0]
 => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 5 - 2
[1,1,0,1,1,0,0,0]
 => [4,3,1,5,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => ? ∊ {4,4,5,5,7,7,7,8,8,8,15} - 2
[1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? ∊ {4,4,5,5,7,7,7,8,8,8,15} - 2
[1,1,1,0,0,1,0,0]
 => [2,5,4,1,3] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => ? ∊ {4,4,5,5,7,7,7,8,8,8,15} - 2
[1,1,1,0,1,0,0,0]
 => [5,3,4,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ? ∊ {4,4,5,5,7,7,7,8,8,8,15} - 2
[1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 4 - 2
[1,0,1,0,1,0,1,0,1,0]
 => [6,1,2,3,4,5] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ? ∊ {5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,9,10,10,10,10,10,10,12,12,12,21} - 2
[1,0,1,0,1,0,1,1,0,0]
 => [5,1,2,3,6,4] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ? ∊ {5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,9,10,10,10,10,10,10,12,12,12,21} - 2
[1,0,1,0,1,1,0,0,1,0]
 => [4,1,2,6,3,5] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ? ∊ {5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,9,10,10,10,10,10,10,12,12,12,21} - 2
[1,0,1,0,1,1,0,1,0,0]
 => [6,1,2,5,3,4] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,9,10,10,10,10,10,10,12,12,12,21} - 2
[1,0,1,0,1,1,1,0,0,0]
 => [4,1,2,5,6,3] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 4 - 2
[1,0,1,1,0,0,1,0,1,0]
 => [3,1,6,2,4,5] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ? ∊ {5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,9,10,10,10,10,10,10,12,12,12,21} - 2
[1,0,1,1,0,0,1,1,0,0]
 => [3,1,5,2,6,4] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => ? ∊ {5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,9,10,10,10,10,10,10,12,12,12,21} - 2
[1,0,1,1,0,1,0,0,1,0]
 => [6,1,4,2,3,5] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,9,10,10,10,10,10,10,12,12,12,21} - 2
[1,0,1,1,0,1,0,1,0,0]
 => [6,1,5,2,3,4] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,9,10,10,10,10,10,10,12,12,12,21} - 2
[1,0,1,1,0,1,1,0,0,0]
 => [5,1,4,2,6,3] => ([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,9,10,10,10,10,10,10,12,12,12,21} - 2
[1,0,1,1,1,0,0,0,1,0]
 => [3,1,4,6,2,5] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ? ∊ {5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,9,10,10,10,10,10,10,12,12,12,21} - 2
[1,0,1,1,1,0,0,1,0,0]
 => [3,1,6,5,2,4] => ([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6)
 => ? ∊ {5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,9,10,10,10,10,10,10,12,12,12,21} - 2
[1,0,1,1,1,0,1,0,0,0]
 => [6,1,4,5,2,3] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ? ∊ {5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,9,10,10,10,10,10,10,12,12,12,21} - 2
[1,0,1,1,1,1,0,0,0,0]
 => [3,1,4,5,6,2] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ? ∊ {5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,9,10,10,10,10,10,10,12,12,12,21} - 2
[1,1,0,0,1,0,1,0,1,0]
 => [2,6,1,3,4,5] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ? ∊ {5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,9,10,10,10,10,10,10,12,12,12,21} - 2
[1,1,0,0,1,0,1,1,0,0]
 => [2,5,1,3,6,4] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ? ∊ {5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,9,10,10,10,10,10,10,12,12,12,21} - 2
[1,1,0,0,1,1,0,0,1,0]
 => [2,4,1,6,3,5] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => ? ∊ {5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,9,10,10,10,10,10,10,12,12,12,21} - 2
[1,1,0,0,1,1,0,1,0,0]
 => [2,6,1,5,3,4] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6)
 => ? ∊ {5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,9,10,10,10,10,10,10,12,12,12,21} - 2
[1,1,0,0,1,1,1,0,0,0]
 => [2,4,1,5,6,3] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ? ∊ {5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,9,10,10,10,10,10,10,12,12,12,21} - 2
[1,1,0,1,0,0,1,0,1,0]
 => [6,3,1,2,4,5] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,9,10,10,10,10,10,10,12,12,12,21} - 2
[1,1,0,1,0,0,1,1,0,0]
 => [5,3,1,2,6,4] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6)
 => ? ∊ {5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,9,10,10,10,10,10,10,12,12,12,21} - 2
[1,1,0,1,0,1,0,0,1,0]
 => [6,4,1,2,3,5] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,9,10,10,10,10,10,10,12,12,12,21} - 2
[1,1,0,1,0,1,0,1,0,0]
 => [5,6,1,2,3,4] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ? ∊ {5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,9,10,10,10,10,10,10,12,12,12,21} - 2
[1,1,0,1,0,1,1,0,0,0]
 => [5,4,1,2,6,3] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,9,10,10,10,10,10,10,12,12,12,21} - 2
[1,1,0,1,1,0,0,0,1,0]
 => [4,3,1,6,2,5] => ([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6)
 => ? ∊ {5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,9,10,10,10,10,10,10,12,12,12,21} - 2
[1,1,0,1,1,0,0,1,0,0]
 => [6,3,1,5,2,4] => ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ? ∊ {5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,9,10,10,10,10,10,10,12,12,12,21} - 2
[1,1,0,1,1,0,1,0,0,0]
 => [6,4,1,5,2,3] => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,9,10,10,10,10,10,10,12,12,12,21} - 2
[1,1,0,1,1,1,0,0,0,0]
 => [4,3,1,5,6,2] => ([(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => ? ∊ {5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,9,10,10,10,10,10,10,12,12,12,21} - 2
[1,1,1,0,0,0,1,0,1,0]
 => [2,3,6,1,4,5] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 4 - 2
[1,1,1,0,0,0,1,1,0,0]
 => [2,3,5,1,6,4] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ? ∊ {5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,9,10,10,10,10,10,10,12,12,12,21} - 2
[1,1,1,0,0,1,0,0,1,0]
 => [2,6,4,1,3,5] => ([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,9,10,10,10,10,10,10,12,12,12,21} - 2
[1,1,1,0,0,1,0,1,0,0]
 => [2,6,5,1,3,4] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,9,10,10,10,10,10,10,12,12,12,21} - 2
[1,1,1,0,0,1,1,0,0,0]
 => [2,5,4,1,6,3] => ([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ? ∊ {5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,9,10,10,10,10,10,10,12,12,12,21} - 2
[1,1,1,0,1,0,0,0,1,0]
 => [6,3,4,1,2,5] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ? ∊ {5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,9,10,10,10,10,10,10,12,12,12,21} - 2
Description
The largest eigenvalue of a graph if it is integral. If a graph is $d$-regular, then its largest eigenvalue equals $d$. One can show that the largest eigenvalue always lies between the average degree and the maximal degree. This statistic is undefined if the largest eigenvalue of the graph is not integral.