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Your data matches 4 different statistics following compositions of up to 3 maps.
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Matching statistic: St000260
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(load all 4 compositions to match this statistic)
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00064: Permutations —reverse⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [1] => ([],1)
=> 0
{{1},{2}}
=> [1,2] => [2,1] => ([(0,1)],2)
=> 1
{{1,2},{3}}
=> [2,1,3] => [3,1,2] => ([(0,2),(1,2)],3)
=> 1
{{1},{2,3}}
=> [1,3,2] => [2,3,1] => ([(0,2),(1,2)],3)
=> 1
{{1},{2},{3}}
=> [1,2,3] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
{{1,2,3},{4}}
=> [2,3,1,4] => [4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
{{1,2},{3,4}}
=> [2,1,4,3] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
{{1,2},{3},{4}}
=> [2,1,3,4] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
{{1,3},{2},{4}}
=> [3,2,1,4] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 1
{{1},{2,3,4}}
=> [1,3,4,2] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
{{1},{2,3},{4}}
=> [1,3,2,4] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
{{1},{2,4},{3}}
=> [1,4,3,2] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
{{1},{2},{3,4}}
=> [1,2,4,3] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
{{1},{2},{3},{4}}
=> [1,2,3,4] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [5,1,4,3,2] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [4,5,1,3,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [5,4,1,3,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [3,1,5,4,2] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 2
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [5,1,3,4,2] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [3,5,4,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [5,3,4,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 1
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [4,5,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 1
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [5,4,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [5,1,4,2,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [2,5,1,4,3] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 2
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [5,2,1,4,3] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 1
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
{{1},{2,3,4,5}}
=> [1,3,4,5,2] => [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
{{1},{2,3,4},{5}}
=> [1,3,4,2,5] => [5,2,4,3,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
{{1},{2,3,5},{4}}
=> [1,3,5,4,2] => [2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
{{1},{2,3},{4,5}}
=> [1,3,2,5,4] => [4,5,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 1
{{1},{2,3},{4},{5}}
=> [1,3,2,4,5] => [5,4,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
{{1,4},{2},{3,5}}
=> [4,2,5,1,3] => [3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2
{{1,4},{2},{3},{5}}
=> [4,2,3,1,5] => [5,1,3,2,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
{{1},{2,4,5},{3}}
=> [1,4,3,5,2] => [2,5,3,4,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
{{1},{2,4},{3,5}}
=> [1,4,5,2,3] => [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 1
{{1},{2,4},{3},{5}}
=> [1,4,3,2,5] => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
{{1},{2,5},{3,4}}
=> [1,5,4,3,2] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
{{1},{2},{3,4,5}}
=> [1,2,4,5,3] => [3,5,4,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
{{1},{2},{3,4},{5}}
=> [1,2,4,3,5] => [5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
{{1},{2,5},{3},{4}}
=> [1,5,3,4,2] => [2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
{{1},{2},{3,5},{4}}
=> [1,2,5,4,3] => [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
{{1},{2},{3},{4,5}}
=> [1,2,3,5,4] => [4,5,3,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
{{1},{2},{3},{4},{5}}
=> [1,2,3,4,5] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
{{1,2,3,4,5},{6}}
=> [2,3,4,5,1,6] => [6,1,5,4,3,2] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
{{1,2,3,4},{5,6}}
=> [2,3,4,1,6,5] => [5,6,1,4,3,2] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> 2
Description
The radius of a connected graph.
This is the minimum eccentricity of any vertex.
Matching statistic: St000455
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 19% ●values known / values provided: 19%●distinct values known / distinct values provided: 25%
Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 19% ●values known / values provided: 19%●distinct values known / distinct values provided: 25%
Values
{{1}}
=> [1] => [1] => ([],1)
=> ? = 0 - 1
{{1},{2}}
=> [1,1] => [2] => ([],2)
=> ? = 1 - 1
{{1,2},{3}}
=> [2,1] => [1,2] => ([(1,2)],3)
=> 0 = 1 - 1
{{1},{2,3}}
=> [1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 0 = 1 - 1
{{1},{2},{3}}
=> [1,1,1] => [3] => ([],3)
=> ? = 1 - 1
{{1,2,3},{4}}
=> [3,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
{{1,2},{3,4}}
=> [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2 - 1
{{1,2},{3},{4}}
=> [2,1,1] => [1,3] => ([(2,3)],4)
=> 0 = 1 - 1
{{1,3},{2},{4}}
=> [2,1,1] => [1,3] => ([(2,3)],4)
=> 0 = 1 - 1
{{1},{2,3,4}}
=> [1,3] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
{{1},{2,3},{4}}
=> [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
=> 0 = 1 - 1
{{1},{2,4},{3}}
=> [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
=> 0 = 1 - 1
{{1},{2},{3,4}}
=> [1,1,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0 = 1 - 1
{{1},{2},{3},{4}}
=> [1,1,1,1] => [4] => ([],4)
=> ? = 1 - 1
{{1,2,3,4},{5}}
=> [4,1] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
{{1,2,3},{4,5}}
=> [3,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
{{1,2,3},{4},{5}}
=> [3,1,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
{{1,2,4},{3,5}}
=> [3,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
{{1,2,4},{3},{5}}
=> [3,1,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
{{1,2},{3,4,5}}
=> [2,3] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
{{1,2},{3,4},{5}}
=> [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
{{1,2},{3,5},{4}}
=> [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
{{1,2},{3},{4,5}}
=> [2,1,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
{{1,2},{3},{4},{5}}
=> [2,1,1,1] => [1,4] => ([(3,4)],5)
=> 0 = 1 - 1
{{1,3,4},{2},{5}}
=> [3,1,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
{{1,3},{2,4,5}}
=> [2,3] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
{{1,3},{2,4},{5}}
=> [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
{{1,3},{2,5},{4}}
=> [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
{{1,3},{2},{4,5}}
=> [2,1,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
{{1,3},{2},{4},{5}}
=> [2,1,1,1] => [1,4] => ([(3,4)],5)
=> 0 = 1 - 1
{{1,4},{2,3},{5}}
=> [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
{{1},{2,3,4,5}}
=> [1,4] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
{{1},{2,3,4},{5}}
=> [1,3,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
{{1},{2,3,5},{4}}
=> [1,3,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
{{1},{2,3},{4,5}}
=> [1,2,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
{{1},{2,3},{4},{5}}
=> [1,2,1,1] => [2,3] => ([(2,4),(3,4)],5)
=> 0 = 1 - 1
{{1,4},{2},{3,5}}
=> [2,1,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
{{1,4},{2},{3},{5}}
=> [2,1,1,1] => [1,4] => ([(3,4)],5)
=> 0 = 1 - 1
{{1},{2,4,5},{3}}
=> [1,3,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
{{1},{2,4},{3,5}}
=> [1,2,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
{{1},{2,4},{3},{5}}
=> [1,2,1,1] => [2,3] => ([(2,4),(3,4)],5)
=> 0 = 1 - 1
{{1},{2,5},{3,4}}
=> [1,2,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
{{1},{2},{3,4,5}}
=> [1,1,3] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
{{1},{2},{3,4},{5}}
=> [1,1,2,1] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
{{1},{2,5},{3},{4}}
=> [1,2,1,1] => [2,3] => ([(2,4),(3,4)],5)
=> 0 = 1 - 1
{{1},{2},{3,5},{4}}
=> [1,1,2,1] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
{{1},{2},{3},{4,5}}
=> [1,1,1,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
{{1},{2},{3},{4},{5}}
=> [1,1,1,1,1] => [5] => ([],5)
=> ? = 1 - 1
{{1,2,3,4,5},{6}}
=> [5,1] => [1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 1 - 1
{{1,2,3,4},{5,6}}
=> [4,2] => [1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
{{1,2,3,4},{5},{6}}
=> [4,1,1] => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 1 - 1
{{1,2,3,5},{4,6}}
=> [4,2] => [1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
{{1,2,3,5},{4},{6}}
=> [4,1,1] => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 1 - 1
{{1,2,3},{4,5,6}}
=> [3,3] => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
{{1,2,3},{4,5},{6}}
=> [3,2,1] => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 - 1
{{1,2,3},{4,6},{5}}
=> [3,2,1] => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
{{1,2,3},{4},{5,6}}
=> [3,1,2] => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 - 1
{{1,2,3},{4},{5},{6}}
=> [3,1,1,1] => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> 0 = 1 - 1
{{1,2,4,5},{3,6}}
=> [4,2] => [1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
{{1,2,4,5},{3},{6}}
=> [4,1,1] => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 1 - 1
{{1,2,4},{3,5,6}}
=> [3,3] => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
{{1,2,4},{3,5},{6}}
=> [3,2,1] => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 - 1
{{1,2,4},{3,6},{5}}
=> [3,2,1] => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
{{1,2,4},{3},{5,6}}
=> [3,1,2] => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
{{1,2,4},{3},{5},{6}}
=> [3,1,1,1] => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> 0 = 1 - 1
{{1,2,5},{3,4,6}}
=> [3,3] => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
{{1,2,5},{3,4},{6}}
=> [3,2,1] => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 - 1
{{1,2},{3,4,5,6}}
=> [2,4] => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
{{1,2},{3,4,5},{6}}
=> [2,3,1] => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 - 1
{{1,2},{3,4,6},{5}}
=> [2,3,1] => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
{{1,2},{3,4},{5,6}}
=> [2,2,2] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
{{1,2},{3,4},{5},{6}}
=> [2,2,1,1] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 - 1
{{1,2,5},{3,6},{4}}
=> [3,2,1] => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
{{1,2,5},{3},{4,6}}
=> [3,1,2] => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
{{1,2,5},{3},{4},{6}}
=> [3,1,1,1] => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> 0 = 1 - 1
{{1,2},{3,5,6},{4}}
=> [2,3,1] => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
{{1,2},{3,5},{4,6}}
=> [2,2,2] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
{{1,2},{3,5},{4},{6}}
=> [2,2,1,1] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 - 1
{{1,2},{3,6},{4,5}}
=> [2,2,2] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
{{1,2},{3},{4,5,6}}
=> [2,1,3] => [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 - 1
{{1,2},{3},{4,5},{6}}
=> [2,1,2,1] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 - 1
{{1,2},{3,6},{4},{5}}
=> [2,2,1,1] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
{{1,2},{3},{4,6},{5}}
=> [2,1,2,1] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 - 1
{{1,2},{3},{4},{5,6}}
=> [2,1,1,2] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 - 1
{{1,2},{3},{4},{5},{6}}
=> [2,1,1,1,1] => [1,5] => ([(4,5)],6)
=> 0 = 1 - 1
{{1,3,4,5},{2},{6}}
=> [4,1,1] => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 1 - 1
{{1,3,4},{2},{5},{6}}
=> [3,1,1,1] => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> 0 = 1 - 1
{{1,3,5},{2},{4},{6}}
=> [3,1,1,1] => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> 0 = 1 - 1
{{1,3},{2},{4},{5},{6}}
=> [2,1,1,1,1] => [1,5] => ([(4,5)],6)
=> 0 = 1 - 1
{{1},{2,3,4,5,6}}
=> [1,5] => [2,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 1 - 1
{{1},{2,3,4,5},{6}}
=> [1,4,1] => [2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 1 - 1
{{1},{2,3,4,6},{5}}
=> [1,4,1] => [2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 1 - 1
{{1},{2,3,4},{5},{6}}
=> [1,3,1,1] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 1 - 1
{{1},{2,3,5,6},{4}}
=> [1,4,1] => [2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 1 - 1
{{1},{2,3,5},{4},{6}}
=> [1,3,1,1] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 1 - 1
{{1},{2,3,6},{4},{5}}
=> [1,3,1,1] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 1 - 1
{{1},{2,3},{4},{5},{6}}
=> [1,2,1,1,1] => [2,4] => ([(3,5),(4,5)],6)
=> 0 = 1 - 1
{{1,4,5},{2},{3},{6}}
=> [3,1,1,1] => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> 0 = 1 - 1
{{1,4},{2},{3},{5},{6}}
=> [2,1,1,1,1] => [1,5] => ([(4,5)],6)
=> 0 = 1 - 1
{{1},{2,4,5,6},{3}}
=> [1,4,1] => [2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 1 - 1
Description
The second largest eigenvalue of a graph if it is integral.
This statistic is undefined if the second largest eigenvalue of the graph is not integral.
Chapter 4 of [1] provides lots of context.
Matching statistic: St001632
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001632: Posets ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 25%
Mp00064: Permutations —reverse⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001632: Posets ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 25%
Values
{{1}}
=> [1] => [1] => ([],1)
=> ? = 0 - 2
{{1},{2}}
=> [1,2] => [2,1] => ([],2)
=> ? = 1 - 2
{{1,2},{3}}
=> [2,1,3] => [3,1,2] => ([(1,2)],3)
=> ? = 1 - 2
{{1},{2,3}}
=> [1,3,2] => [2,3,1] => ([(1,2)],3)
=> ? = 1 - 2
{{1},{2},{3}}
=> [1,2,3] => [3,2,1] => ([],3)
=> ? = 1 - 2
{{1,2,3},{4}}
=> [2,3,1,4] => [4,1,3,2] => ([(1,2),(1,3)],4)
=> ? = 1 - 2
{{1,2},{3,4}}
=> [2,1,4,3] => [3,4,1,2] => ([(0,3),(1,2)],4)
=> ? = 2 - 2
{{1,2},{3},{4}}
=> [2,1,3,4] => [4,3,1,2] => ([(2,3)],4)
=> ? = 1 - 2
{{1,3},{2},{4}}
=> [3,2,1,4] => [4,1,2,3] => ([(1,2),(2,3)],4)
=> ? = 1 - 2
{{1},{2,3,4}}
=> [1,3,4,2] => [2,4,3,1] => ([(1,2),(1,3)],4)
=> ? = 1 - 2
{{1},{2,3},{4}}
=> [1,3,2,4] => [4,2,3,1] => ([(2,3)],4)
=> ? = 1 - 2
{{1},{2,4},{3}}
=> [1,4,3,2] => [2,3,4,1] => ([(1,2),(2,3)],4)
=> ? = 1 - 2
{{1},{2},{3,4}}
=> [1,2,4,3] => [3,4,2,1] => ([(2,3)],4)
=> ? = 1 - 2
{{1},{2},{3},{4}}
=> [1,2,3,4] => [4,3,2,1] => ([],4)
=> ? = 1 - 2
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [5,1,4,3,2] => ([(1,2),(1,3),(1,4)],5)
=> ? = 1 - 2
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [4,5,1,3,2] => ([(0,4),(1,2),(1,3)],5)
=> ? = 2 - 2
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [5,4,1,3,2] => ([(2,3),(2,4)],5)
=> ? = 1 - 2
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [3,1,5,4,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> 0 = 2 - 2
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [5,1,3,4,2] => ([(1,3),(1,4),(4,2)],5)
=> ? = 1 - 2
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [3,5,4,1,2] => ([(0,4),(1,2),(1,3)],5)
=> ? = 2 - 2
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [5,3,4,1,2] => ([(1,4),(2,3)],5)
=> ? = 1 - 2
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [3,4,5,1,2] => ([(0,3),(1,4),(4,2)],5)
=> ? = 2 - 2
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [4,5,3,1,2] => ([(1,4),(2,3)],5)
=> ? = 1 - 2
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [5,4,3,1,2] => ([(3,4)],5)
=> ? = 1 - 2
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [5,1,4,2,3] => ([(1,3),(1,4),(4,2)],5)
=> ? = 1 - 2
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [2,5,1,4,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> 0 = 2 - 2
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [5,2,1,4,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ? = 1 - 2
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [2,4,1,5,3] => ([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> 0 = 2 - 2
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [4,5,1,2,3] => ([(0,3),(1,4),(4,2)],5)
=> ? = 2 - 2
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [5,4,1,2,3] => ([(2,3),(3,4)],5)
=> ? = 1 - 2
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5)
=> ? = 1 - 2
{{1},{2,3,4,5}}
=> [1,3,4,5,2] => [2,5,4,3,1] => ([(1,2),(1,3),(1,4)],5)
=> ? = 1 - 2
{{1},{2,3,4},{5}}
=> [1,3,4,2,5] => [5,2,4,3,1] => ([(2,3),(2,4)],5)
=> ? = 1 - 2
{{1},{2,3,5},{4}}
=> [1,3,5,4,2] => [2,4,5,3,1] => ([(1,3),(1,4),(4,2)],5)
=> ? = 1 - 2
{{1},{2,3},{4,5}}
=> [1,3,2,5,4] => [4,5,2,3,1] => ([(1,4),(2,3)],5)
=> ? = 1 - 2
{{1},{2,3},{4},{5}}
=> [1,3,2,4,5] => [5,4,2,3,1] => ([(3,4)],5)
=> ? = 1 - 2
{{1,4},{2},{3,5}}
=> [4,2,5,1,3] => [3,1,5,2,4] => ([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> 0 = 2 - 2
{{1,4},{2},{3},{5}}
=> [4,2,3,1,5] => [5,1,3,2,4] => ([(1,2),(1,3),(2,4),(3,4)],5)
=> ? = 1 - 2
{{1},{2,4,5},{3}}
=> [1,4,3,5,2] => [2,5,3,4,1] => ([(1,3),(1,4),(4,2)],5)
=> ? = 1 - 2
{{1},{2,4},{3,5}}
=> [1,4,5,2,3] => [3,2,5,4,1] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ? = 1 - 2
{{1},{2,4},{3},{5}}
=> [1,4,3,2,5] => [5,2,3,4,1] => ([(2,3),(3,4)],5)
=> ? = 1 - 2
{{1},{2,5},{3,4}}
=> [1,5,4,3,2] => [2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
=> ? = 1 - 2
{{1},{2},{3,4,5}}
=> [1,2,4,5,3] => [3,5,4,2,1] => ([(2,3),(2,4)],5)
=> ? = 1 - 2
{{1},{2},{3,4},{5}}
=> [1,2,4,3,5] => [5,3,4,2,1] => ([(3,4)],5)
=> ? = 1 - 2
{{1},{2,5},{3},{4}}
=> [1,5,3,4,2] => [2,4,3,5,1] => ([(1,2),(1,3),(2,4),(3,4)],5)
=> ? = 1 - 2
{{1},{2},{3,5},{4}}
=> [1,2,5,4,3] => [3,4,5,2,1] => ([(2,3),(3,4)],5)
=> ? = 1 - 2
{{1},{2},{3},{4,5}}
=> [1,2,3,5,4] => [4,5,3,2,1] => ([(3,4)],5)
=> ? = 1 - 2
{{1},{2},{3},{4},{5}}
=> [1,2,3,4,5] => [5,4,3,2,1] => ([],5)
=> ? = 1 - 2
{{1,2,3,4,5},{6}}
=> [2,3,4,5,1,6] => [6,1,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5)],6)
=> ? = 1 - 2
{{1,2,3,4},{5,6}}
=> [2,3,4,1,6,5] => [5,6,1,4,3,2] => ([(0,5),(1,2),(1,3),(1,4)],6)
=> ? = 2 - 2
{{1,2,3,4},{5},{6}}
=> [2,3,4,1,5,6] => [6,5,1,4,3,2] => ([(2,3),(2,4),(2,5)],6)
=> ? = 1 - 2
{{1,2,3,5},{4,6}}
=> [2,3,5,6,1,4] => [4,1,6,5,3,2] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5)],6)
=> 0 = 2 - 2
{{1,2,3,5},{4},{6}}
=> [2,3,5,4,1,6] => [6,1,4,5,3,2] => ([(1,3),(1,4),(1,5),(5,2)],6)
=> ? = 1 - 2
{{1,2,3},{4,5,6}}
=> [2,3,1,5,6,4] => [4,6,5,1,3,2] => ([(0,4),(0,5),(1,2),(1,3)],6)
=> ? = 2 - 2
{{1,2,3},{4,5},{6}}
=> [2,3,1,5,4,6] => [6,4,5,1,3,2] => ([(1,5),(2,3),(2,4)],6)
=> ? = 1 - 2
{{1,2,4,5},{3,6}}
=> [2,4,6,5,1,3] => [3,1,5,6,4,2] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(5,2)],6)
=> 0 = 2 - 2
{{1,2,4},{3,5,6}}
=> [2,4,5,1,6,3] => [3,6,1,5,4,2] => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5)],6)
=> 0 = 2 - 2
{{1,2,4},{3,6},{5}}
=> [2,4,6,1,5,3] => [3,5,1,6,4,2] => ([(0,3),(0,4),(1,2),(1,4),(1,5),(3,5)],6)
=> 0 = 2 - 2
{{1,2,5},{3,4,6}}
=> [2,5,4,6,1,3] => [3,1,6,4,5,2] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(5,2)],6)
=> 0 = 2 - 2
{{1,2,5},{3,6},{4}}
=> [2,5,6,4,1,3] => [3,1,4,6,5,2] => ([(0,5),(1,2),(1,5),(5,3),(5,4)],6)
=> 0 = 2 - 2
{{1,2,5},{3},{4,6}}
=> [2,5,3,6,1,4] => [4,1,6,3,5,2] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(3,5)],6)
=> 0 = 2 - 2
{{1,3,4},{2,5,6}}
=> [3,5,4,1,6,2] => [2,6,1,4,5,3] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(5,2)],6)
=> 0 = 2 - 2
{{1,3,4},{2,6},{5}}
=> [3,6,4,1,5,2] => [2,5,1,4,6,3] => ([(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(5,3)],6)
=> 0 = 2 - 2
{{1,3},{2,4,5,6}}
=> [3,4,1,5,6,2] => [2,6,5,1,4,3] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5)],6)
=> 0 = 2 - 2
{{1,3},{2,4,6},{5}}
=> [3,4,1,6,5,2] => [2,5,6,1,4,3] => ([(0,4),(0,5),(1,2),(1,4),(1,5),(2,3)],6)
=> 0 = 2 - 2
{{1,3,5},{2},{4,6}}
=> [3,2,5,6,1,4] => [4,1,6,5,2,3] => ([(0,4),(0,5),(1,2),(1,4),(1,5),(2,3)],6)
=> 0 = 2 - 2
{{1,3},{2,5,6},{4}}
=> [3,5,1,4,6,2] => [2,6,4,1,5,3] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(3,5)],6)
=> 0 = 2 - 2
{{1,3},{2,5},{4,6}}
=> [3,5,1,6,2,4] => [4,2,6,1,5,3] => ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5)],6)
=> 0 = 2 - 2
{{1,3},{2,6},{4,5}}
=> [3,6,1,5,4,2] => [2,4,5,1,6,3] => ([(0,4),(0,5),(1,2),(1,4),(2,3),(3,5)],6)
=> 0 = 2 - 2
{{1,3},{2,6},{4},{5}}
=> [3,6,1,4,5,2] => [2,5,4,1,6,3] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(2,5),(3,5)],6)
=> 0 = 2 - 2
{{1,4},{2,3,5,6}}
=> [4,3,5,1,6,2] => [2,6,1,5,3,4] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(5,2)],6)
=> 0 = 2 - 2
{{1,4},{2,3,6},{5}}
=> [4,3,6,1,5,2] => [2,5,1,6,3,4] => ([(0,4),(0,5),(1,2),(1,4),(2,5),(4,3)],6)
=> 0 = 2 - 2
{{1,5},{2,3},{4,6}}
=> [5,3,2,6,1,4] => [4,1,6,2,3,5] => ([(0,4),(0,5),(1,2),(1,4),(2,3),(3,5)],6)
=> 0 = 2 - 2
{{1,4,5},{2},{3,6}}
=> [4,2,6,5,1,3] => [3,1,5,6,2,4] => ([(0,4),(0,5),(1,2),(1,4),(2,5),(4,3)],6)
=> 0 = 2 - 2
{{1,4},{2,5,6},{3}}
=> [4,5,3,1,6,2] => [2,6,1,3,5,4] => ([(0,5),(1,2),(1,5),(5,3),(5,4)],6)
=> 0 = 2 - 2
{{1,4},{2},{3,5,6}}
=> [4,2,5,1,6,3] => [3,6,1,5,2,4] => ([(0,3),(0,4),(1,2),(1,4),(1,5),(3,5)],6)
=> 0 = 2 - 2
{{1,4},{2,6},{3},{5}}
=> [4,6,3,1,5,2] => [2,5,1,3,6,4] => ([(0,4),(1,2),(1,4),(2,5),(4,3),(4,5)],6)
=> 0 = 2 - 2
{{1,4},{2},{3,6},{5}}
=> [4,2,6,1,5,3] => [3,5,1,6,2,4] => ([(0,3),(0,5),(1,2),(1,4),(2,5),(3,4)],6)
=> 0 = 2 - 2
{{1,5},{2},{3,4,6}}
=> [5,2,4,6,1,3] => [3,1,6,4,2,5] => ([(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(5,3)],6)
=> 0 = 2 - 2
{{1,5},{2},{3,6},{4}}
=> [5,2,6,4,1,3] => [3,1,4,6,2,5] => ([(0,4),(1,2),(1,4),(2,5),(4,3),(4,5)],6)
=> 0 = 2 - 2
{{1,5},{2},{3},{4,6}}
=> [5,2,3,6,1,4] => [4,1,6,3,2,5] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(2,5),(3,5)],6)
=> 0 = 2 - 2
Description
The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset.
Matching statistic: St001876
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
Mp00196: Lattices —The modular quotient of a lattice.⟶ Lattices
St001876: Lattices ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 25%
Mp00208: Permutations —lattice of intervals⟶ Lattices
Mp00196: Lattices —The modular quotient of a lattice.⟶ Lattices
St001876: Lattices ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 25%
Values
{{1}}
=> [1] => ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 0
{{1},{2}}
=> [1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
{{1,2},{3}}
=> [2,1,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> ? = 1
{{1},{2,3}}
=> [1,3,2] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> ? = 1
{{1},{2},{3}}
=> [1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
{{1,2,3},{4}}
=> [2,3,1,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> ([(0,1)],2)
=> ? = 1
{{1,2},{3,4}}
=> [2,1,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ([],1)
=> ? = 2
{{1,2},{3},{4}}
=> [2,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> ([(0,1)],2)
=> ? = 1
{{1,3},{2},{4}}
=> [3,2,1,4] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> ([(0,1)],2)
=> ? = 1
{{1},{2,3,4}}
=> [1,3,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> ([(0,1)],2)
=> ? = 1
{{1},{2,3},{4}}
=> [1,3,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,7),(4,6),(5,6),(5,7),(6,8),(7,8)],9)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
{{1},{2,4},{3}}
=> [1,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> ([(0,1)],2)
=> ? = 1
{{1},{2},{3,4}}
=> [1,2,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> ([(0,1)],2)
=> ? = 1
{{1},{2},{3},{4}}
=> [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
=> ([(0,1)],2)
=> ? = 1
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(8,9)],10)
=> ([],1)
=> ? = 2
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,7),(4,7),(5,6),(5,9),(6,10),(7,8),(8,9),(9,10)],11)
=> ([(0,1)],2)
=> ? = 1
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> ([],1)
=> ? = 2
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,9),(5,7),(6,9),(8,7),(9,8)],10)
=> ([(0,1)],2)
=> ? = 1
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(8,9)],10)
=> ([],1)
=> ? = 2
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(7,9),(8,10),(9,10)],11)
=> ([(0,1)],2)
=> ? = 1
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,6),(4,6),(5,7),(5,8),(6,10),(7,9),(8,9),(9,10)],11)
=> ([],1)
=> ? = 2
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(1,9),(2,7),(3,7),(4,6),(5,6),(6,9),(7,8),(8,10),(9,10)],11)
=> ([],1)
=> ? = 1
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,11),(3,10),(4,9),(4,12),(5,10),(5,12),(7,6),(8,6),(9,7),(10,8),(11,9),(12,7),(12,8)],13)
=> ([(0,1)],2)
=> ? = 1
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,9),(5,7),(6,9),(8,7),(9,8)],10)
=> ([(0,1)],2)
=> ? = 1
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> ([],1)
=> ? = 2
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,7),(4,6),(5,6),(6,9),(7,9),(9,8)],10)
=> ([(0,1)],2)
=> ? = 1
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 2
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,6),(4,6),(5,7),(5,8),(6,10),(7,9),(8,9),(9,10)],11)
=> ([],1)
=> ? = 2
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,9),(3,11),(4,9),(4,10),(5,8),(5,11),(7,8),(8,6),(9,7),(10,7),(11,6)],12)
=> ([(0,1)],2)
=> ? = 1
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ([(0,1)],2)
=> ? = 1
{{1},{2,3,4,5}}
=> [1,3,4,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
=> ([(0,1)],2)
=> ? = 1
{{1},{2,3,4},{5}}
=> [1,3,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
{{1},{2,3,5},{4}}
=> [1,3,5,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,9),(5,7),(6,9),(8,7),(9,8)],10)
=> ([(0,1)],2)
=> ? = 1
{{1},{2,3},{4,5}}
=> [1,3,2,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(7,9),(8,10),(9,10)],11)
=> ([(0,1)],2)
=> ? = 1
{{1},{2,3},{4},{5}}
=> [1,3,2,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,12),(3,12),(4,9),(5,10),(5,11),(7,6),(8,6),(9,8),(10,7),(11,7),(11,8),(12,9),(12,11)],13)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
{{1,4},{2},{3,5}}
=> [4,2,5,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 2
{{1,4},{2},{3},{5}}
=> [4,2,3,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,9),(4,8),(5,7),(6,8),(6,9),(8,10),(9,10),(10,7)],11)
=> ([(0,1)],2)
=> ? = 1
{{1},{2,4,5},{3}}
=> [1,4,3,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,9),(5,7),(6,9),(8,7),(9,8)],10)
=> ([(0,1)],2)
=> ? = 1
{{1},{2,4},{3,5}}
=> [1,4,5,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,7),(4,6),(5,6),(6,9),(7,9),(9,8)],10)
=> ([(0,1)],2)
=> ? = 1
{{1},{2,4},{3},{5}}
=> [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,10),(4,9),(5,9),(5,10),(7,6),(8,6),(9,11),(10,11),(11,7),(11,8)],12)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
{{1},{2,5},{3,4}}
=> [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ([(0,1)],2)
=> ? = 1
{{1},{2},{3,4,5}}
=> [1,2,4,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,7),(4,7),(5,6),(5,9),(6,10),(7,8),(8,9),(9,10)],11)
=> ([(0,1)],2)
=> ? = 1
{{1},{2},{3,4},{5}}
=> [1,2,4,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,12),(3,12),(4,9),(5,10),(5,11),(7,6),(8,6),(9,8),(10,7),(11,7),(11,8),(12,9),(12,11)],13)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
{{1},{2,5},{3},{4}}
=> [1,5,3,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,9),(4,8),(5,7),(6,8),(6,9),(8,10),(9,10),(10,7)],11)
=> ([(0,1)],2)
=> ? = 1
{{1},{2},{3,5},{4}}
=> [1,2,5,4,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,9),(3,11),(4,9),(4,10),(5,8),(5,11),(7,8),(8,6),(9,7),(10,7),(11,6)],12)
=> ([(0,1)],2)
=> ? = 1
{{1},{2},{3},{4,5}}
=> [1,2,3,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,11),(3,10),(4,9),(4,12),(5,10),(5,12),(7,6),(8,6),(9,7),(10,8),(11,9),(12,7),(12,8)],13)
=> ([(0,1)],2)
=> ? = 1
{{1},{2},{3},{4},{5}}
=> [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
{{1,2,3,4,5},{6}}
=> [2,3,4,5,1,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,13),(3,12),(4,7),(5,12),(5,14),(6,13),(6,14),(8,11),(9,8),(10,8),(11,7),(12,9),(13,10),(14,9),(14,10)],15)
=> ([(0,1)],2)
=> ? = 1
{{1,2,3,4},{5,6}}
=> [2,3,4,1,6,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,8),(4,7),(5,7),(6,8),(6,9),(7,12),(8,11),(9,11),(10,12),(11,10)],13)
=> ([],1)
=> ? = 2
{{1,2,3,4},{5},{6}}
=> [2,3,4,1,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,12),(2,11),(3,13),(4,10),(5,11),(5,12),(6,9),(6,13),(8,10),(9,7),(10,9),(11,8),(12,8),(13,7)],14)
=> ([(0,1)],2)
=> ? = 1
{{1,2,3,5},{4,6}}
=> [2,3,5,6,1,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,8),(4,8),(5,7),(6,7),(7,9),(8,9)],10)
=> ([],1)
=> ? = 2
{{1,2,3,5},{4},{6}}
=> [2,3,5,4,1,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,7),(4,9),(5,11),(6,7),(6,10),(7,12),(8,10),(10,12),(11,9),(12,11)],13)
=> ([(0,1)],2)
=> ? = 1
{{1,2,3},{4,5,6}}
=> [2,3,1,5,6,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,7),(4,7),(5,8),(6,8),(7,9),(8,10),(9,11),(10,11)],12)
=> ([],1)
=> ? = 2
{{1,2,3},{4,5},{6}}
=> [2,3,1,5,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,10),(3,7),(4,7),(5,8),(6,8),(7,10),(8,9),(8,11),(9,12),(10,11),(11,12)],13)
=> ([(0,1)],2)
=> ? = 1
{{1,2,3},{4,6},{5}}
=> [2,3,1,6,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,8),(3,7),(4,9),(5,9),(6,7),(6,8),(7,11),(8,11),(9,10),(10,12),(11,12)],13)
=> ([],1)
=> ? = 2
{{1,2,3},{4},{5,6}}
=> [2,3,1,4,6,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,8),(4,8),(5,11),(6,9),(6,10),(7,10),(8,11),(9,12),(10,12),(11,9)],13)
=> ([],1)
=> ? = 1
{{1,2,3},{4},{5},{6}}
=> [2,3,1,4,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,12),(2,13),(3,13),(4,11),(5,12),(5,14),(6,10),(6,14),(8,7),(9,7),(10,9),(11,10),(12,8),(13,11),(14,8),(14,9)],15)
=> ([(0,1)],2)
=> ? = 1
{{1,2,4,5},{3,6}}
=> [2,4,6,5,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,9),(3,9),(4,9),(5,7),(6,7),(7,8),(8,9)],10)
=> ([],1)
=> ? = 2
{{1},{2,3,4,5},{6}}
=> [1,3,4,5,2,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,12),(4,11),(5,13),(6,11),(6,12),(8,13),(9,7),(10,7),(11,8),(12,8),(13,9),(13,10)],14)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
{{1},{2,3,4},{5},{6}}
=> [1,3,4,2,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,14),(3,12),(4,12),(5,10),(6,11),(6,13),(8,7),(9,7),(10,9),(11,8),(12,14),(13,8),(13,9),(14,10),(14,13)],15)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
{{1},{2,3,5},{4},{6}}
=> [1,3,5,4,2,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,9),(4,8),(5,11),(6,10),(7,10),(8,12),(9,12),(10,11),(11,8),(11,9)],13)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
{{1},{2,3},{4,5},{6}}
=> [1,3,2,5,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,10),(3,13),(4,13),(5,14),(6,14),(8,7),(9,7),(10,8),(11,9),(12,8),(12,9),(13,10),(13,12),(14,11),(14,12)],15)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
{{1},{2,3},{4},{5},{6}}
=> [1,3,2,4,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(2,17),(3,17),(4,12),(5,15),(5,16),(6,13),(6,16),(8,10),(9,11),(10,7),(11,7),(12,9),(13,8),(14,10),(14,11),(15,9),(15,14),(16,8),(16,14),(17,12),(17,15)],18)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
{{1},{2,4,5},{3},{6}}
=> [1,4,3,5,2,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,9),(4,8),(5,11),(6,10),(7,10),(8,12),(9,12),(10,11),(11,8),(11,9)],13)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
{{1},{2,4},{3,5},{6}}
=> [1,4,5,2,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,8),(4,8),(5,7),(6,7),(7,11),(8,11),(9,12),(10,12),(11,9),(11,10)],13)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
{{1},{2,4},{3},{5},{6}}
=> [1,4,3,2,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(2,12),(3,11),(4,10),(5,12),(5,13),(6,11),(6,15),(8,7),(9,7),(10,9),(11,8),(12,14),(13,14),(14,10),(14,15),(15,8),(15,9)],16)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
{{1},{2,5},{3,4},{6}}
=> [1,5,4,3,2,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,10),(3,13),(4,12),(5,12),(5,15),(6,13),(6,15),(8,14),(9,14),(10,7),(11,7),(12,8),(13,9),(14,10),(14,11),(15,8),(15,9)],16)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
{{1},{2},{3,4,5},{6}}
=> [1,2,4,5,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,14),(3,12),(4,12),(5,10),(6,11),(6,13),(8,7),(9,7),(10,9),(11,8),(12,14),(13,8),(13,9),(14,10),(14,13)],15)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
{{1},{2},{3,4},{5},{6}}
=> [1,2,4,3,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,17),(2,17),(3,13),(4,12),(5,12),(5,15),(6,13),(6,16),(8,10),(9,11),(10,7),(11,7),(12,8),(13,9),(14,10),(14,11),(15,8),(15,14),(16,9),(16,14),(17,15),(17,16)],18)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
{{1},{2,5},{3},{4},{6}}
=> [1,5,3,4,2,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,8),(3,11),(4,10),(5,13),(6,13),(8,12),(9,12),(10,7),(11,7),(12,10),(12,11),(13,8),(13,9)],14)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
{{1},{2},{3,5},{4},{6}}
=> [1,2,5,4,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(2,12),(3,11),(4,10),(5,12),(5,13),(6,11),(6,15),(8,7),(9,7),(10,9),(11,8),(12,14),(13,14),(14,10),(14,15),(15,8),(15,9)],16)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
{{1},{2},{3},{4,5},{6}}
=> [1,2,3,5,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(2,17),(3,17),(4,12),(5,15),(5,16),(6,13),(6,16),(8,10),(9,11),(10,7),(11,7),(12,9),(13,8),(14,10),(14,11),(15,9),(15,14),(16,8),(16,14),(17,12),(17,15)],18)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
Description
The number of 2-regular simple modules in the incidence algebra of the lattice.
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