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Your data matches 9 different statistics following compositions of up to 3 maps.
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Matching statistic: St000260
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Values
[1] => ([],1)
=> ([],1)
=> 0
[2,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[2,3,1] => ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 1
[3,1,2] => ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 1
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 1
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 1
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2
[2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2
[2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 2
[2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2
[2,5,1,4,3] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 2
[2,5,3,1,4] => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 2
[2,5,3,4,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[2,5,4,1,3] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 2
[2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2
[3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2
[3,1,5,4,2] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 2
[3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[3,2,5,1,4] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 2
[3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 1
[3,4,1,5,2] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 2
[3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 2
[3,5,1,4,2] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 2
[3,5,2,1,4] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 2
[3,5,2,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[3,5,4,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[3,5,4,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2
[4,1,3,5,2] => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 2
[4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 2
Description
The radius of a connected graph.
This is the minimum eccentricity of any vertex.
Matching statistic: St000456
Mp00064: Permutations —reverse⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00274: Graphs —block-cut tree⟶ Graphs
St000456: Graphs ⟶ ℤResult quality: 25% ●values known / values provided: 34%●distinct values known / distinct values provided: 25%
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00274: Graphs —block-cut tree⟶ Graphs
St000456: Graphs ⟶ ℤResult quality: 25% ●values known / values provided: 34%●distinct values known / distinct values provided: 25%
Values
[1] => [1] => ([],1)
=> ([],1)
=> ? = 0 - 1
[2,1] => [1,2] => ([],2)
=> ([],2)
=> ? = 1 - 1
[2,3,1] => [1,3,2] => ([(1,2)],3)
=> ([],2)
=> ? = 1 - 1
[3,1,2] => [2,1,3] => ([(1,2)],3)
=> ([],2)
=> ? = 1 - 1
[3,2,1] => [1,2,3] => ([],3)
=> ([],3)
=> ? = 1 - 1
[2,3,4,1] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ([],2)
=> ? = 1 - 1
[2,4,1,3] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 2 - 1
[2,4,3,1] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? = 1 - 1
[3,1,4,2] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 2 - 1
[3,2,4,1] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? = 1 - 1
[3,4,1,2] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> ([],2)
=> ? = 2 - 1
[3,4,2,1] => [1,2,4,3] => ([(2,3)],4)
=> ([],3)
=> ? = 1 - 1
[4,1,2,3] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ([],2)
=> ? = 1 - 1
[4,1,3,2] => [2,3,1,4] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? = 1 - 1
[4,2,1,3] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? = 1 - 1
[4,2,3,1] => [1,3,2,4] => ([(2,3)],4)
=> ([],3)
=> ? = 1 - 1
[4,3,1,2] => [2,1,3,4] => ([(2,3)],4)
=> ([],3)
=> ? = 1 - 1
[4,3,2,1] => [1,2,3,4] => ([],4)
=> ([],4)
=> ? = 1 - 1
[2,3,4,5,1] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> ? = 1 - 1
[2,3,5,1,4] => [4,1,5,3,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 1 = 2 - 1
[2,3,5,4,1] => [1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> ? = 1 - 1
[2,4,1,5,3] => [3,5,1,4,2] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ? = 2 - 1
[2,4,3,5,1] => [1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> ? = 1 - 1
[2,4,5,1,3] => [3,1,5,4,2] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 2 - 1
[2,4,5,3,1] => [1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ? = 1 - 1
[2,5,1,3,4] => [4,3,1,5,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 1 = 2 - 1
[2,5,1,4,3] => [3,4,1,5,2] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 1 = 2 - 1
[2,5,3,1,4] => [4,1,3,5,2] => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 2 - 1
[2,5,3,4,1] => [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ? = 1 - 1
[2,5,4,1,3] => [3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 1 = 2 - 1
[2,5,4,3,1] => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ? = 1 - 1
[3,1,4,5,2] => [2,5,4,1,3] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 1 = 2 - 1
[3,1,5,2,4] => [4,2,5,1,3] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ? = 2 - 1
[3,1,5,4,2] => [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 1 = 2 - 1
[3,2,4,5,1] => [1,5,4,2,3] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> ? = 1 - 1
[3,2,5,1,4] => [4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 1 = 2 - 1
[3,2,5,4,1] => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([],2)
=> ? = 1 - 1
[3,4,1,5,2] => [2,5,1,4,3] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 2 - 1
[3,4,2,5,1] => [1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ? = 1 - 1
[3,4,5,1,2] => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> ? = 2 - 1
[3,4,5,2,1] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([],3)
=> ? = 1 - 1
[3,5,1,2,4] => [4,2,1,5,3] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 2 - 1
[3,5,1,4,2] => [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 1 = 2 - 1
[3,5,2,1,4] => [4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 1 = 2 - 1
[3,5,2,4,1] => [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ? = 1 - 1
[3,5,4,1,2] => [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ? = 2 - 1
[3,5,4,2,1] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ? = 1 - 1
[4,1,2,5,3] => [3,5,2,1,4] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 1 = 2 - 1
[4,1,3,5,2] => [2,5,3,1,4] => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 2 - 1
[4,1,5,2,3] => [3,2,5,1,4] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 2 - 1
[4,1,5,3,2] => [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 1 = 2 - 1
[4,2,1,5,3] => [3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 1 = 2 - 1
[4,2,3,5,1] => [1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ? = 1 - 1
[4,2,5,1,3] => [3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 1 = 2 - 1
[4,2,5,3,1] => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ? = 1 - 1
[4,3,1,5,2] => [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 1 = 2 - 1
[4,3,2,5,1] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ? = 1 - 1
[4,3,5,1,2] => [2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ? = 2 - 1
[4,3,5,2,1] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ? = 1 - 1
[4,5,1,2,3] => [3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> ? = 2 - 1
[4,5,1,3,2] => [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ? = 2 - 1
[4,5,2,1,3] => [3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ? = 2 - 1
[4,5,2,3,1] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ([],3)
=> ? = 1 - 1
[4,5,3,1,2] => [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> ([],3)
=> ? = 1 - 1
[4,5,3,2,1] => [1,2,3,5,4] => ([(3,4)],5)
=> ([],4)
=> ? = 2 - 1
[5,1,2,3,4] => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> ? = 1 - 1
[5,1,2,4,3] => [3,4,2,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> ? = 1 - 1
[5,1,3,2,4] => [4,2,3,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> ? = 1 - 1
[5,1,3,4,2] => [2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ? = 1 - 1
[5,1,4,2,3] => [3,2,4,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ? = 1 - 1
[5,1,4,3,2] => [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ? = 1 - 1
[5,2,1,3,4] => [4,3,1,2,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> ? = 1 - 1
[2,3,4,6,1,5] => [5,1,6,4,3,2] => ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 1 = 2 - 1
[2,3,5,6,1,4] => [4,1,6,5,3,2] => ([(0,3),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 1 = 2 - 1
[2,3,6,4,1,5] => [5,1,4,6,3,2] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 1 = 2 - 1
[2,3,6,5,1,4] => [4,1,5,6,3,2] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 1 = 2 - 1
[2,4,3,6,1,5] => [5,1,6,3,4,2] => ([(0,5),(1,2),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 1 = 2 - 1
[2,4,5,6,1,3] => [3,1,6,5,4,2] => ([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 2 - 1
[2,4,6,3,1,5] => [5,1,3,6,4,2] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 1 = 2 - 1
[2,4,6,5,1,3] => [3,1,5,6,4,2] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 2 - 1
[2,5,3,6,1,4] => [4,1,6,3,5,2] => ([(0,4),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 1 = 2 - 1
[2,5,4,6,1,3] => [3,1,6,4,5,2] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 2 - 1
[2,5,6,1,3,4] => [4,3,1,6,5,2] => ([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 1 = 2 - 1
[2,5,6,1,4,3] => [3,4,1,6,5,2] => ([(0,3),(0,4),(1,2),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 1 = 2 - 1
[2,5,6,3,1,4] => [4,1,3,6,5,2] => ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 2 - 1
[2,5,6,4,1,3] => [3,1,4,6,5,2] => ([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 1 = 2 - 1
[2,6,1,3,4,5] => [5,4,3,1,6,2] => ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 1 = 2 - 1
[2,6,1,3,5,4] => [4,5,3,1,6,2] => ([(0,5),(1,2),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 1 = 2 - 1
[2,6,1,4,3,5] => [5,3,4,1,6,2] => ([(0,5),(1,2),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 1 = 2 - 1
[2,6,1,4,5,3] => [3,5,4,1,6,2] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,2),(1,2)],3)
=> 1 = 2 - 1
[2,6,1,5,3,4] => [4,3,5,1,6,2] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,2),(1,2)],3)
=> 1 = 2 - 1
[2,6,1,5,4,3] => [3,4,5,1,6,2] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,2),(1,2)],3)
=> 1 = 2 - 1
[2,6,3,1,4,5] => [5,4,1,3,6,2] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 1 = 2 - 1
[2,6,3,1,5,4] => [4,5,1,3,6,2] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 1 = 2 - 1
[2,6,3,4,1,5] => [5,1,4,3,6,2] => ([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 2 - 1
[2,6,3,5,1,4] => [4,1,5,3,6,2] => ([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 2 - 1
[2,6,4,1,3,5] => [5,3,1,4,6,2] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 1 = 2 - 1
[2,6,4,1,5,3] => [3,5,1,4,6,2] => ([(0,5),(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 1 = 2 - 1
[2,6,4,3,1,5] => [5,1,3,4,6,2] => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 2 - 1
[2,6,4,5,1,3] => [3,1,5,4,6,2] => ([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 1 = 2 - 1
Description
The monochromatic index of a connected graph.
This is the maximal number of colours such that there is a colouring of the edges where any two vertices can be joined by a monochromatic path.
For example, a circle graph other than the triangle can be coloured with at most two colours: one edge blue, all the others red.
Matching statistic: St001545
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Values
[1] => ([],1)
=> ([],1)
=> ([],1)
=> ? = 0 + 22
[2,1] => ([(0,1)],2)
=> ([],1)
=> ([],1)
=> ? = 1 + 22
[2,3,1] => ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ? = 1 + 22
[3,1,2] => ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ? = 1 + 22
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> ([],1)
=> ([],1)
=> ? = 1 + 22
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 1 + 22
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 24 = 2 + 22
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ? = 1 + 22
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 24 = 2 + 22
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ? = 1 + 22
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([],1)
=> ([],1)
=> ? = 2 + 22
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],1)
=> ([],1)
=> ? = 1 + 22
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 1 + 22
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ? = 1 + 22
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ? = 1 + 22
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],1)
=> ([],1)
=> ? = 1 + 22
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],1)
=> ([],1)
=> ? = 1 + 22
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],1)
=> ([],1)
=> ? = 1 + 22
[2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 22
[2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 22
[2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 1 + 22
[2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ([(0,1),(0,3),(0,4),(0,6),(1,2),(1,4),(1,5),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 22
[2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 1 + 22
[2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ? = 2 + 22
[2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ? = 1 + 22
[2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 22
[2,5,1,4,3] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 24 = 2 + 22
[2,5,3,1,4] => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 24 = 2 + 22
[2,5,3,4,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ? = 1 + 22
[2,5,4,1,3] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ? = 2 + 22
[2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ? = 1 + 22
[3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 22
[3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ([(0,1),(0,3),(0,4),(0,6),(1,2),(1,4),(1,5),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 22
[3,1,5,4,2] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 24 = 2 + 22
[3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 1 + 22
[3,2,5,1,4] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 24 = 2 + 22
[3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ? = 1 + 22
[3,4,1,5,2] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ? = 2 + 22
[3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ? = 1 + 22
[3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([],1)
=> ([],1)
=> ? = 2 + 22
[3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> ? = 1 + 22
[3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ? = 2 + 22
[3,5,1,4,2] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> ? = 2 + 22
[3,5,2,1,4] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ? = 2 + 22
[3,5,2,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> ? = 1 + 22
[3,5,4,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([],1)
=> ([],1)
=> ? = 2 + 22
[3,5,4,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> ? = 1 + 22
[4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 22
[4,1,3,5,2] => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 24 = 2 + 22
[4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ? = 2 + 22
[4,1,5,3,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ? = 2 + 22
[4,2,1,5,3] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 24 = 2 + 22
[4,2,3,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ? = 1 + 22
[4,2,5,1,3] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> ? = 2 + 22
[4,2,5,3,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> ? = 1 + 22
[4,3,1,5,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ? = 2 + 22
[4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ? = 1 + 22
[4,3,5,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([],1)
=> ([],1)
=> ? = 2 + 22
[2,4,5,1,6,3] => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 24 = 2 + 22
[2,4,6,1,3,5] => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 24 = 2 + 22
[2,4,6,3,1,5] => ([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 24 = 2 + 22
[2,5,1,6,3,4] => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 24 = 2 + 22
[2,5,1,6,4,3] => ([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 24 = 2 + 22
[2,5,4,1,6,3] => ([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 24 = 2 + 22
[2,6,1,4,5,3] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 24 = 2 + 22
[2,6,1,5,3,4] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 24 = 2 + 22
[2,6,1,5,4,3] => ([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 24 = 2 + 22
[2,6,3,1,5,4] => ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 24 = 2 + 22
[2,6,3,4,1,5] => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 24 = 2 + 22
[2,6,4,1,3,5] => ([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 24 = 2 + 22
[2,6,4,3,1,5] => ([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 24 = 2 + 22
[3,1,5,6,2,4] => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 24 = 2 + 22
[3,1,5,6,4,2] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 24 = 2 + 22
[3,1,6,4,5,2] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 24 = 2 + 22
[3,1,6,5,2,4] => ([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 24 = 2 + 22
[3,1,6,5,4,2] => ([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 24 = 2 + 22
[3,2,6,1,5,4] => ([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 24 = 2 + 22
[3,2,6,4,1,5] => ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 24 = 2 + 22
[3,4,1,6,2,5] => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 24 = 2 + 22
[3,4,2,6,1,5] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 24 = 2 + 22
[3,5,1,2,6,4] => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 24 = 2 + 22
[3,5,2,1,6,4] => ([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 24 = 2 + 22
[4,1,3,6,5,2] => ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 24 = 2 + 22
[4,1,5,2,6,3] => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 24 = 2 + 22
[4,1,5,3,6,2] => ([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 24 = 2 + 22
[4,1,6,2,3,5] => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 24 = 2 + 22
[4,1,6,3,2,5] => ([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 24 = 2 + 22
[4,2,1,6,5,3] => ([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 24 = 2 + 22
[4,2,3,6,1,5] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 24 = 2 + 22
[4,3,1,6,2,5] => ([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 24 = 2 + 22
[4,3,2,6,1,5] => ([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 24 = 2 + 22
[5,1,3,4,6,2] => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 24 = 2 + 22
[5,1,4,2,6,3] => ([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 24 = 2 + 22
[5,1,4,3,6,2] => ([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 24 = 2 + 22
[5,2,1,4,6,3] => ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 24 = 2 + 22
[5,2,3,1,6,4] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 24 = 2 + 22
[5,3,1,2,6,4] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 24 = 2 + 22
[5,3,2,1,6,4] => ([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 24 = 2 + 22
[2,4,5,1,7,6,3] => ([(0,5),(1,2),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 24 = 2 + 22
[2,4,5,6,1,7,3] => ([(0,6),(1,5),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 24 = 2 + 22
Description
The second Elser number of a connected graph.
For a connected graph $G$ the $k$-th Elser number is
$$
els_k(G) = (-1)^{|V(G)|+1} \sum_N (-1)^{|E(N)|} |V(N)|^k
$$
where the sum is over all nuclei of $G$, that is, the connected subgraphs of $G$ whose vertex set is a vertex cover of $G$.
It is clear that this number is even. It was shown in [1] that it is non-negative.
Matching statistic: St001632
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Values
[1] => ([],1)
=> ([],1)
=> ([],1)
=> ? = 0 - 2
[2,1] => ([(0,1)],2)
=> ([],2)
=> ([],2)
=> ? = 1 - 2
[2,3,1] => ([(0,2),(1,2)],3)
=> ([],2)
=> ([],2)
=> ? = 1 - 2
[3,1,2] => ([(0,2),(1,2)],3)
=> ([],2)
=> ([],2)
=> ? = 1 - 2
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ([],3)
=> ? = 1 - 2
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> ([],2)
=> ([],2)
=> ? = 1 - 2
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ? = 2 - 2
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,2),(1,3)],4)
=> ? = 1 - 2
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ? = 2 - 2
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,2),(1,3)],4)
=> ? = 1 - 2
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([],2)
=> ([],2)
=> ? = 2 - 2
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],3)
=> ([],3)
=> ? = 1 - 2
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ([],2)
=> ([],2)
=> ? = 1 - 2
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,2),(1,3)],4)
=> ? = 1 - 2
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,2),(1,3)],4)
=> ? = 1 - 2
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],3)
=> ([],3)
=> ? = 1 - 2
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],3)
=> ([],3)
=> ? = 1 - 2
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> ([],4)
=> ? = 1 - 2
[2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([],2)
=> ([],2)
=> ? = 1 - 2
[2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ? = 2 - 2
[2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(1,2),(1,3)],4)
=> ? = 1 - 2
[2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(2,3),(2,4)],5)
=> ([(2,4),(3,4)],5)
=> ? = 2 - 2
[2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(1,2),(1,3)],4)
=> ? = 1 - 2
[2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ? = 2 - 2
[2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(1,2),(1,3)],4)
=> ? = 1 - 2
[2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ? = 2 - 2
[2,5,1,4,3] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(3,4)],5)
=> ([(3,4)],5)
=> ? = 2 - 2
[2,5,3,1,4] => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4)],5)
=> 0 = 2 - 2
[2,5,3,4,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(1,2),(1,3)],4)
=> ? = 1 - 2
[2,5,4,1,3] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3)],5)
=> ([(0,4),(1,2),(1,3)],5)
=> ? = 2 - 2
[2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4)],5)
=> ? = 1 - 2
[3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ? = 2 - 2
[3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(2,3),(2,4)],5)
=> ([(2,4),(3,4)],5)
=> ? = 2 - 2
[3,1,5,4,2] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(3,4)],5)
=> ([(3,4)],5)
=> ? = 2 - 2
[3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(1,2),(1,3)],4)
=> ? = 1 - 2
[3,2,5,1,4] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(3,4)],5)
=> ([(3,4)],5)
=> ? = 2 - 2
[3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([],5)
=> ([],5)
=> ? = 1 - 2
[3,4,1,5,2] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ? = 2 - 2
[3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(1,2),(1,3)],4)
=> ? = 1 - 2
[3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([],2)
=> ([],2)
=> ? = 2 - 2
[3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],3)
=> ([],3)
=> ? = 1 - 2
[3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ? = 2 - 2
[3,5,1,4,2] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ? = 2 - 2
[3,5,2,1,4] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3)],5)
=> ([(0,4),(1,2),(1,3)],5)
=> ? = 2 - 2
[3,5,2,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ? = 1 - 2
[3,5,4,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(1,2),(1,3)],4)
=> ? = 2 - 2
[3,5,4,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ([(2,3),(2,4)],5)
=> ? = 1 - 2
[4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ? = 2 - 2
[4,1,3,5,2] => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4)],5)
=> 0 = 2 - 2
[4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ? = 2 - 2
[4,1,5,3,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3)],5)
=> ([(0,4),(1,2),(1,3)],5)
=> ? = 2 - 2
[4,2,1,5,3] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(3,4)],5)
=> ([(3,4)],5)
=> ? = 2 - 2
[2,3,6,4,1,5] => ([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4)],5)
=> 0 = 2 - 2
[2,4,6,3,1,5] => ([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,4),(1,5),(2,3),(3,5)],6)
=> ([(0,3),(0,5),(1,4),(1,5),(4,2)],6)
=> 0 = 2 - 2
[2,5,6,3,1,4] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4)],5)
=> 0 = 2 - 2
[2,6,3,1,4,5] => ([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4)],5)
=> 0 = 2 - 2
[2,6,3,4,1,5] => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4)],5)
=> 0 = 2 - 2
[2,6,4,1,3,5] => ([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,4),(1,5),(2,3),(3,5)],6)
=> ([(0,3),(0,5),(1,4),(1,5),(4,2)],6)
=> 0 = 2 - 2
[2,6,4,3,1,5] => ([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5)],6)
=> 0 = 2 - 2
[2,6,5,3,1,4] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,4),(3,4),(3,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5)],6)
=> 0 = 2 - 2
[3,6,4,1,2,5] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4)],5)
=> 0 = 2 - 2
[3,6,4,2,1,5] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,4),(3,4),(3,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5)],6)
=> 0 = 2 - 2
[4,1,3,5,6,2] => ([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4)],5)
=> 0 = 2 - 2
[4,1,5,3,6,2] => ([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,4),(1,5),(2,3),(3,5)],6)
=> ([(0,3),(0,5),(1,4),(1,5),(4,2)],6)
=> 0 = 2 - 2
[4,5,1,3,6,2] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4)],5)
=> 0 = 2 - 2
[5,1,2,4,6,3] => ([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4)],5)
=> 0 = 2 - 2
[5,1,3,4,6,2] => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4)],5)
=> 0 = 2 - 2
[5,1,4,2,6,3] => ([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,4),(1,5),(2,3),(3,5)],6)
=> ([(0,3),(0,5),(1,4),(1,5),(4,2)],6)
=> 0 = 2 - 2
[5,1,4,3,6,2] => ([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5)],6)
=> 0 = 2 - 2
[5,1,4,6,2,3] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4)],5)
=> 0 = 2 - 2
[5,1,4,6,3,2] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,4),(3,4),(3,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5)],6)
=> 0 = 2 - 2
[5,4,1,3,6,2] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,4),(3,4),(3,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5)],6)
=> 0 = 2 - 2
[2,3,4,7,5,1,6] => ([(0,6),(1,6),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4)],5)
=> 0 = 2 - 2
[2,3,5,7,4,1,6] => ([(0,6),(1,6),(2,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(1,4),(1,5),(2,3),(3,5)],6)
=> ([(0,3),(0,5),(1,4),(1,5),(4,2)],6)
=> 0 = 2 - 2
[2,3,6,7,4,1,5] => ([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4)],5)
=> 0 = 2 - 2
[2,3,7,4,1,5,6] => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4)],5)
=> 0 = 2 - 2
[2,3,7,4,5,1,6] => ([(0,6),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4)],5)
=> 0 = 2 - 2
[2,3,7,5,1,4,6] => ([(0,6),(1,5),(2,5),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(1,4),(1,5),(2,3),(3,5)],6)
=> ([(0,3),(0,5),(1,4),(1,5),(4,2)],6)
=> 0 = 2 - 2
[2,3,7,5,4,1,6] => ([(0,6),(1,6),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5)],6)
=> 0 = 2 - 2
[2,3,7,6,4,1,5] => ([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(1,5),(2,4),(3,4),(3,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5)],6)
=> 0 = 2 - 2
[2,4,5,7,3,1,6] => ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(1,4),(1,5),(2,3),(3,5)],6)
=> ([(0,3),(0,5),(1,4),(1,5),(4,2)],6)
=> 0 = 2 - 2
[2,4,6,7,3,1,5] => ([(0,6),(1,5),(1,6),(2,3),(2,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(1,4),(1,5),(2,3),(3,5)],6)
=> ([(0,3),(0,5),(1,4),(1,5),(4,2)],6)
=> 0 = 2 - 2
[2,4,7,3,1,5,6] => ([(0,6),(1,5),(2,5),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(1,4),(1,5),(2,3),(3,5)],6)
=> ([(0,3),(0,5),(1,4),(1,5),(4,2)],6)
=> 0 = 2 - 2
[2,5,6,7,3,1,4] => ([(0,6),(1,2),(1,3),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4)],5)
=> 0 = 2 - 2
[2,5,7,3,4,1,6] => ([(0,6),(1,5),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(1,4),(1,5),(2,3),(3,5)],6)
=> ([(0,3),(0,5),(1,4),(1,5),(4,2)],6)
=> 0 = 2 - 2
[2,6,7,3,1,4,5] => ([(0,6),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4)],5)
=> 0 = 2 - 2
[2,6,7,3,4,1,5] => ([(0,6),(1,4),(1,5),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4)],5)
=> 0 = 2 - 2
[2,6,7,4,1,3,5] => ([(0,6),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ([(0,4),(1,4),(1,5),(2,3),(3,5)],6)
=> ([(0,3),(0,5),(1,4),(1,5),(4,2)],6)
=> 0 = 2 - 2
[2,6,7,4,3,1,5] => ([(0,6),(1,4),(1,5),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5)],6)
=> 0 = 2 - 2
[2,6,7,5,3,1,4] => ([(0,6),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(1,5),(2,4),(3,4),(3,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5)],6)
=> 0 = 2 - 2
[2,7,3,1,4,5,6] => ([(0,6),(1,6),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4)],5)
=> 0 = 2 - 2
[2,7,3,4,1,5,6] => ([(0,6),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4)],5)
=> 0 = 2 - 2
[2,7,3,4,5,1,6] => ([(0,6),(1,5),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4)],5)
=> 0 = 2 - 2
[2,7,4,1,3,5,6] => ([(0,6),(1,6),(2,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(1,4),(1,5),(2,3),(3,5)],6)
=> ([(0,3),(0,5),(1,4),(1,5),(4,2)],6)
=> 0 = 2 - 2
[2,7,4,3,1,5,6] => ([(0,6),(1,6),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5)],6)
=> 0 = 2 - 2
[2,7,4,5,1,3,6] => ([(0,6),(1,5),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(1,4),(1,5),(2,3),(3,5)],6)
=> ([(0,3),(0,5),(1,4),(1,5),(4,2)],6)
=> 0 = 2 - 2
[2,7,4,5,3,1,6] => ([(0,6),(1,5),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5)],6)
=> 0 = 2 - 2
[2,7,5,1,3,4,6] => ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(1,4),(1,5),(2,3),(3,5)],6)
=> ([(0,3),(0,5),(1,4),(1,5),(4,2)],6)
=> 0 = 2 - 2
[2,7,5,3,4,1,6] => ([(0,6),(1,5),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5)],6)
=> 0 = 2 - 2
[2,7,5,6,3,1,4] => ([(0,6),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(1,5),(2,4),(3,4),(3,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5)],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: St000259
Values
[1] => ([],1)
=> ([],1)
=> ([],1)
=> 0
[2,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> ? = 1
[2,3,1] => ([(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1
[3,1,2] => ([(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> ? = 1
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> ? = 2
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> ? = 1
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> ? = 2
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> ? = 1
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> ? = 2
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> ? = 1
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> ? = 1
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> ? = 1
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> ? = 1
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> ? = 1
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> ? = 1
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> ? = 1
[2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],5)
=> ? = 1
[2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ? = 2
[2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],5)
=> ? = 1
[2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ? = 2
[2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],5)
=> ? = 1
[2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(3,4)],5)
=> ? = 2
[2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],5)
=> ? = 1
[2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ? = 2
[2,5,1,4,3] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ? = 2
[2,5,3,1,4] => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(3,4)],5)
=> ? = 2
[2,5,3,4,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],5)
=> ? = 1
[2,5,4,1,3] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(3,4)],5)
=> ? = 2
[2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],5)
=> ? = 1
[3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ? = 2
[3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ? = 2
[3,1,5,4,2] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ? = 2
[3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],5)
=> ? = 1
[3,2,5,1,4] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ? = 2
[3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],5)
=> ? = 1
[3,4,1,5,2] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(3,4)],5)
=> ? = 2
[3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],5)
=> ? = 1
[3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],5)
=> ? = 2
[3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],5)
=> ? = 1
[3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(3,4)],5)
=> ? = 2
[3,5,1,4,2] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],5)
=> ? = 2
[3,5,2,1,4] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(3,4)],5)
=> ? = 2
[3,5,2,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],5)
=> ? = 1
[3,5,4,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],5)
=> ? = 2
[3,5,4,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],5)
=> ? = 1
[4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ? = 2
[4,1,3,5,2] => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(3,4)],5)
=> ? = 2
[4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(3,4)],5)
=> ? = 2
[4,1,5,3,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(3,4)],5)
=> ? = 2
[2,4,1,6,3,5] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> 3
[3,1,5,2,6,4] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> 3
[2,3,5,1,7,4,6] => ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> ([(0,3),(0,5),(1,2),(1,4),(1,6),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,4),(1,5),(2,3),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> 3
[2,4,1,5,7,3,6] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(0,4),(0,5),(1,2),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,5),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(5,6)],7)
=> 3
[2,4,1,6,3,7,5] => ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ([(0,3),(0,6),(1,2),(1,5),(2,4),(2,5),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,4),(1,6),(2,3),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
=> 3
[2,4,1,6,7,3,5] => ([(0,5),(1,2),(1,3),(2,6),(3,6),(4,5),(4,6)],7)
=> ([(0,2),(0,6),(1,3),(1,4),(1,5),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> 3
[2,4,1,7,3,5,6] => ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> ([(0,3),(0,5),(1,2),(1,4),(1,6),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,4),(1,5),(2,3),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> 3
[2,4,1,7,3,6,5] => ([(0,5),(1,4),(1,5),(2,3),(2,6),(3,6),(4,6)],7)
=> ([(0,3),(0,5),(1,2),(1,4),(1,6),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,4),(1,5),(2,3),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> 3
[2,4,1,7,5,3,6] => ([(0,6),(1,4),(2,5),(2,6),(3,4),(3,5),(5,6)],7)
=> ([(0,2),(0,6),(1,3),(1,4),(1,5),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> 3
[2,4,1,7,6,3,5] => ([(0,3),(1,3),(1,4),(2,5),(2,6),(4,5),(4,6),(5,6)],7)
=> ([(0,2),(0,6),(1,3),(1,4),(1,5),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> 3
[2,4,5,1,7,3,6] => ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5)],7)
=> ([(0,3),(0,6),(1,2),(1,4),(1,5),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,6),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
=> 3
[2,5,1,3,7,4,6] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(0,4),(0,5),(1,2),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,5),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(5,6)],7)
=> 3
[2,5,1,4,7,3,6] => ([(0,4),(1,3),(2,5),(2,6),(3,5),(4,6),(5,6)],7)
=> ([(0,4),(0,6),(1,3),(1,5),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,5),(2,3),(2,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
[2,5,1,7,3,4,6] => ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5)],7)
=> ([(0,3),(0,6),(1,2),(1,4),(1,5),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,6),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
=> 3
[2,5,1,7,4,3,6] => ([(0,6),(1,2),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,3),(0,6),(1,2),(1,4),(1,5),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,6),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
=> 3
[2,5,3,1,7,4,6] => ([(0,6),(1,4),(2,5),(2,6),(3,4),(3,5),(5,6)],7)
=> ([(0,2),(0,6),(1,3),(1,4),(1,5),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> 3
[2,5,4,1,7,3,6] => ([(0,6),(1,2),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,3),(0,6),(1,2),(1,4),(1,5),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,6),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
=> 3
[3,1,4,6,2,7,5] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(0,4),(0,5),(1,2),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,5),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(5,6)],7)
=> 3
[3,1,5,2,6,7,4] => ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> ([(0,3),(0,5),(1,2),(1,4),(1,6),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,4),(1,5),(2,3),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> 3
[3,1,5,2,7,4,6] => ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ([(0,3),(0,6),(1,2),(1,5),(2,4),(2,5),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,4),(1,6),(2,3),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
=> 3
[3,1,5,2,7,6,4] => ([(0,5),(1,4),(1,5),(2,3),(2,6),(3,6),(4,6)],7)
=> ([(0,3),(0,5),(1,2),(1,4),(1,6),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,4),(1,5),(2,3),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> 3
[3,1,5,6,2,7,4] => ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5)],7)
=> ([(0,3),(0,6),(1,2),(1,4),(1,5),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,6),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
=> 3
[3,1,6,2,4,7,5] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(0,4),(0,5),(1,2),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,5),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(5,6)],7)
=> 3
[3,1,6,2,5,7,4] => ([(0,6),(1,4),(2,5),(2,6),(3,4),(3,5),(5,6)],7)
=> ([(0,2),(0,6),(1,3),(1,4),(1,5),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> 3
[3,1,6,2,7,4,5] => ([(0,5),(1,2),(1,3),(2,6),(3,6),(4,5),(4,6)],7)
=> ([(0,2),(0,6),(1,3),(1,4),(1,5),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> 3
[3,1,6,2,7,5,4] => ([(0,3),(1,3),(1,4),(2,5),(2,6),(4,5),(4,6),(5,6)],7)
=> ([(0,2),(0,6),(1,3),(1,4),(1,5),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> 3
[3,1,6,4,2,7,5] => ([(0,4),(1,3),(2,5),(2,6),(3,5),(4,6),(5,6)],7)
=> ([(0,4),(0,6),(1,3),(1,5),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,5),(2,3),(2,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
[3,1,6,5,2,7,4] => ([(0,6),(1,2),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,3),(0,6),(1,2),(1,4),(1,5),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,6),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
=> 3
[3,2,5,1,7,4,6] => ([(0,5),(1,4),(1,5),(2,3),(2,6),(3,6),(4,6)],7)
=> ([(0,3),(0,5),(1,2),(1,4),(1,6),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,4),(1,5),(2,3),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> 3
[3,4,1,6,2,7,5] => ([(0,5),(1,2),(1,3),(2,6),(3,6),(4,5),(4,6)],7)
=> ([(0,2),(0,6),(1,3),(1,4),(1,5),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> 3
[3,5,1,2,7,4,6] => ([(0,5),(1,2),(1,3),(2,6),(3,6),(4,5),(4,6)],7)
=> ([(0,2),(0,6),(1,3),(1,4),(1,5),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> 3
[3,5,2,1,7,4,6] => ([(0,3),(1,3),(1,4),(2,5),(2,6),(4,5),(4,6),(5,6)],7)
=> ([(0,2),(0,6),(1,3),(1,4),(1,5),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> 3
[4,1,2,6,3,7,5] => ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> ([(0,3),(0,5),(1,2),(1,4),(1,6),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,4),(1,5),(2,3),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> 3
[4,1,3,6,2,7,5] => ([(0,6),(1,4),(2,5),(2,6),(3,4),(3,5),(5,6)],7)
=> ([(0,2),(0,6),(1,3),(1,4),(1,5),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> 3
[4,1,6,2,3,7,5] => ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5)],7)
=> ([(0,3),(0,6),(1,2),(1,4),(1,5),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,6),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
=> 3
[4,1,6,3,2,7,5] => ([(0,6),(1,2),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,3),(0,6),(1,2),(1,4),(1,5),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,6),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
=> 3
[4,2,1,6,3,7,5] => ([(0,5),(1,4),(1,5),(2,3),(2,6),(3,6),(4,6)],7)
=> ([(0,3),(0,5),(1,2),(1,4),(1,6),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,4),(1,5),(2,3),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> 3
[4,3,1,6,2,7,5] => ([(0,3),(1,3),(1,4),(2,5),(2,6),(4,5),(4,6),(5,6)],7)
=> ([(0,2),(0,6),(1,3),(1,4),(1,5),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> 3
Description
The diameter of a connected graph.
This is the greatest distance between any pair of vertices.
Matching statistic: St001876
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00088: Permutations —Kreweras complement⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
Mp00196: Lattices —The modular quotient of a lattice.⟶ Lattices
St001876: Lattices ⟶ ℤResult quality: 1% ●values known / values provided: 1%●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: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 25%
Values
[1] => [1] => ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 0
[2,1] => [1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[2,3,1] => [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
[3,1,2] => [3,1,2] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> ? = 1
[3,2,1] => [1,3,2] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> ? = 1
[2,3,4,1] => [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
[2,4,1,3] => [4,2,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> ([(0,1)],2)
=> ? = 2
[2,4,3,1] => [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
[3,1,4,2] => [3,1,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> ([(0,1)],2)
=> ? = 2
[3,2,4,1] => [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
[3,4,1,2] => [4,1,2,3] => ([(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)
=> ? = 2
[3,4,2,1] => [1,4,2,3] => ([(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
[4,1,2,3] => [3,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ([],1)
=> ? = 1
[4,1,3,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 1
[4,2,1,3] => [4,3,1,2] => ([(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
[4,2,3,1] => [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
[4,3,1,2] => [4,1,3,2] => ([(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
[4,3,2,1] => [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
[2,3,4,5,1] => [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
[2,3,5,1,4] => [5,2,3,1,4] => ([(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)
=> ? = 2
[2,3,5,4,1] => [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
[2,4,1,5,3] => [4,2,1,3,5] => ([(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)
=> ? = 2
[2,4,3,5,1] => [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
[2,4,5,1,3] => [5,2,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,6),(3,7),(4,7),(5,6),(5,8),(6,10),(7,8),(8,10),(10,9)],11)
=> ([(0,1)],2)
=> ? = 2
[2,4,5,3,1] => [1,2,5,3,4] => ([(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
[2,5,1,3,4] => [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
[2,5,1,4,3] => [4,2,1,5,3] => ([(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
[2,5,3,1,4] => [5,2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
=> ([(0,1)],2)
=> ? = 2
[2,5,3,4,1] => [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
[2,5,4,1,3] => [5,2,1,4,3] => ([(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)
=> ? = 2
[2,5,4,3,1] => [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
[3,1,4,5,2] => [3,1,2,4,5] => ([(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)
=> ? = 2
[3,1,5,2,4] => [3,5,2,1,4] => ([(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
[3,1,5,4,2] => [3,1,2,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(8,9)],10)
=> ([],1)
=> ? = 2
[3,2,4,5,1] => [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
[3,2,5,1,4] => [5,3,2,1,4] => ([(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)
=> ? = 2
[3,2,5,4,1] => [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
[3,4,1,5,2] => [4,1,2,3,5] => ([(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)
=> ? = 2
[3,4,2,5,1] => [1,4,2,3,5] => ([(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
[3,4,5,1,2] => [5,1,2,3,4] => ([(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)
=> ? = 2
[3,4,5,2,1] => [1,5,2,3,4] => ([(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
[3,5,1,2,4] => [4,5,2,1,3] => ([(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
[3,5,1,4,2] => [4,1,2,5,3] => ([(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
[3,5,2,1,4] => [5,4,2,1,3] => ([(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)
=> ? = 2
[3,5,2,4,1] => [1,4,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
=> ([(0,1)],2)
=> ? = 1
[3,5,4,1,2] => [5,1,2,4,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,6),(3,7),(4,7),(5,6),(5,8),(6,10),(7,8),(8,10),(10,9)],11)
=> ([(0,1)],2)
=> ? = 2
[3,5,4,2,1] => [1,5,2,4,3] => ([(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
[4,1,2,5,3] => [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)
=> ? = 2
[4,1,3,5,2] => [3,1,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
=> ([(0,1)],2)
=> ? = 2
[4,1,5,2,3] => [3,5,1,2,4] => ([(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
[4,1,5,3,2] => [3,1,5,2,4] => ([(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
[4,2,1,5,3] => [4,3,1,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,6),(3,7),(4,7),(5,6),(5,8),(6,10),(7,8),(8,10),(10,9)],11)
=> ([(0,1)],2)
=> ? = 2
[4,2,3,5,1] => [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
[4,2,5,1,3] => [5,3,1,2,4] => ([(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)
=> ? = 2
[4,2,5,3,1] => [1,3,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
=> ([(0,1)],2)
=> ? = 1
[4,3,1,5,2] => [4,1,3,2,5] => ([(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)
=> ? = 2
[4,3,2,5,1] => [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
[4,3,5,1,2] => [5,1,3,2,4] => ([(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)
=> ? = 2
[4,3,5,2,1] => [1,5,3,2,4] => ([(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
[4,5,1,2,3] => [4,5,1,2,3] => ([(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
[2,3,5,4,6,1] => [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
[2,4,3,5,6,1] => [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
[2,4,5,3,6,1] => [1,2,5,3,4,6] => ([(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
[2,5,3,4,6,1] => [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
[2,5,4,3,6,1] => [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
[3,2,4,5,6,1] => [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
[3,2,5,4,6,1] => [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
[3,4,2,5,6,1] => [1,4,2,3,5,6] => ([(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
[3,4,5,2,6,1] => [1,5,2,3,4,6] => ([(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
[3,5,2,4,6,1] => [1,4,2,5,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,10),(3,10),(4,10),(5,8),(6,7),(7,9),(8,9),(10,7),(10,8)],11)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[3,5,4,2,6,1] => [1,5,2,4,3,6] => ([(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
[4,2,3,5,6,1] => [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
[4,2,5,3,6,1] => [1,3,5,2,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,10),(3,10),(4,10),(5,8),(6,7),(7,9),(8,9),(10,7),(10,8)],11)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[4,3,2,5,6,1] => [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
[4,3,5,2,6,1] => [1,5,3,2,4,6] => ([(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
[4,5,2,3,6,1] => [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
[4,5,3,2,6,1] => [1,5,4,2,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,11),(4,12),(5,12),(6,8),(6,11),(8,13),(9,7),(10,7),(11,13),(12,8),(13,9),(13,10)],14)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[5,2,3,4,6,1] => [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
[5,2,4,3,6,1] => [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
[5,3,2,4,6,1] => [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
[5,3,4,2,6,1] => [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
[5,4,2,3,6,1] => [1,4,5,3,2,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,11),(4,12),(5,12),(6,8),(6,11),(8,13),(9,7),(10,7),(11,13),(12,8),(13,9),(13,10)],14)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[5,4,3,2,6,1] => [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
Description
The number of 2-regular simple modules in the incidence algebra of the lattice.
Matching statistic: St001060
Values
[1] => ([],1)
=> ([(0,1)],2)
=> ([],2)
=> ? = 0
[2,1] => ([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1
[2,3,1] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> ? = 1
[3,1,2] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> ? = 1
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> ? = 1
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ? = 1
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> 2
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ? = 1
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> 2
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ? = 1
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,3)],5)
=> ? = 2
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(3,4)],5)
=> ? = 1
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ? = 1
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ? = 1
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ? = 1
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(3,4)],5)
=> ? = 1
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(3,4)],5)
=> ? = 1
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],5)
=> ? = 1
[2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1
[2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> 2
[2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1
[2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> 2
[2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1
[2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> 2
[2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1
[2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> 2
[2,5,1,4,3] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 2
[2,5,3,1,4] => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> 2
[2,5,3,4,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1
[2,5,4,1,3] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,5),(3,4),(4,5)],6)
=> 2
[2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(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)
=> ([(2,5),(3,5),(4,5)],6)
=> ? = 1
[3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> 2
[3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> 2
[3,1,5,4,2] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 2
[3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1
[3,2,5,1,4] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 2
[3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 1
[3,4,1,5,2] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> 2
[3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1
[3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(1,2),(3,4),(3,5),(4,5)],6)
=> ? = 2
[3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(3,4),(3,5),(4,5)],6)
=> ? = 1
[3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> 2
[3,5,1,4,2] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 2
[3,5,2,1,4] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,5),(3,4),(4,5)],6)
=> 2
[3,5,2,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> ? = 1
[3,5,4,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(1,2),(3,5),(4,5)],6)
=> ? = 2
[3,5,4,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(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)
=> ([(3,5),(4,5)],6)
=> ? = 1
[4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> 2
[4,1,3,5,2] => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> 2
[4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> 2
[4,1,5,3,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,5),(3,4),(4,5)],6)
=> 2
[4,2,1,5,3] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 2
[4,2,3,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1
[4,2,5,1,3] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 2
[4,2,5,3,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> ? = 1
[4,3,1,5,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,5),(3,4),(4,5)],6)
=> 2
[4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(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)
=> ([(2,5),(3,5),(4,5)],6)
=> ? = 1
[4,3,5,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(1,2),(3,5),(4,5)],6)
=> ? = 2
[4,3,5,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(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)
=> ([(3,5),(4,5)],6)
=> ? = 1
[4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(1,2),(3,4),(3,5),(4,5)],6)
=> ? = 2
[4,5,1,3,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(1,2),(3,5),(4,5)],6)
=> ? = 2
[4,5,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(1,2),(3,5),(4,5)],6)
=> ? = 2
[4,5,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,5),(3,4)],6)
=> ? = 1
[4,5,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,5),(3,4)],6)
=> ? = 1
[4,5,3,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(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)
=> ([(4,5)],6)
=> ? = 2
[5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1
[5,1,2,4,3] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1
[5,1,3,2,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1
[5,1,3,4,2] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1
[5,1,4,2,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1
[5,1,4,3,2] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(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)
=> ([(2,5),(3,5),(4,5)],6)
=> ? = 1
[5,2,1,3,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1
[5,2,1,4,3] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 1
[5,2,3,1,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1
Description
The distinguishing index of a graph.
This is the smallest number of colours such that there is a colouring of the edges which is not preserved by any automorphism.
If the graph has a connected component which is a single edge, or at least two isolated vertices, this statistic is undefined.
Matching statistic: St001199
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001199: Dyck paths ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 25%
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001199: Dyck paths ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 25%
Values
[1] => [1] => [1] => [1,0]
=> ? = 0 - 2
[2,1] => [2,1] => [2,1] => [1,1,0,0]
=> ? = 1 - 2
[2,3,1] => [3,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> ? = 1 - 2
[3,1,2] => [3,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> ? = 1 - 2
[3,2,1] => [3,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> ? = 1 - 2
[2,3,4,1] => [4,2,3,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 1 - 2
[2,4,1,3] => [3,4,1,2] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 2 - 2
[2,4,3,1] => [4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 1 - 2
[3,1,4,2] => [4,2,3,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 2 - 2
[3,2,4,1] => [4,2,3,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 1 - 2
[3,4,1,2] => [4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 2 - 2
[3,4,2,1] => [4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 1 - 2
[4,1,2,3] => [4,2,3,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 1 - 2
[4,1,3,2] => [4,2,3,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 1 - 2
[4,2,1,3] => [4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 1 - 2
[4,2,3,1] => [4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 1 - 2
[4,3,1,2] => [4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 1 - 2
[4,3,2,1] => [4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 1 - 2
[2,3,4,5,1] => [5,2,3,4,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 2
[2,3,5,1,4] => [4,2,5,1,3] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 2 - 2
[2,3,5,4,1] => [5,2,4,3,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 2
[2,4,1,5,3] => [3,5,1,4,2] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 2 - 2
[2,4,3,5,1] => [5,3,2,4,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 2
[2,4,5,1,3] => [4,5,3,1,2] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 2 - 2
[2,4,5,3,1] => [5,4,3,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 2
[2,5,1,3,4] => [3,5,1,4,2] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 2 - 2
[2,5,1,4,3] => [3,5,1,4,2] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 2 - 2
[2,5,3,1,4] => [4,5,3,1,2] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 2 - 2
[2,5,3,4,1] => [5,4,3,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 2
[2,5,4,1,3] => [4,5,3,1,2] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 2 - 2
[2,5,4,3,1] => [5,4,3,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 2
[3,1,4,5,2] => [5,2,3,4,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 2 - 2
[3,1,5,2,4] => [4,2,5,1,3] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 2 - 2
[3,1,5,4,2] => [5,2,4,3,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 2 - 2
[3,2,4,5,1] => [5,2,3,4,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 2
[3,2,5,1,4] => [4,2,5,1,3] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 2 - 2
[3,2,5,4,1] => [5,2,4,3,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 2
[3,4,1,5,2] => [5,3,2,4,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 2 - 2
[3,4,2,5,1] => [5,3,2,4,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 2
[3,4,5,1,2] => [5,4,3,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 2 - 2
[3,4,5,2,1] => [5,4,3,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 2
[3,5,1,2,4] => [4,5,3,1,2] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 2 - 2
[3,5,1,4,2] => [5,4,3,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 2 - 2
[3,5,2,1,4] => [4,5,3,1,2] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 2 - 2
[3,5,2,4,1] => [5,4,3,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 2
[3,5,4,1,2] => [5,4,3,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 2 - 2
[3,5,4,2,1] => [5,4,3,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 2
[4,1,2,5,3] => [5,2,3,4,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 2 - 2
[4,1,3,5,2] => [5,2,3,4,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 2 - 2
[4,1,5,2,3] => [5,2,4,3,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 2 - 2
[2,4,1,6,3,5] => [3,5,1,6,2,4] => [5,6,3,4,1,2] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> 1 = 3 - 2
[2,3,5,1,7,4,6] => [4,2,6,1,7,3,5] => [6,4,7,2,5,1,3] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> 1 = 3 - 2
[2,4,1,5,7,3,6] => [3,6,1,4,7,2,5] => [6,7,3,4,5,1,2] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> 1 = 3 - 2
[2,4,1,6,3,7,5] => [3,5,1,7,2,6,4] => [5,7,3,6,1,4,2] => [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> 1 = 3 - 2
[2,4,1,6,7,3,5] => [3,6,1,7,5,2,4] => [6,7,3,5,4,1,2] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> 1 = 3 - 2
[2,4,1,7,3,5,6] => [3,5,1,7,2,6,4] => [5,7,3,6,1,4,2] => [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> 1 = 3 - 2
[2,4,1,7,3,6,5] => [3,5,1,7,2,6,4] => [5,7,3,6,1,4,2] => [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> 1 = 3 - 2
[2,4,1,7,5,3,6] => [3,6,1,7,5,2,4] => [6,7,3,5,4,1,2] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> 1 = 3 - 2
[2,4,1,7,6,3,5] => [3,6,1,7,5,2,4] => [6,7,3,5,4,1,2] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> 1 = 3 - 2
[2,4,5,1,7,3,6] => [4,6,3,1,7,2,5] => [6,7,4,3,5,1,2] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> 1 = 3 - 2
[2,5,1,3,7,4,6] => [3,6,1,4,7,2,5] => [6,7,3,4,5,1,2] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> 1 = 3 - 2
[2,5,1,4,7,3,6] => [3,6,1,4,7,2,5] => [6,7,3,4,5,1,2] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> 1 = 3 - 2
[2,5,1,7,3,4,6] => [3,6,1,7,5,2,4] => [6,7,3,5,4,1,2] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> 1 = 3 - 2
[2,5,1,7,4,3,6] => [3,6,1,7,5,2,4] => [6,7,3,5,4,1,2] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> 1 = 3 - 2
[2,5,3,1,7,4,6] => [4,6,3,1,7,2,5] => [6,7,4,3,5,1,2] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> 1 = 3 - 2
[2,5,4,1,7,3,6] => [4,6,3,1,7,2,5] => [6,7,4,3,5,1,2] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> 1 = 3 - 2
[3,1,5,2,7,4,6] => [4,2,6,1,7,3,5] => [6,4,7,2,5,1,3] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> 1 = 3 - 2
[3,2,5,1,7,4,6] => [4,2,6,1,7,3,5] => [6,4,7,2,5,1,3] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> 1 = 3 - 2
[3,5,1,2,7,4,6] => [4,6,3,1,7,2,5] => [6,7,4,3,5,1,2] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> 1 = 3 - 2
[3,5,2,1,7,4,6] => [4,6,3,1,7,2,5] => [6,7,4,3,5,1,2] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> 1 = 3 - 2
Description
The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$.
Matching statistic: St001498
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001498: Dyck paths ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 25%
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001498: Dyck paths ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 25%
Values
[1] => [1] => [1] => [1,0]
=> ? = 0 - 3
[2,1] => [2,1] => [2,1] => [1,1,0,0]
=> ? = 1 - 3
[2,3,1] => [3,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> ? = 1 - 3
[3,1,2] => [3,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> ? = 1 - 3
[3,2,1] => [3,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> ? = 1 - 3
[2,3,4,1] => [4,2,3,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 1 - 3
[2,4,1,3] => [3,4,1,2] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 2 - 3
[2,4,3,1] => [4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 1 - 3
[3,1,4,2] => [4,2,3,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 2 - 3
[3,2,4,1] => [4,2,3,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 1 - 3
[3,4,1,2] => [4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 2 - 3
[3,4,2,1] => [4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 1 - 3
[4,1,2,3] => [4,2,3,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 1 - 3
[4,1,3,2] => [4,2,3,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 1 - 3
[4,2,1,3] => [4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 1 - 3
[4,2,3,1] => [4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 1 - 3
[4,3,1,2] => [4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 1 - 3
[4,3,2,1] => [4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 1 - 3
[2,3,4,5,1] => [5,2,3,4,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 3
[2,3,5,1,4] => [4,2,5,1,3] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 2 - 3
[2,3,5,4,1] => [5,2,4,3,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 3
[2,4,1,5,3] => [3,5,1,4,2] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 2 - 3
[2,4,3,5,1] => [5,3,2,4,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 3
[2,4,5,1,3] => [4,5,3,1,2] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 2 - 3
[2,4,5,3,1] => [5,4,3,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 3
[2,5,1,3,4] => [3,5,1,4,2] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 2 - 3
[2,5,1,4,3] => [3,5,1,4,2] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 2 - 3
[2,5,3,1,4] => [4,5,3,1,2] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 2 - 3
[2,5,3,4,1] => [5,4,3,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 3
[2,5,4,1,3] => [4,5,3,1,2] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 2 - 3
[2,5,4,3,1] => [5,4,3,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 3
[3,1,4,5,2] => [5,2,3,4,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 2 - 3
[3,1,5,2,4] => [4,2,5,1,3] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 2 - 3
[3,1,5,4,2] => [5,2,4,3,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 2 - 3
[3,2,4,5,1] => [5,2,3,4,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 3
[3,2,5,1,4] => [4,2,5,1,3] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 2 - 3
[3,2,5,4,1] => [5,2,4,3,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 3
[3,4,1,5,2] => [5,3,2,4,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 2 - 3
[3,4,2,5,1] => [5,3,2,4,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 3
[3,4,5,1,2] => [5,4,3,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 2 - 3
[3,4,5,2,1] => [5,4,3,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 3
[3,5,1,2,4] => [4,5,3,1,2] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 2 - 3
[3,5,1,4,2] => [5,4,3,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 2 - 3
[3,5,2,1,4] => [4,5,3,1,2] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 2 - 3
[3,5,2,4,1] => [5,4,3,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 3
[3,5,4,1,2] => [5,4,3,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 2 - 3
[3,5,4,2,1] => [5,4,3,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 3
[4,1,2,5,3] => [5,2,3,4,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 2 - 3
[4,1,3,5,2] => [5,2,3,4,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 2 - 3
[4,1,5,2,3] => [5,2,4,3,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 2 - 3
[2,4,1,6,3,5] => [3,5,1,6,2,4] => [5,6,3,4,1,2] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> 0 = 3 - 3
[2,3,5,1,7,4,6] => [4,2,6,1,7,3,5] => [6,4,7,2,5,1,3] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> 0 = 3 - 3
[2,4,1,5,7,3,6] => [3,6,1,4,7,2,5] => [6,7,3,4,5,1,2] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> 0 = 3 - 3
[2,4,1,6,3,7,5] => [3,5,1,7,2,6,4] => [5,7,3,6,1,4,2] => [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> 0 = 3 - 3
[2,4,1,6,7,3,5] => [3,6,1,7,5,2,4] => [6,7,3,5,4,1,2] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> 0 = 3 - 3
[2,4,1,7,3,5,6] => [3,5,1,7,2,6,4] => [5,7,3,6,1,4,2] => [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> 0 = 3 - 3
[2,4,1,7,3,6,5] => [3,5,1,7,2,6,4] => [5,7,3,6,1,4,2] => [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> 0 = 3 - 3
[2,4,1,7,5,3,6] => [3,6,1,7,5,2,4] => [6,7,3,5,4,1,2] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> 0 = 3 - 3
[2,4,1,7,6,3,5] => [3,6,1,7,5,2,4] => [6,7,3,5,4,1,2] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> 0 = 3 - 3
[2,4,5,1,7,3,6] => [4,6,3,1,7,2,5] => [6,7,4,3,5,1,2] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> 0 = 3 - 3
[2,5,1,3,7,4,6] => [3,6,1,4,7,2,5] => [6,7,3,4,5,1,2] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> 0 = 3 - 3
[2,5,1,4,7,3,6] => [3,6,1,4,7,2,5] => [6,7,3,4,5,1,2] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> 0 = 3 - 3
[2,5,1,7,3,4,6] => [3,6,1,7,5,2,4] => [6,7,3,5,4,1,2] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> 0 = 3 - 3
[2,5,1,7,4,3,6] => [3,6,1,7,5,2,4] => [6,7,3,5,4,1,2] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> 0 = 3 - 3
[2,5,3,1,7,4,6] => [4,6,3,1,7,2,5] => [6,7,4,3,5,1,2] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> 0 = 3 - 3
[2,5,4,1,7,3,6] => [4,6,3,1,7,2,5] => [6,7,4,3,5,1,2] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> 0 = 3 - 3
[3,1,5,2,7,4,6] => [4,2,6,1,7,3,5] => [6,4,7,2,5,1,3] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> 0 = 3 - 3
[3,2,5,1,7,4,6] => [4,2,6,1,7,3,5] => [6,4,7,2,5,1,3] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> 0 = 3 - 3
[3,5,1,2,7,4,6] => [4,6,3,1,7,2,5] => [6,7,4,3,5,1,2] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> 0 = 3 - 3
[3,5,2,1,7,4,6] => [4,6,3,1,7,2,5] => [6,7,4,3,5,1,2] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> 0 = 3 - 3
Description
The normalised height of a Nakayama algebra with magnitude 1.
We use the bijection (see code) suggested by Christian Stump, to have a bijection between such Nakayama algebras with magnitude 1 and Dyck paths. The normalised height is the height of the (periodic) Dyck path given by the top of the Auslander-Reiten quiver. Thus when having a CNakayama algebra it is the Loewy length minus the number of simple modules and for the LNakayama algebras it is the usual height.
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