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Your data matches 57 different statistics following compositions of up to 3 maps.
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Matching statistic: St001060
Mp00252: Permutations —restriction⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001060: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001060: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[3,1,4,2,5] => [3,1,4,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> 2
[3,1,4,5,2] => [3,1,4,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> 2
[3,1,5,4,2] => [3,1,4,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> 2
[3,2,4,1,5] => [3,2,4,1] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> 2
[3,2,4,5,1] => [3,2,4,1] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> 2
[3,2,5,4,1] => [3,2,4,1] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> 2
[3,5,1,4,2] => [3,1,4,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> 2
[3,5,2,4,1] => [3,2,4,1] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> 2
[4,1,2,3,5] => [4,1,2,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 3
[4,1,2,5,3] => [4,1,2,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 3
[4,1,3,2,5] => [4,1,3,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 2
[4,1,3,5,2] => [4,1,3,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 2
[4,1,5,2,3] => [4,1,2,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 3
[4,1,5,3,2] => [4,1,3,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 2
[4,2,1,3,5] => [4,2,1,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 3
[4,2,1,5,3] => [4,2,1,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 3
[4,2,3,1,5] => [4,2,3,1] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 2
[4,2,3,5,1] => [4,2,3,1] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 2
[4,2,5,1,3] => [4,2,1,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 3
[4,2,5,3,1] => [4,2,3,1] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 2
[4,3,1,2,5] => [4,3,1,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 2
[4,3,1,5,2] => [4,3,1,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 2
[4,3,2,1,5] => [4,3,2,1] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 2
[4,3,2,5,1] => [4,3,2,1] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 2
[4,3,5,1,2] => [4,3,1,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 2
[4,3,5,2,1] => [4,3,2,1] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 2
[4,5,1,2,3] => [4,1,2,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 3
[4,5,1,3,2] => [4,1,3,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 2
[4,5,2,1,3] => [4,2,1,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 3
[4,5,2,3,1] => [4,2,3,1] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 2
[4,5,3,1,2] => [4,3,1,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 2
[4,5,3,2,1] => [4,3,2,1] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 2
[5,3,1,4,2] => [3,1,4,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> 2
[5,3,2,4,1] => [3,2,4,1] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> 2
[5,4,1,2,3] => [4,1,2,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 3
[5,4,1,3,2] => [4,1,3,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 2
[5,4,2,1,3] => [4,2,1,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 3
[5,4,2,3,1] => [4,2,3,1] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 2
[5,4,3,1,2] => [4,3,1,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 2
[5,4,3,2,1] => [4,3,2,1] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 2
[3,1,4,5,2,6] => [3,1,4,5,2] => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> 3
[3,1,4,5,6,2] => [3,1,4,5,2] => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> 3
[3,1,4,6,5,2] => [3,1,4,5,2] => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> 3
[3,1,5,2,4,6] => [3,1,5,2,4] => [1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[3,1,5,2,6,4] => [3,1,5,2,4] => [1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[3,1,5,4,2,6] => [3,1,5,4,2] => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> 2
[3,1,5,4,6,2] => [3,1,5,4,2] => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> 2
[3,1,5,6,2,4] => [3,1,5,2,4] => [1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[3,1,5,6,4,2] => [3,1,5,4,2] => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> 2
[3,1,6,4,5,2] => [3,1,4,5,2] => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> 3
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: St000259
Mp00064: Permutations —reverse⟶ Permutations
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000259: Graphs ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 33%
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000259: Graphs ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 33%
Values
[3,1,4,2,5] => [5,2,4,1,3] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 2
[3,1,4,5,2] => [2,5,4,1,3] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 2
[3,1,5,4,2] => [2,4,5,1,3] => [5] => ([],5)
=> ? = 2
[3,2,4,1,5] => [5,1,4,2,3] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 2
[3,2,4,5,1] => [1,5,4,2,3] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ? = 2
[3,2,5,4,1] => [1,4,5,2,3] => [1,4] => ([(3,4)],5)
=> ? = 2
[3,5,1,4,2] => [2,4,1,5,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2
[3,5,2,4,1] => [1,4,2,5,3] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2
[4,1,2,3,5] => [5,3,2,1,4] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3
[4,1,2,5,3] => [3,5,2,1,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 3
[4,1,3,2,5] => [5,2,3,1,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2
[4,1,3,5,2] => [2,5,3,1,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2
[4,1,5,2,3] => [3,2,5,1,4] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 3
[4,1,5,3,2] => [2,3,5,1,4] => [5] => ([],5)
=> ? = 2
[4,2,1,3,5] => [5,3,1,2,4] => [1,4] => ([(3,4)],5)
=> ? = 3
[4,2,1,5,3] => [3,5,1,2,4] => [5] => ([],5)
=> ? = 3
[4,2,3,1,5] => [5,1,3,2,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2
[4,2,3,5,1] => [1,5,3,2,4] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2
[4,2,5,1,3] => [3,1,5,2,4] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 3
[4,2,5,3,1] => [1,3,5,2,4] => [1,4] => ([(3,4)],5)
=> ? = 2
[4,3,1,2,5] => [5,2,1,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 2
[4,3,1,5,2] => [2,5,1,3,4] => [5] => ([],5)
=> ? = 2
[4,3,2,1,5] => [5,1,2,3,4] => [5] => ([],5)
=> ? = 2
[4,3,2,5,1] => [1,5,2,3,4] => [1,4] => ([(3,4)],5)
=> ? = 2
[4,3,5,1,2] => [2,1,5,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 2
[4,3,5,2,1] => [1,2,5,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 2
[4,5,1,2,3] => [3,2,1,5,4] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3
[4,5,1,3,2] => [2,3,1,5,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2
[4,5,2,1,3] => [3,1,2,5,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 3
[4,5,2,3,1] => [1,3,2,5,4] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2
[4,5,3,1,2] => [2,1,3,5,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2
[4,5,3,2,1] => [1,2,3,5,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2
[5,3,1,4,2] => [2,4,1,3,5] => [5] => ([],5)
=> ? = 2
[5,3,2,4,1] => [1,4,2,3,5] => [1,4] => ([(3,4)],5)
=> ? = 2
[5,4,1,2,3] => [3,2,1,4,5] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 3
[5,4,1,3,2] => [2,3,1,4,5] => [5] => ([],5)
=> ? = 2
[5,4,2,1,3] => [3,1,2,4,5] => [5] => ([],5)
=> ? = 3
[5,4,2,3,1] => [1,3,2,4,5] => [1,4] => ([(3,4)],5)
=> ? = 2
[5,4,3,1,2] => [2,1,3,4,5] => [5] => ([],5)
=> ? = 2
[5,4,3,2,1] => [1,2,3,4,5] => [5] => ([],5)
=> ? = 2
[3,1,4,5,2,6] => [6,2,5,4,1,3] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
[3,1,4,5,6,2] => [2,6,5,4,1,3] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
[3,1,4,6,5,2] => [2,5,6,4,1,3] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 3
[3,1,5,2,4,6] => [6,4,2,5,1,3] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2
[3,1,5,2,6,4] => [4,6,2,5,1,3] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? = 2
[3,1,5,4,2,6] => [6,2,4,5,1,3] => [2,4] => ([(3,5),(4,5)],6)
=> ? = 2
[3,1,5,4,6,2] => [2,6,4,5,1,3] => [2,4] => ([(3,5),(4,5)],6)
=> ? = 2
[3,1,5,6,2,4] => [4,2,6,5,1,3] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2
[3,1,5,6,4,2] => [2,4,6,5,1,3] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? = 2
[3,1,6,4,5,2] => [2,5,4,6,1,3] => [2,4] => ([(3,5),(4,5)],6)
=> ? = 3
[3,4,5,1,2,6] => [6,2,1,5,4,3] => [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[3,4,5,1,6,2] => [2,6,1,5,4,3] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[3,4,5,2,1,6] => [6,1,2,5,4,3] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[3,4,5,2,6,1] => [1,6,2,5,4,3] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[3,4,5,6,1,2] => [2,1,6,5,4,3] => [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[3,4,5,6,2,1] => [1,2,6,5,4,3] => [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[3,4,6,5,1,2] => [2,1,5,6,4,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[3,4,6,5,2,1] => [1,2,5,6,4,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[4,5,1,2,3,6] => [6,3,2,1,5,4] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[4,5,1,2,6,3] => [3,6,2,1,5,4] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[4,5,1,6,2,3] => [3,2,6,1,5,4] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[4,5,2,1,3,6] => [6,3,1,2,5,4] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[4,5,2,1,6,3] => [3,6,1,2,5,4] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[4,5,2,6,1,3] => [3,1,6,2,5,4] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[4,5,3,1,2,6] => [6,2,1,3,5,4] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[4,5,3,1,6,2] => [2,6,1,3,5,4] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[4,5,3,2,1,6] => [6,1,2,3,5,4] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[4,5,3,2,6,1] => [1,6,2,3,5,4] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[4,5,3,6,1,2] => [2,1,6,3,5,4] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[4,5,3,6,2,1] => [1,2,6,3,5,4] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[4,5,6,1,2,3] => [3,2,1,6,5,4] => [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[4,5,6,2,1,3] => [3,1,2,6,5,4] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[4,5,6,3,1,2] => [2,1,3,6,5,4] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[4,5,6,3,2,1] => [1,2,3,6,5,4] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[3,4,5,1,6,2,7] => [7,2,6,1,5,4,3] => [2,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
[3,4,5,1,6,7,2] => [2,7,6,1,5,4,3] => [2,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
[3,4,5,1,7,6,2] => [2,6,7,1,5,4,3] => [5,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
[3,4,5,2,6,1,7] => [7,1,6,2,5,4,3] => [2,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
[3,4,5,2,6,7,1] => [1,7,6,2,5,4,3] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
[3,4,5,2,7,6,1] => [1,6,7,2,5,4,3] => [1,4,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
[3,4,5,7,1,6,2] => [2,6,1,7,5,4,3] => [3,2,1,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
[3,4,5,7,2,6,1] => [1,6,2,7,5,4,3] => [1,2,2,1,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
[3,4,6,1,2,5,7] => [7,5,2,1,6,4,3] => [1,2,1,2,1] => ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
[3,4,6,1,2,7,5] => [5,7,2,1,6,4,3] => [3,1,2,1] => ([(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
[3,4,6,1,5,2,7] => [7,2,5,1,6,4,3] => [2,2,2,1] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
[3,4,6,1,5,7,2] => [2,7,5,1,6,4,3] => [2,2,2,1] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
[3,4,6,1,7,2,5] => [5,2,7,1,6,4,3] => [2,2,2,1] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
[3,4,6,1,7,5,2] => [2,5,7,1,6,4,3] => [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
[3,4,6,2,1,5,7] => [7,5,1,2,6,4,3] => [1,3,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
[3,4,6,2,1,7,5] => [5,7,1,2,6,4,3] => [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
[3,4,6,2,5,1,7] => [7,1,5,2,6,4,3] => [2,2,2,1] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
[3,4,6,2,5,7,1] => [1,7,5,2,6,4,3] => [1,1,2,2,1] => ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
[3,4,6,2,7,1,5] => [5,1,7,2,6,4,3] => [2,2,2,1] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
[3,4,6,2,7,5,1] => [1,5,7,2,6,4,3] => [1,3,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
[3,4,6,5,1,2,7] => [7,2,1,5,6,4,3] => [2,1,3,1] => ([(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
[3,4,6,5,1,7,2] => [2,7,1,5,6,4,3] => [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
[3,4,6,5,2,1,7] => [7,1,2,5,6,4,3] => [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
[3,4,6,5,2,7,1] => [1,7,2,5,6,4,3] => [1,2,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
[3,4,6,5,7,1,2] => [2,1,7,5,6,4,3] => [2,1,3,1] => ([(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
[3,4,6,5,7,2,1] => [1,2,7,5,6,4,3] => [2,1,3,1] => ([(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
Description
The diameter of a connected graph.
This is the greatest distance between any pair of vertices.
Matching statistic: St000260
Mp00064: Permutations —reverse⟶ Permutations
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 33%
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 33%
Values
[3,1,4,2,5] => [5,2,4,1,3] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 2 - 1
[3,1,4,5,2] => [2,5,4,1,3] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 2 - 1
[3,1,5,4,2] => [2,4,5,1,3] => [5] => ([],5)
=> ? = 2 - 1
[3,2,4,1,5] => [5,1,4,2,3] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 2 - 1
[3,2,4,5,1] => [1,5,4,2,3] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[3,2,5,4,1] => [1,4,5,2,3] => [1,4] => ([(3,4)],5)
=> ? = 2 - 1
[3,5,1,4,2] => [2,4,1,5,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2 - 1
[3,5,2,4,1] => [1,4,2,5,3] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[4,1,2,3,5] => [5,3,2,1,4] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 - 1
[4,1,2,5,3] => [3,5,2,1,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 3 - 1
[4,1,3,2,5] => [5,2,3,1,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2 - 1
[4,1,3,5,2] => [2,5,3,1,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2 - 1
[4,1,5,2,3] => [3,2,5,1,4] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 3 - 1
[4,1,5,3,2] => [2,3,5,1,4] => [5] => ([],5)
=> ? = 2 - 1
[4,2,1,3,5] => [5,3,1,2,4] => [1,4] => ([(3,4)],5)
=> ? = 3 - 1
[4,2,1,5,3] => [3,5,1,2,4] => [5] => ([],5)
=> ? = 3 - 1
[4,2,3,1,5] => [5,1,3,2,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2 - 1
[4,2,3,5,1] => [1,5,3,2,4] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[4,2,5,1,3] => [3,1,5,2,4] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 3 - 1
[4,2,5,3,1] => [1,3,5,2,4] => [1,4] => ([(3,4)],5)
=> ? = 2 - 1
[4,3,1,2,5] => [5,2,1,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 2 - 1
[4,3,1,5,2] => [2,5,1,3,4] => [5] => ([],5)
=> ? = 2 - 1
[4,3,2,1,5] => [5,1,2,3,4] => [5] => ([],5)
=> ? = 2 - 1
[4,3,2,5,1] => [1,5,2,3,4] => [1,4] => ([(3,4)],5)
=> ? = 2 - 1
[4,3,5,1,2] => [2,1,5,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 2 - 1
[4,3,5,2,1] => [1,2,5,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 2 - 1
[4,5,1,2,3] => [3,2,1,5,4] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 - 1
[4,5,1,3,2] => [2,3,1,5,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2 - 1
[4,5,2,1,3] => [3,1,2,5,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 3 - 1
[4,5,2,3,1] => [1,3,2,5,4] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[4,5,3,1,2] => [2,1,3,5,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2 - 1
[4,5,3,2,1] => [1,2,3,5,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2 - 1
[5,3,1,4,2] => [2,4,1,3,5] => [5] => ([],5)
=> ? = 2 - 1
[5,3,2,4,1] => [1,4,2,3,5] => [1,4] => ([(3,4)],5)
=> ? = 2 - 1
[5,4,1,2,3] => [3,2,1,4,5] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 3 - 1
[5,4,1,3,2] => [2,3,1,4,5] => [5] => ([],5)
=> ? = 2 - 1
[5,4,2,1,3] => [3,1,2,4,5] => [5] => ([],5)
=> ? = 3 - 1
[5,4,2,3,1] => [1,3,2,4,5] => [1,4] => ([(3,4)],5)
=> ? = 2 - 1
[5,4,3,1,2] => [2,1,3,4,5] => [5] => ([],5)
=> ? = 2 - 1
[5,4,3,2,1] => [1,2,3,4,5] => [5] => ([],5)
=> ? = 2 - 1
[3,1,4,5,2,6] => [6,2,5,4,1,3] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 1
[3,1,4,5,6,2] => [2,6,5,4,1,3] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 1
[3,1,4,6,5,2] => [2,5,6,4,1,3] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 3 - 1
[3,1,5,2,4,6] => [6,4,2,5,1,3] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
[3,1,5,2,6,4] => [4,6,2,5,1,3] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? = 2 - 1
[3,1,5,4,2,6] => [6,2,4,5,1,3] => [2,4] => ([(3,5),(4,5)],6)
=> ? = 2 - 1
[3,1,5,4,6,2] => [2,6,4,5,1,3] => [2,4] => ([(3,5),(4,5)],6)
=> ? = 2 - 1
[3,1,5,6,2,4] => [4,2,6,5,1,3] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
[3,1,5,6,4,2] => [2,4,6,5,1,3] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? = 2 - 1
[3,1,6,4,5,2] => [2,5,4,6,1,3] => [2,4] => ([(3,5),(4,5)],6)
=> ? = 3 - 1
[3,4,5,1,2,6] => [6,2,1,5,4,3] => [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[3,4,5,1,6,2] => [2,6,1,5,4,3] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[3,4,5,2,1,6] => [6,1,2,5,4,3] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[3,4,5,2,6,1] => [1,6,2,5,4,3] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[3,4,5,6,1,2] => [2,1,6,5,4,3] => [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[3,4,5,6,2,1] => [1,2,6,5,4,3] => [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[3,4,6,5,1,2] => [2,1,5,6,4,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[3,4,6,5,2,1] => [1,2,5,6,4,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[4,5,1,2,3,6] => [6,3,2,1,5,4] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[4,5,1,2,6,3] => [3,6,2,1,5,4] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[4,5,1,6,2,3] => [3,2,6,1,5,4] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[4,5,2,1,3,6] => [6,3,1,2,5,4] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[4,5,2,1,6,3] => [3,6,1,2,5,4] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 1 = 2 - 1
[4,5,2,6,1,3] => [3,1,6,2,5,4] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[4,5,3,1,2,6] => [6,2,1,3,5,4] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[4,5,3,1,6,2] => [2,6,1,3,5,4] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 1 = 2 - 1
[4,5,3,2,1,6] => [6,1,2,3,5,4] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 1 = 2 - 1
[4,5,3,2,6,1] => [1,6,2,3,5,4] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[4,5,3,6,1,2] => [2,1,6,3,5,4] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[4,5,3,6,2,1] => [1,2,6,3,5,4] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[4,5,6,1,2,3] => [3,2,1,6,5,4] => [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[4,5,6,2,1,3] => [3,1,2,6,5,4] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[4,5,6,3,1,2] => [2,1,3,6,5,4] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[4,5,6,3,2,1] => [1,2,3,6,5,4] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[3,4,5,1,6,2,7] => [7,2,6,1,5,4,3] => [2,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1 = 2 - 1
[3,4,5,1,6,7,2] => [2,7,6,1,5,4,3] => [2,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1 = 2 - 1
[3,4,5,1,7,6,2] => [2,6,7,1,5,4,3] => [5,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1 = 2 - 1
[3,4,5,2,6,1,7] => [7,1,6,2,5,4,3] => [2,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1 = 2 - 1
[3,4,5,2,6,7,1] => [1,7,6,2,5,4,3] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1 = 2 - 1
[3,4,5,2,7,6,1] => [1,6,7,2,5,4,3] => [1,4,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1 = 2 - 1
[3,4,5,7,1,6,2] => [2,6,1,7,5,4,3] => [3,2,1,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1 = 2 - 1
[3,4,5,7,2,6,1] => [1,6,2,7,5,4,3] => [1,2,2,1,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1 = 2 - 1
[3,4,6,1,2,5,7] => [7,5,2,1,6,4,3] => [1,2,1,2,1] => ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1 = 2 - 1
[3,4,6,1,2,7,5] => [5,7,2,1,6,4,3] => [3,1,2,1] => ([(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1 = 2 - 1
[3,4,6,1,5,2,7] => [7,2,5,1,6,4,3] => [2,2,2,1] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1 = 2 - 1
[3,4,6,1,5,7,2] => [2,7,5,1,6,4,3] => [2,2,2,1] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1 = 2 - 1
[3,4,6,1,7,2,5] => [5,2,7,1,6,4,3] => [2,2,2,1] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1 = 2 - 1
[3,4,6,1,7,5,2] => [2,5,7,1,6,4,3] => [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1 = 2 - 1
[3,4,6,2,1,5,7] => [7,5,1,2,6,4,3] => [1,3,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1 = 2 - 1
[3,4,6,2,1,7,5] => [5,7,1,2,6,4,3] => [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1 = 2 - 1
[3,4,6,2,5,1,7] => [7,1,5,2,6,4,3] => [2,2,2,1] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1 = 2 - 1
[3,4,6,2,5,7,1] => [1,7,5,2,6,4,3] => [1,1,2,2,1] => ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1 = 2 - 1
[3,4,6,2,7,1,5] => [5,1,7,2,6,4,3] => [2,2,2,1] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1 = 2 - 1
[3,4,6,2,7,5,1] => [1,5,7,2,6,4,3] => [1,3,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1 = 2 - 1
[3,4,6,5,1,2,7] => [7,2,1,5,6,4,3] => [2,1,3,1] => ([(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1 = 2 - 1
[3,4,6,5,1,7,2] => [2,7,1,5,6,4,3] => [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1 = 2 - 1
[3,4,6,5,2,1,7] => [7,1,2,5,6,4,3] => [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1 = 2 - 1
[3,4,6,5,2,7,1] => [1,7,2,5,6,4,3] => [1,2,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1 = 2 - 1
[3,4,6,5,7,1,2] => [2,1,7,5,6,4,3] => [2,1,3,1] => ([(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1 = 2 - 1
[3,4,6,5,7,2,1] => [1,2,7,5,6,4,3] => [2,1,3,1] => ([(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1 = 2 - 1
Description
The radius of a connected graph.
This is the minimum eccentricity of any vertex.
Matching statistic: St001198
Mp00252: Permutations —restriction⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001198: Dyck paths ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 33%
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001198: Dyck paths ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 33%
Values
[3,1,4,2,5] => [3,1,4,2] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? = 2
[3,1,4,5,2] => [3,1,4,2] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? = 2
[3,1,5,4,2] => [3,1,4,2] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? = 2
[3,2,4,1,5] => [3,2,4,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? = 2
[3,2,4,5,1] => [3,2,4,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? = 2
[3,2,5,4,1] => [3,2,4,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? = 2
[3,5,1,4,2] => [3,1,4,2] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? = 2
[3,5,2,4,1] => [3,2,4,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? = 2
[4,1,2,3,5] => [4,1,2,3] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? = 3
[4,1,2,5,3] => [4,1,2,3] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? = 3
[4,1,3,2,5] => [4,1,3,2] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? = 2
[4,1,3,5,2] => [4,1,3,2] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? = 2
[4,1,5,2,3] => [4,1,2,3] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? = 3
[4,1,5,3,2] => [4,1,3,2] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? = 2
[4,2,1,3,5] => [4,2,1,3] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 3
[4,2,1,5,3] => [4,2,1,3] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 3
[4,2,3,1,5] => [4,2,3,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 2
[4,2,3,5,1] => [4,2,3,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 2
[4,2,5,1,3] => [4,2,1,3] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 3
[4,2,5,3,1] => [4,2,3,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 2
[4,3,1,2,5] => [4,3,1,2] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 2
[4,3,1,5,2] => [4,3,1,2] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 2
[4,3,2,1,5] => [4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 2
[4,3,2,5,1] => [4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 2
[4,3,5,1,2] => [4,3,1,2] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 2
[4,3,5,2,1] => [4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 2
[4,5,1,2,3] => [4,1,2,3] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? = 3
[4,5,1,3,2] => [4,1,3,2] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? = 2
[4,5,2,1,3] => [4,2,1,3] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 3
[4,5,2,3,1] => [4,2,3,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 2
[4,5,3,1,2] => [4,3,1,2] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 2
[4,5,3,2,1] => [4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 2
[5,3,1,4,2] => [3,1,4,2] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? = 2
[5,3,2,4,1] => [3,2,4,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? = 2
[5,4,1,2,3] => [4,1,2,3] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? = 3
[5,4,1,3,2] => [4,1,3,2] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? = 2
[5,4,2,1,3] => [4,2,1,3] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 3
[5,4,2,3,1] => [4,2,3,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 2
[5,4,3,1,2] => [4,3,1,2] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 2
[5,4,3,2,1] => [4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 2
[3,1,4,5,2,6] => [3,1,4,5,2] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 3
[3,1,4,5,6,2] => [3,1,4,5,2] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 3
[3,1,4,6,5,2] => [3,1,4,5,2] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 3
[3,1,5,2,4,6] => [3,1,5,2,4] => [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> 2
[3,1,5,2,6,4] => [3,1,5,2,4] => [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> 2
[3,1,5,4,2,6] => [3,1,5,4,2] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 2
[3,1,5,4,6,2] => [3,1,5,4,2] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 2
[3,1,5,6,2,4] => [3,1,5,2,4] => [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> 2
[3,1,5,6,4,2] => [3,1,5,4,2] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 2
[3,1,6,4,5,2] => [3,1,4,5,2] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 3
[3,1,6,5,2,4] => [3,1,5,2,4] => [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> 2
[3,1,6,5,4,2] => [3,1,5,4,2] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 2
[3,2,4,5,1,6] => [3,2,4,5,1] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 3
[3,2,4,5,6,1] => [3,2,4,5,1] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 3
[3,2,5,1,4,6] => [3,2,5,1,4] => [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> 2
[3,2,5,1,6,4] => [3,2,5,1,4] => [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> 2
[3,2,5,6,1,4] => [3,2,5,1,4] => [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> 2
[3,2,6,5,1,4] => [3,2,5,1,4] => [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> 2
[3,6,1,5,2,4] => [3,1,5,2,4] => [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> 2
[3,6,2,5,1,4] => [3,2,5,1,4] => [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> 2
[6,3,1,5,2,4] => [3,1,5,2,4] => [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> 2
[6,3,2,5,1,4] => [3,2,5,1,4] => [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> 2
[3,1,4,6,2,5,7] => [3,1,4,6,2,5] => [5,2,3,6,1,4] => [1,1,1,1,1,0,0,0,1,0,0,0]
=> 2
[3,1,4,6,2,7,5] => [3,1,4,6,2,5] => [5,2,3,6,1,4] => [1,1,1,1,1,0,0,0,1,0,0,0]
=> 2
[3,1,4,6,7,2,5] => [3,1,4,6,2,5] => [5,2,3,6,1,4] => [1,1,1,1,1,0,0,0,1,0,0,0]
=> 2
[3,1,4,7,6,2,5] => [3,1,4,6,2,5] => [5,2,3,6,1,4] => [1,1,1,1,1,0,0,0,1,0,0,0]
=> 2
[3,1,5,2,6,4,7] => [3,1,5,2,6,4] => [4,2,6,1,5,3] => [1,1,1,1,0,0,1,1,0,0,0,0]
=> 2
[3,1,5,2,6,7,4] => [3,1,5,2,6,4] => [4,2,6,1,5,3] => [1,1,1,1,0,0,1,1,0,0,0,0]
=> 2
[3,1,5,2,7,6,4] => [3,1,5,2,6,4] => [4,2,6,1,5,3] => [1,1,1,1,0,0,1,1,0,0,0,0]
=> 2
[3,1,5,7,2,6,4] => [3,1,5,2,6,4] => [4,2,6,1,5,3] => [1,1,1,1,0,0,1,1,0,0,0,0]
=> 2
[3,1,6,2,4,5,7] => [3,1,6,2,4,5] => [4,2,6,1,5,3] => [1,1,1,1,0,0,1,1,0,0,0,0]
=> 2
[3,1,6,2,4,7,5] => [3,1,6,2,4,5] => [4,2,6,1,5,3] => [1,1,1,1,0,0,1,1,0,0,0,0]
=> 2
[3,1,6,2,5,4,7] => [3,1,6,2,5,4] => [4,2,6,1,5,3] => [1,1,1,1,0,0,1,1,0,0,0,0]
=> 2
[3,1,6,2,5,7,4] => [3,1,6,2,5,4] => [4,2,6,1,5,3] => [1,1,1,1,0,0,1,1,0,0,0,0]
=> 2
[3,1,6,2,7,4,5] => [3,1,6,2,4,5] => [4,2,6,1,5,3] => [1,1,1,1,0,0,1,1,0,0,0,0]
=> 2
[3,1,6,2,7,5,4] => [3,1,6,2,5,4] => [4,2,6,1,5,3] => [1,1,1,1,0,0,1,1,0,0,0,0]
=> 2
[3,1,6,4,2,5,7] => [3,1,6,4,2,5] => [5,2,6,4,1,3] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> 2
[3,1,6,4,2,7,5] => [3,1,6,4,2,5] => [5,2,6,4,1,3] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> 2
[3,1,6,4,7,2,5] => [3,1,6,4,2,5] => [5,2,6,4,1,3] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> 2
[3,1,6,5,2,4,7] => [3,1,6,5,2,4] => [5,2,6,4,1,3] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> 2
[3,1,6,5,2,7,4] => [3,1,6,5,2,4] => [5,2,6,4,1,3] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> 2
[3,1,6,5,7,2,4] => [3,1,6,5,2,4] => [5,2,6,4,1,3] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> 2
[3,1,6,7,2,4,5] => [3,1,6,2,4,5] => [4,2,6,1,5,3] => [1,1,1,1,0,0,1,1,0,0,0,0]
=> 2
[3,1,6,7,2,5,4] => [3,1,6,2,5,4] => [4,2,6,1,5,3] => [1,1,1,1,0,0,1,1,0,0,0,0]
=> 2
[3,1,6,7,4,2,5] => [3,1,6,4,2,5] => [5,2,6,4,1,3] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> 2
[3,1,6,7,5,2,4] => [3,1,6,5,2,4] => [5,2,6,4,1,3] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> 2
[3,1,7,4,6,2,5] => [3,1,4,6,2,5] => [5,2,3,6,1,4] => [1,1,1,1,1,0,0,0,1,0,0,0]
=> 2
[3,1,7,5,2,6,4] => [3,1,5,2,6,4] => [4,2,6,1,5,3] => [1,1,1,1,0,0,1,1,0,0,0,0]
=> 2
[3,1,7,6,2,4,5] => [3,1,6,2,4,5] => [4,2,6,1,5,3] => [1,1,1,1,0,0,1,1,0,0,0,0]
=> 2
[3,1,7,6,2,5,4] => [3,1,6,2,5,4] => [4,2,6,1,5,3] => [1,1,1,1,0,0,1,1,0,0,0,0]
=> 2
[3,1,7,6,4,2,5] => [3,1,6,4,2,5] => [5,2,6,4,1,3] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> 2
[3,1,7,6,5,2,4] => [3,1,6,5,2,4] => [5,2,6,4,1,3] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> 2
[3,2,4,6,1,5,7] => [3,2,4,6,1,5] => [5,2,3,6,1,4] => [1,1,1,1,1,0,0,0,1,0,0,0]
=> 2
[3,2,4,6,1,7,5] => [3,2,4,6,1,5] => [5,2,3,6,1,4] => [1,1,1,1,1,0,0,0,1,0,0,0]
=> 2
[3,2,4,6,7,1,5] => [3,2,4,6,1,5] => [5,2,3,6,1,4] => [1,1,1,1,1,0,0,0,1,0,0,0]
=> 2
[3,2,4,7,6,1,5] => [3,2,4,6,1,5] => [5,2,3,6,1,4] => [1,1,1,1,1,0,0,0,1,0,0,0]
=> 2
[3,2,5,1,6,4,7] => [3,2,5,1,6,4] => [4,2,6,1,5,3] => [1,1,1,1,0,0,1,1,0,0,0,0]
=> 2
[3,2,5,1,6,7,4] => [3,2,5,1,6,4] => [4,2,6,1,5,3] => [1,1,1,1,0,0,1,1,0,0,0,0]
=> 2
[3,2,5,1,7,6,4] => [3,2,5,1,6,4] => [4,2,6,1,5,3] => [1,1,1,1,0,0,1,1,0,0,0,0]
=> 2
[3,2,5,7,1,6,4] => [3,2,5,1,6,4] => [4,2,6,1,5,3] => [1,1,1,1,0,0,1,1,0,0,0,0]
=> 2
Description
The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$.
Matching statistic: St001206
Mp00252: Permutations —restriction⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001206: Dyck paths ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 33%
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001206: Dyck paths ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 33%
Values
[3,1,4,2,5] => [3,1,4,2] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? = 2
[3,1,4,5,2] => [3,1,4,2] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? = 2
[3,1,5,4,2] => [3,1,4,2] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? = 2
[3,2,4,1,5] => [3,2,4,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? = 2
[3,2,4,5,1] => [3,2,4,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? = 2
[3,2,5,4,1] => [3,2,4,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? = 2
[3,5,1,4,2] => [3,1,4,2] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? = 2
[3,5,2,4,1] => [3,2,4,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? = 2
[4,1,2,3,5] => [4,1,2,3] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? = 3
[4,1,2,5,3] => [4,1,2,3] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? = 3
[4,1,3,2,5] => [4,1,3,2] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? = 2
[4,1,3,5,2] => [4,1,3,2] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? = 2
[4,1,5,2,3] => [4,1,2,3] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? = 3
[4,1,5,3,2] => [4,1,3,2] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? = 2
[4,2,1,3,5] => [4,2,1,3] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 3
[4,2,1,5,3] => [4,2,1,3] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 3
[4,2,3,1,5] => [4,2,3,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 2
[4,2,3,5,1] => [4,2,3,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 2
[4,2,5,1,3] => [4,2,1,3] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 3
[4,2,5,3,1] => [4,2,3,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 2
[4,3,1,2,5] => [4,3,1,2] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 2
[4,3,1,5,2] => [4,3,1,2] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 2
[4,3,2,1,5] => [4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 2
[4,3,2,5,1] => [4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 2
[4,3,5,1,2] => [4,3,1,2] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 2
[4,3,5,2,1] => [4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 2
[4,5,1,2,3] => [4,1,2,3] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? = 3
[4,5,1,3,2] => [4,1,3,2] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? = 2
[4,5,2,1,3] => [4,2,1,3] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 3
[4,5,2,3,1] => [4,2,3,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 2
[4,5,3,1,2] => [4,3,1,2] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 2
[4,5,3,2,1] => [4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 2
[5,3,1,4,2] => [3,1,4,2] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? = 2
[5,3,2,4,1] => [3,2,4,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? = 2
[5,4,1,2,3] => [4,1,2,3] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? = 3
[5,4,1,3,2] => [4,1,3,2] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? = 2
[5,4,2,1,3] => [4,2,1,3] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 3
[5,4,2,3,1] => [4,2,3,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 2
[5,4,3,1,2] => [4,3,1,2] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 2
[5,4,3,2,1] => [4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 2
[3,1,4,5,2,6] => [3,1,4,5,2] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 3
[3,1,4,5,6,2] => [3,1,4,5,2] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 3
[3,1,4,6,5,2] => [3,1,4,5,2] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 3
[3,1,5,2,4,6] => [3,1,5,2,4] => [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> 2
[3,1,5,2,6,4] => [3,1,5,2,4] => [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> 2
[3,1,5,4,2,6] => [3,1,5,4,2] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 2
[3,1,5,4,6,2] => [3,1,5,4,2] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 2
[3,1,5,6,2,4] => [3,1,5,2,4] => [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> 2
[3,1,5,6,4,2] => [3,1,5,4,2] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 2
[3,1,6,4,5,2] => [3,1,4,5,2] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 3
[3,1,6,5,2,4] => [3,1,5,2,4] => [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> 2
[3,1,6,5,4,2] => [3,1,5,4,2] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 2
[3,2,4,5,1,6] => [3,2,4,5,1] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 3
[3,2,4,5,6,1] => [3,2,4,5,1] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 3
[3,2,5,1,4,6] => [3,2,5,1,4] => [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> 2
[3,2,5,1,6,4] => [3,2,5,1,4] => [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> 2
[3,2,5,6,1,4] => [3,2,5,1,4] => [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> 2
[3,2,6,5,1,4] => [3,2,5,1,4] => [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> 2
[3,6,1,5,2,4] => [3,1,5,2,4] => [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> 2
[3,6,2,5,1,4] => [3,2,5,1,4] => [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> 2
[6,3,1,5,2,4] => [3,1,5,2,4] => [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> 2
[6,3,2,5,1,4] => [3,2,5,1,4] => [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> 2
[3,1,4,6,2,5,7] => [3,1,4,6,2,5] => [5,2,3,6,1,4] => [1,1,1,1,1,0,0,0,1,0,0,0]
=> 2
[3,1,4,6,2,7,5] => [3,1,4,6,2,5] => [5,2,3,6,1,4] => [1,1,1,1,1,0,0,0,1,0,0,0]
=> 2
[3,1,4,6,7,2,5] => [3,1,4,6,2,5] => [5,2,3,6,1,4] => [1,1,1,1,1,0,0,0,1,0,0,0]
=> 2
[3,1,4,7,6,2,5] => [3,1,4,6,2,5] => [5,2,3,6,1,4] => [1,1,1,1,1,0,0,0,1,0,0,0]
=> 2
[3,1,5,2,6,4,7] => [3,1,5,2,6,4] => [4,2,6,1,5,3] => [1,1,1,1,0,0,1,1,0,0,0,0]
=> 2
[3,1,5,2,6,7,4] => [3,1,5,2,6,4] => [4,2,6,1,5,3] => [1,1,1,1,0,0,1,1,0,0,0,0]
=> 2
[3,1,5,2,7,6,4] => [3,1,5,2,6,4] => [4,2,6,1,5,3] => [1,1,1,1,0,0,1,1,0,0,0,0]
=> 2
[3,1,5,7,2,6,4] => [3,1,5,2,6,4] => [4,2,6,1,5,3] => [1,1,1,1,0,0,1,1,0,0,0,0]
=> 2
[3,1,6,2,4,5,7] => [3,1,6,2,4,5] => [4,2,6,1,5,3] => [1,1,1,1,0,0,1,1,0,0,0,0]
=> 2
[3,1,6,2,4,7,5] => [3,1,6,2,4,5] => [4,2,6,1,5,3] => [1,1,1,1,0,0,1,1,0,0,0,0]
=> 2
[3,1,6,2,5,4,7] => [3,1,6,2,5,4] => [4,2,6,1,5,3] => [1,1,1,1,0,0,1,1,0,0,0,0]
=> 2
[3,1,6,2,5,7,4] => [3,1,6,2,5,4] => [4,2,6,1,5,3] => [1,1,1,1,0,0,1,1,0,0,0,0]
=> 2
[3,1,6,2,7,4,5] => [3,1,6,2,4,5] => [4,2,6,1,5,3] => [1,1,1,1,0,0,1,1,0,0,0,0]
=> 2
[3,1,6,2,7,5,4] => [3,1,6,2,5,4] => [4,2,6,1,5,3] => [1,1,1,1,0,0,1,1,0,0,0,0]
=> 2
[3,1,6,4,2,5,7] => [3,1,6,4,2,5] => [5,2,6,4,1,3] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> 2
[3,1,6,4,2,7,5] => [3,1,6,4,2,5] => [5,2,6,4,1,3] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> 2
[3,1,6,4,7,2,5] => [3,1,6,4,2,5] => [5,2,6,4,1,3] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> 2
[3,1,6,5,2,4,7] => [3,1,6,5,2,4] => [5,2,6,4,1,3] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> 2
[3,1,6,5,2,7,4] => [3,1,6,5,2,4] => [5,2,6,4,1,3] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> 2
[3,1,6,5,7,2,4] => [3,1,6,5,2,4] => [5,2,6,4,1,3] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> 2
[3,1,6,7,2,4,5] => [3,1,6,2,4,5] => [4,2,6,1,5,3] => [1,1,1,1,0,0,1,1,0,0,0,0]
=> 2
[3,1,6,7,2,5,4] => [3,1,6,2,5,4] => [4,2,6,1,5,3] => [1,1,1,1,0,0,1,1,0,0,0,0]
=> 2
[3,1,6,7,4,2,5] => [3,1,6,4,2,5] => [5,2,6,4,1,3] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> 2
[3,1,6,7,5,2,4] => [3,1,6,5,2,4] => [5,2,6,4,1,3] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> 2
[3,1,7,4,6,2,5] => [3,1,4,6,2,5] => [5,2,3,6,1,4] => [1,1,1,1,1,0,0,0,1,0,0,0]
=> 2
[3,1,7,5,2,6,4] => [3,1,5,2,6,4] => [4,2,6,1,5,3] => [1,1,1,1,0,0,1,1,0,0,0,0]
=> 2
[3,1,7,6,2,4,5] => [3,1,6,2,4,5] => [4,2,6,1,5,3] => [1,1,1,1,0,0,1,1,0,0,0,0]
=> 2
[3,1,7,6,2,5,4] => [3,1,6,2,5,4] => [4,2,6,1,5,3] => [1,1,1,1,0,0,1,1,0,0,0,0]
=> 2
[3,1,7,6,4,2,5] => [3,1,6,4,2,5] => [5,2,6,4,1,3] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> 2
[3,1,7,6,5,2,4] => [3,1,6,5,2,4] => [5,2,6,4,1,3] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> 2
[3,2,4,6,1,5,7] => [3,2,4,6,1,5] => [5,2,3,6,1,4] => [1,1,1,1,1,0,0,0,1,0,0,0]
=> 2
[3,2,4,6,1,7,5] => [3,2,4,6,1,5] => [5,2,3,6,1,4] => [1,1,1,1,1,0,0,0,1,0,0,0]
=> 2
[3,2,4,6,7,1,5] => [3,2,4,6,1,5] => [5,2,3,6,1,4] => [1,1,1,1,1,0,0,0,1,0,0,0]
=> 2
[3,2,4,7,6,1,5] => [3,2,4,6,1,5] => [5,2,3,6,1,4] => [1,1,1,1,1,0,0,0,1,0,0,0]
=> 2
[3,2,5,1,6,4,7] => [3,2,5,1,6,4] => [4,2,6,1,5,3] => [1,1,1,1,0,0,1,1,0,0,0,0]
=> 2
[3,2,5,1,6,7,4] => [3,2,5,1,6,4] => [4,2,6,1,5,3] => [1,1,1,1,0,0,1,1,0,0,0,0]
=> 2
[3,2,5,1,7,6,4] => [3,2,5,1,6,4] => [4,2,6,1,5,3] => [1,1,1,1,0,0,1,1,0,0,0,0]
=> 2
[3,2,5,7,1,6,4] => [3,2,5,1,6,4] => [4,2,6,1,5,3] => [1,1,1,1,0,0,1,1,0,0,0,0]
=> 2
Description
The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$.
Matching statistic: St001199
Mp00252: Permutations —restriction⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001199: Dyck paths ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 33%
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001199: Dyck paths ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 33%
Values
[3,1,4,2,5] => [3,1,4,2] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? = 2 - 1
[3,1,4,5,2] => [3,1,4,2] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? = 2 - 1
[3,1,5,4,2] => [3,1,4,2] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? = 2 - 1
[3,2,4,1,5] => [3,2,4,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? = 2 - 1
[3,2,4,5,1] => [3,2,4,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? = 2 - 1
[3,2,5,4,1] => [3,2,4,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? = 2 - 1
[3,5,1,4,2] => [3,1,4,2] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? = 2 - 1
[3,5,2,4,1] => [3,2,4,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? = 2 - 1
[4,1,2,3,5] => [4,1,2,3] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? = 3 - 1
[4,1,2,5,3] => [4,1,2,3] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? = 3 - 1
[4,1,3,2,5] => [4,1,3,2] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? = 2 - 1
[4,1,3,5,2] => [4,1,3,2] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? = 2 - 1
[4,1,5,2,3] => [4,1,2,3] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? = 3 - 1
[4,1,5,3,2] => [4,1,3,2] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? = 2 - 1
[4,2,1,3,5] => [4,2,1,3] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 3 - 1
[4,2,1,5,3] => [4,2,1,3] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 3 - 1
[4,2,3,1,5] => [4,2,3,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 2 - 1
[4,2,3,5,1] => [4,2,3,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 2 - 1
[4,2,5,1,3] => [4,2,1,3] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 3 - 1
[4,2,5,3,1] => [4,2,3,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 2 - 1
[4,3,1,2,5] => [4,3,1,2] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 2 - 1
[4,3,1,5,2] => [4,3,1,2] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 2 - 1
[4,3,2,1,5] => [4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 2 - 1
[4,3,2,5,1] => [4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 2 - 1
[4,3,5,1,2] => [4,3,1,2] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 2 - 1
[4,3,5,2,1] => [4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 2 - 1
[4,5,1,2,3] => [4,1,2,3] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? = 3 - 1
[4,5,1,3,2] => [4,1,3,2] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? = 2 - 1
[4,5,2,1,3] => [4,2,1,3] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 3 - 1
[4,5,2,3,1] => [4,2,3,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 2 - 1
[4,5,3,1,2] => [4,3,1,2] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 2 - 1
[4,5,3,2,1] => [4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 2 - 1
[5,3,1,4,2] => [3,1,4,2] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? = 2 - 1
[5,3,2,4,1] => [3,2,4,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? = 2 - 1
[5,4,1,2,3] => [4,1,2,3] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? = 3 - 1
[5,4,1,3,2] => [4,1,3,2] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? = 2 - 1
[5,4,2,1,3] => [4,2,1,3] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 3 - 1
[5,4,2,3,1] => [4,2,3,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 2 - 1
[5,4,3,1,2] => [4,3,1,2] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 2 - 1
[5,4,3,2,1] => [4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 2 - 1
[3,1,4,5,2,6] => [3,1,4,5,2] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 3 - 1
[3,1,4,5,6,2] => [3,1,4,5,2] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 3 - 1
[3,1,4,6,5,2] => [3,1,4,5,2] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 3 - 1
[3,1,5,2,4,6] => [3,1,5,2,4] => [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> 1 = 2 - 1
[3,1,5,2,6,4] => [3,1,5,2,4] => [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> 1 = 2 - 1
[3,1,5,4,2,6] => [3,1,5,4,2] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 2 - 1
[3,1,5,4,6,2] => [3,1,5,4,2] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 2 - 1
[3,1,5,6,2,4] => [3,1,5,2,4] => [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> 1 = 2 - 1
[3,1,5,6,4,2] => [3,1,5,4,2] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 2 - 1
[3,1,6,4,5,2] => [3,1,4,5,2] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 3 - 1
[3,1,6,5,2,4] => [3,1,5,2,4] => [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> 1 = 2 - 1
[3,1,6,5,4,2] => [3,1,5,4,2] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 2 - 1
[3,2,4,5,1,6] => [3,2,4,5,1] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 3 - 1
[3,2,4,5,6,1] => [3,2,4,5,1] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 3 - 1
[3,2,5,1,4,6] => [3,2,5,1,4] => [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> 1 = 2 - 1
[3,2,5,1,6,4] => [3,2,5,1,4] => [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> 1 = 2 - 1
[3,2,5,6,1,4] => [3,2,5,1,4] => [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> 1 = 2 - 1
[3,2,6,5,1,4] => [3,2,5,1,4] => [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> 1 = 2 - 1
[3,6,1,5,2,4] => [3,1,5,2,4] => [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> 1 = 2 - 1
[3,6,2,5,1,4] => [3,2,5,1,4] => [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> 1 = 2 - 1
[6,3,1,5,2,4] => [3,1,5,2,4] => [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> 1 = 2 - 1
[6,3,2,5,1,4] => [3,2,5,1,4] => [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> 1 = 2 - 1
[3,1,4,6,2,5,7] => [3,1,4,6,2,5] => [5,2,3,6,1,4] => [1,1,1,1,1,0,0,0,1,0,0,0]
=> 1 = 2 - 1
[3,1,4,6,2,7,5] => [3,1,4,6,2,5] => [5,2,3,6,1,4] => [1,1,1,1,1,0,0,0,1,0,0,0]
=> 1 = 2 - 1
[3,1,4,6,7,2,5] => [3,1,4,6,2,5] => [5,2,3,6,1,4] => [1,1,1,1,1,0,0,0,1,0,0,0]
=> 1 = 2 - 1
[3,1,4,7,6,2,5] => [3,1,4,6,2,5] => [5,2,3,6,1,4] => [1,1,1,1,1,0,0,0,1,0,0,0]
=> 1 = 2 - 1
[3,1,5,2,6,4,7] => [3,1,5,2,6,4] => [4,2,6,1,5,3] => [1,1,1,1,0,0,1,1,0,0,0,0]
=> 1 = 2 - 1
[3,1,5,2,6,7,4] => [3,1,5,2,6,4] => [4,2,6,1,5,3] => [1,1,1,1,0,0,1,1,0,0,0,0]
=> 1 = 2 - 1
[3,1,5,2,7,6,4] => [3,1,5,2,6,4] => [4,2,6,1,5,3] => [1,1,1,1,0,0,1,1,0,0,0,0]
=> 1 = 2 - 1
[3,1,5,7,2,6,4] => [3,1,5,2,6,4] => [4,2,6,1,5,3] => [1,1,1,1,0,0,1,1,0,0,0,0]
=> 1 = 2 - 1
[3,1,6,2,4,5,7] => [3,1,6,2,4,5] => [4,2,6,1,5,3] => [1,1,1,1,0,0,1,1,0,0,0,0]
=> 1 = 2 - 1
[3,1,6,2,4,7,5] => [3,1,6,2,4,5] => [4,2,6,1,5,3] => [1,1,1,1,0,0,1,1,0,0,0,0]
=> 1 = 2 - 1
[3,1,6,2,5,4,7] => [3,1,6,2,5,4] => [4,2,6,1,5,3] => [1,1,1,1,0,0,1,1,0,0,0,0]
=> 1 = 2 - 1
[3,1,6,2,5,7,4] => [3,1,6,2,5,4] => [4,2,6,1,5,3] => [1,1,1,1,0,0,1,1,0,0,0,0]
=> 1 = 2 - 1
[3,1,6,2,7,4,5] => [3,1,6,2,4,5] => [4,2,6,1,5,3] => [1,1,1,1,0,0,1,1,0,0,0,0]
=> 1 = 2 - 1
[3,1,6,2,7,5,4] => [3,1,6,2,5,4] => [4,2,6,1,5,3] => [1,1,1,1,0,0,1,1,0,0,0,0]
=> 1 = 2 - 1
[3,1,6,4,2,5,7] => [3,1,6,4,2,5] => [5,2,6,4,1,3] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> 1 = 2 - 1
[3,1,6,4,2,7,5] => [3,1,6,4,2,5] => [5,2,6,4,1,3] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> 1 = 2 - 1
[3,1,6,4,7,2,5] => [3,1,6,4,2,5] => [5,2,6,4,1,3] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> 1 = 2 - 1
[3,1,6,5,2,4,7] => [3,1,6,5,2,4] => [5,2,6,4,1,3] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> 1 = 2 - 1
[3,1,6,5,2,7,4] => [3,1,6,5,2,4] => [5,2,6,4,1,3] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> 1 = 2 - 1
[3,1,6,5,7,2,4] => [3,1,6,5,2,4] => [5,2,6,4,1,3] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> 1 = 2 - 1
[3,1,6,7,2,4,5] => [3,1,6,2,4,5] => [4,2,6,1,5,3] => [1,1,1,1,0,0,1,1,0,0,0,0]
=> 1 = 2 - 1
[3,1,6,7,2,5,4] => [3,1,6,2,5,4] => [4,2,6,1,5,3] => [1,1,1,1,0,0,1,1,0,0,0,0]
=> 1 = 2 - 1
[3,1,6,7,4,2,5] => [3,1,6,4,2,5] => [5,2,6,4,1,3] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> 1 = 2 - 1
[3,1,6,7,5,2,4] => [3,1,6,5,2,4] => [5,2,6,4,1,3] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> 1 = 2 - 1
[3,1,7,4,6,2,5] => [3,1,4,6,2,5] => [5,2,3,6,1,4] => [1,1,1,1,1,0,0,0,1,0,0,0]
=> 1 = 2 - 1
[3,1,7,5,2,6,4] => [3,1,5,2,6,4] => [4,2,6,1,5,3] => [1,1,1,1,0,0,1,1,0,0,0,0]
=> 1 = 2 - 1
[3,1,7,6,2,4,5] => [3,1,6,2,4,5] => [4,2,6,1,5,3] => [1,1,1,1,0,0,1,1,0,0,0,0]
=> 1 = 2 - 1
[3,1,7,6,2,5,4] => [3,1,6,2,5,4] => [4,2,6,1,5,3] => [1,1,1,1,0,0,1,1,0,0,0,0]
=> 1 = 2 - 1
[3,1,7,6,4,2,5] => [3,1,6,4,2,5] => [5,2,6,4,1,3] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> 1 = 2 - 1
[3,1,7,6,5,2,4] => [3,1,6,5,2,4] => [5,2,6,4,1,3] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> 1 = 2 - 1
[3,2,4,6,1,5,7] => [3,2,4,6,1,5] => [5,2,3,6,1,4] => [1,1,1,1,1,0,0,0,1,0,0,0]
=> 1 = 2 - 1
[3,2,4,6,1,7,5] => [3,2,4,6,1,5] => [5,2,3,6,1,4] => [1,1,1,1,1,0,0,0,1,0,0,0]
=> 1 = 2 - 1
[3,2,4,6,7,1,5] => [3,2,4,6,1,5] => [5,2,3,6,1,4] => [1,1,1,1,1,0,0,0,1,0,0,0]
=> 1 = 2 - 1
[3,2,4,7,6,1,5] => [3,2,4,6,1,5] => [5,2,3,6,1,4] => [1,1,1,1,1,0,0,0,1,0,0,0]
=> 1 = 2 - 1
[3,2,5,1,6,4,7] => [3,2,5,1,6,4] => [4,2,6,1,5,3] => [1,1,1,1,0,0,1,1,0,0,0,0]
=> 1 = 2 - 1
[3,2,5,1,6,7,4] => [3,2,5,1,6,4] => [4,2,6,1,5,3] => [1,1,1,1,0,0,1,1,0,0,0,0]
=> 1 = 2 - 1
[3,2,5,1,7,6,4] => [3,2,5,1,6,4] => [4,2,6,1,5,3] => [1,1,1,1,0,0,1,1,0,0,0,0]
=> 1 = 2 - 1
[3,2,5,7,1,6,4] => [3,2,5,1,6,4] => [4,2,6,1,5,3] => [1,1,1,1,0,0,1,1,0,0,0,0]
=> 1 = 2 - 1
Description
The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$.
Matching statistic: St001498
Mp00252: Permutations —restriction⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001498: Dyck paths ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 33%
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001498: Dyck paths ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 33%
Values
[3,1,4,2,5] => [3,1,4,2] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? = 2 - 2
[3,1,4,5,2] => [3,1,4,2] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? = 2 - 2
[3,1,5,4,2] => [3,1,4,2] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? = 2 - 2
[3,2,4,1,5] => [3,2,4,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? = 2 - 2
[3,2,4,5,1] => [3,2,4,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? = 2 - 2
[3,2,5,4,1] => [3,2,4,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? = 2 - 2
[3,5,1,4,2] => [3,1,4,2] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? = 2 - 2
[3,5,2,4,1] => [3,2,4,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? = 2 - 2
[4,1,2,3,5] => [4,1,2,3] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? = 3 - 2
[4,1,2,5,3] => [4,1,2,3] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? = 3 - 2
[4,1,3,2,5] => [4,1,3,2] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? = 2 - 2
[4,1,3,5,2] => [4,1,3,2] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? = 2 - 2
[4,1,5,2,3] => [4,1,2,3] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? = 3 - 2
[4,1,5,3,2] => [4,1,3,2] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? = 2 - 2
[4,2,1,3,5] => [4,2,1,3] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 3 - 2
[4,2,1,5,3] => [4,2,1,3] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 3 - 2
[4,2,3,1,5] => [4,2,3,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 2 - 2
[4,2,3,5,1] => [4,2,3,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 2 - 2
[4,2,5,1,3] => [4,2,1,3] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 3 - 2
[4,2,5,3,1] => [4,2,3,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 2 - 2
[4,3,1,2,5] => [4,3,1,2] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 2 - 2
[4,3,1,5,2] => [4,3,1,2] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 2 - 2
[4,3,2,1,5] => [4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 2 - 2
[4,3,2,5,1] => [4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 2 - 2
[4,3,5,1,2] => [4,3,1,2] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 2 - 2
[4,3,5,2,1] => [4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 2 - 2
[4,5,1,2,3] => [4,1,2,3] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? = 3 - 2
[4,5,1,3,2] => [4,1,3,2] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? = 2 - 2
[4,5,2,1,3] => [4,2,1,3] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 3 - 2
[4,5,2,3,1] => [4,2,3,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 2 - 2
[4,5,3,1,2] => [4,3,1,2] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 2 - 2
[4,5,3,2,1] => [4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 2 - 2
[5,3,1,4,2] => [3,1,4,2] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? = 2 - 2
[5,3,2,4,1] => [3,2,4,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? = 2 - 2
[5,4,1,2,3] => [4,1,2,3] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? = 3 - 2
[5,4,1,3,2] => [4,1,3,2] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? = 2 - 2
[5,4,2,1,3] => [4,2,1,3] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 3 - 2
[5,4,2,3,1] => [4,2,3,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 2 - 2
[5,4,3,1,2] => [4,3,1,2] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 2 - 2
[5,4,3,2,1] => [4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 2 - 2
[3,1,4,5,2,6] => [3,1,4,5,2] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 3 - 2
[3,1,4,5,6,2] => [3,1,4,5,2] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 3 - 2
[3,1,4,6,5,2] => [3,1,4,5,2] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 3 - 2
[3,1,5,2,4,6] => [3,1,5,2,4] => [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> 0 = 2 - 2
[3,1,5,2,6,4] => [3,1,5,2,4] => [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> 0 = 2 - 2
[3,1,5,4,2,6] => [3,1,5,4,2] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 2 - 2
[3,1,5,4,6,2] => [3,1,5,4,2] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 2 - 2
[3,1,5,6,2,4] => [3,1,5,2,4] => [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> 0 = 2 - 2
[3,1,5,6,4,2] => [3,1,5,4,2] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 2 - 2
[3,1,6,4,5,2] => [3,1,4,5,2] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 3 - 2
[3,1,6,5,2,4] => [3,1,5,2,4] => [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> 0 = 2 - 2
[3,1,6,5,4,2] => [3,1,5,4,2] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 2 - 2
[3,2,4,5,1,6] => [3,2,4,5,1] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 3 - 2
[3,2,4,5,6,1] => [3,2,4,5,1] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 3 - 2
[3,2,5,1,4,6] => [3,2,5,1,4] => [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> 0 = 2 - 2
[3,2,5,1,6,4] => [3,2,5,1,4] => [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> 0 = 2 - 2
[3,2,5,6,1,4] => [3,2,5,1,4] => [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> 0 = 2 - 2
[3,2,6,5,1,4] => [3,2,5,1,4] => [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> 0 = 2 - 2
[3,6,1,5,2,4] => [3,1,5,2,4] => [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> 0 = 2 - 2
[3,6,2,5,1,4] => [3,2,5,1,4] => [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> 0 = 2 - 2
[6,3,1,5,2,4] => [3,1,5,2,4] => [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> 0 = 2 - 2
[6,3,2,5,1,4] => [3,2,5,1,4] => [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> 0 = 2 - 2
[3,1,4,6,2,5,7] => [3,1,4,6,2,5] => [5,2,3,6,1,4] => [1,1,1,1,1,0,0,0,1,0,0,0]
=> 0 = 2 - 2
[3,1,4,6,2,7,5] => [3,1,4,6,2,5] => [5,2,3,6,1,4] => [1,1,1,1,1,0,0,0,1,0,0,0]
=> 0 = 2 - 2
[3,1,4,6,7,2,5] => [3,1,4,6,2,5] => [5,2,3,6,1,4] => [1,1,1,1,1,0,0,0,1,0,0,0]
=> 0 = 2 - 2
[3,1,4,7,6,2,5] => [3,1,4,6,2,5] => [5,2,3,6,1,4] => [1,1,1,1,1,0,0,0,1,0,0,0]
=> 0 = 2 - 2
[3,1,5,2,6,4,7] => [3,1,5,2,6,4] => [4,2,6,1,5,3] => [1,1,1,1,0,0,1,1,0,0,0,0]
=> 0 = 2 - 2
[3,1,5,2,6,7,4] => [3,1,5,2,6,4] => [4,2,6,1,5,3] => [1,1,1,1,0,0,1,1,0,0,0,0]
=> 0 = 2 - 2
[3,1,5,2,7,6,4] => [3,1,5,2,6,4] => [4,2,6,1,5,3] => [1,1,1,1,0,0,1,1,0,0,0,0]
=> 0 = 2 - 2
[3,1,5,7,2,6,4] => [3,1,5,2,6,4] => [4,2,6,1,5,3] => [1,1,1,1,0,0,1,1,0,0,0,0]
=> 0 = 2 - 2
[3,1,6,2,4,5,7] => [3,1,6,2,4,5] => [4,2,6,1,5,3] => [1,1,1,1,0,0,1,1,0,0,0,0]
=> 0 = 2 - 2
[3,1,6,2,4,7,5] => [3,1,6,2,4,5] => [4,2,6,1,5,3] => [1,1,1,1,0,0,1,1,0,0,0,0]
=> 0 = 2 - 2
[3,1,6,2,5,4,7] => [3,1,6,2,5,4] => [4,2,6,1,5,3] => [1,1,1,1,0,0,1,1,0,0,0,0]
=> 0 = 2 - 2
[3,1,6,2,5,7,4] => [3,1,6,2,5,4] => [4,2,6,1,5,3] => [1,1,1,1,0,0,1,1,0,0,0,0]
=> 0 = 2 - 2
[3,1,6,2,7,4,5] => [3,1,6,2,4,5] => [4,2,6,1,5,3] => [1,1,1,1,0,0,1,1,0,0,0,0]
=> 0 = 2 - 2
[3,1,6,2,7,5,4] => [3,1,6,2,5,4] => [4,2,6,1,5,3] => [1,1,1,1,0,0,1,1,0,0,0,0]
=> 0 = 2 - 2
[3,1,6,4,2,5,7] => [3,1,6,4,2,5] => [5,2,6,4,1,3] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> 0 = 2 - 2
[3,1,6,4,2,7,5] => [3,1,6,4,2,5] => [5,2,6,4,1,3] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> 0 = 2 - 2
[3,1,6,4,7,2,5] => [3,1,6,4,2,5] => [5,2,6,4,1,3] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> 0 = 2 - 2
[3,1,6,5,2,4,7] => [3,1,6,5,2,4] => [5,2,6,4,1,3] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> 0 = 2 - 2
[3,1,6,5,2,7,4] => [3,1,6,5,2,4] => [5,2,6,4,1,3] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> 0 = 2 - 2
[3,1,6,5,7,2,4] => [3,1,6,5,2,4] => [5,2,6,4,1,3] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> 0 = 2 - 2
[3,1,6,7,2,4,5] => [3,1,6,2,4,5] => [4,2,6,1,5,3] => [1,1,1,1,0,0,1,1,0,0,0,0]
=> 0 = 2 - 2
[3,1,6,7,2,5,4] => [3,1,6,2,5,4] => [4,2,6,1,5,3] => [1,1,1,1,0,0,1,1,0,0,0,0]
=> 0 = 2 - 2
[3,1,6,7,4,2,5] => [3,1,6,4,2,5] => [5,2,6,4,1,3] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> 0 = 2 - 2
[3,1,6,7,5,2,4] => [3,1,6,5,2,4] => [5,2,6,4,1,3] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> 0 = 2 - 2
[3,1,7,4,6,2,5] => [3,1,4,6,2,5] => [5,2,3,6,1,4] => [1,1,1,1,1,0,0,0,1,0,0,0]
=> 0 = 2 - 2
[3,1,7,5,2,6,4] => [3,1,5,2,6,4] => [4,2,6,1,5,3] => [1,1,1,1,0,0,1,1,0,0,0,0]
=> 0 = 2 - 2
[3,1,7,6,2,4,5] => [3,1,6,2,4,5] => [4,2,6,1,5,3] => [1,1,1,1,0,0,1,1,0,0,0,0]
=> 0 = 2 - 2
[3,1,7,6,2,5,4] => [3,1,6,2,5,4] => [4,2,6,1,5,3] => [1,1,1,1,0,0,1,1,0,0,0,0]
=> 0 = 2 - 2
[3,1,7,6,4,2,5] => [3,1,6,4,2,5] => [5,2,6,4,1,3] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> 0 = 2 - 2
[3,1,7,6,5,2,4] => [3,1,6,5,2,4] => [5,2,6,4,1,3] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> 0 = 2 - 2
[3,2,4,6,1,5,7] => [3,2,4,6,1,5] => [5,2,3,6,1,4] => [1,1,1,1,1,0,0,0,1,0,0,0]
=> 0 = 2 - 2
[3,2,4,6,1,7,5] => [3,2,4,6,1,5] => [5,2,3,6,1,4] => [1,1,1,1,1,0,0,0,1,0,0,0]
=> 0 = 2 - 2
[3,2,4,6,7,1,5] => [3,2,4,6,1,5] => [5,2,3,6,1,4] => [1,1,1,1,1,0,0,0,1,0,0,0]
=> 0 = 2 - 2
[3,2,4,7,6,1,5] => [3,2,4,6,1,5] => [5,2,3,6,1,4] => [1,1,1,1,1,0,0,0,1,0,0,0]
=> 0 = 2 - 2
[3,2,5,1,6,4,7] => [3,2,5,1,6,4] => [4,2,6,1,5,3] => [1,1,1,1,0,0,1,1,0,0,0,0]
=> 0 = 2 - 2
[3,2,5,1,6,7,4] => [3,2,5,1,6,4] => [4,2,6,1,5,3] => [1,1,1,1,0,0,1,1,0,0,0,0]
=> 0 = 2 - 2
[3,2,5,1,7,6,4] => [3,2,5,1,6,4] => [4,2,6,1,5,3] => [1,1,1,1,0,0,1,1,0,0,0,0]
=> 0 = 2 - 2
[3,2,5,7,1,6,4] => [3,2,5,1,6,4] => [4,2,6,1,5,3] => [1,1,1,1,0,0,1,1,0,0,0,0]
=> 0 = 2 - 2
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.
Matching statistic: St000772
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000772: Graphs ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 33%
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000772: Graphs ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 33%
Values
[3,1,4,2,5] => [4,2,3,1,5] => [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ? = 2 - 1
[3,1,4,5,2] => [5,2,3,4,1] => [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2 - 1
[3,1,5,4,2] => [5,2,4,3,1] => [2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[3,2,4,1,5] => [4,2,3,1,5] => [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ? = 2 - 1
[3,2,4,5,1] => [5,2,3,4,1] => [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2 - 1
[3,2,5,4,1] => [5,2,4,3,1] => [2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[3,5,1,4,2] => [5,4,3,2,1] => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[3,5,2,4,1] => [5,4,3,2,1] => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[4,1,2,3,5] => [4,2,3,1,5] => [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ? = 3 - 1
[4,1,2,5,3] => [5,2,3,4,1] => [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> ? = 3 - 1
[4,1,3,2,5] => [4,2,3,1,5] => [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ? = 2 - 1
[4,1,3,5,2] => [5,2,3,4,1] => [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2 - 1
[4,1,5,2,3] => [5,2,4,3,1] => [2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 - 1
[4,1,5,3,2] => [5,2,4,3,1] => [2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[4,2,1,3,5] => [4,3,2,1,5] => [3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ? = 3 - 1
[4,2,1,5,3] => [5,3,2,4,1] => [3,2,4,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 - 1
[4,2,3,1,5] => [4,3,2,1,5] => [3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[4,2,3,5,1] => [5,3,2,4,1] => [3,2,4,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[4,2,5,1,3] => [5,4,3,2,1] => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 - 1
[4,2,5,3,1] => [5,4,3,2,1] => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[4,3,1,2,5] => [4,3,2,1,5] => [3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[4,3,1,5,2] => [5,3,2,4,1] => [3,2,4,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[4,3,2,1,5] => [4,3,2,1,5] => [3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[4,3,2,5,1] => [5,3,2,4,1] => [3,2,4,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[4,3,5,1,2] => [5,4,3,2,1] => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[4,3,5,2,1] => [5,4,3,2,1] => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[4,5,1,2,3] => [5,4,3,2,1] => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 - 1
[4,5,1,3,2] => [5,4,3,2,1] => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[4,5,2,1,3] => [5,4,3,2,1] => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 - 1
[4,5,2,3,1] => [5,4,3,2,1] => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[4,5,3,1,2] => [5,4,3,2,1] => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[4,5,3,2,1] => [5,4,3,2,1] => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[5,3,1,4,2] => [5,4,3,2,1] => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[5,3,2,4,1] => [5,4,3,2,1] => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[5,4,1,2,3] => [5,4,3,2,1] => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 - 1
[5,4,1,3,2] => [5,4,3,2,1] => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[5,4,2,1,3] => [5,4,3,2,1] => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 - 1
[5,4,2,3,1] => [5,4,3,2,1] => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[5,4,3,1,2] => [5,4,3,2,1] => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[5,4,3,2,1] => [5,4,3,2,1] => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[3,1,4,5,2,6] => [5,2,3,4,1,6] => [2,3,4,1,6,5] => ([(0,1),(2,5),(3,5),(4,5)],6)
=> ? = 3 - 1
[3,1,4,5,6,2] => [6,2,3,4,5,1] => [2,3,4,5,1,6] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 3 - 1
[3,1,4,6,5,2] => [6,2,3,5,4,1] => [2,3,5,4,1,6] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 1
[3,1,5,2,4,6] => [4,2,5,1,3,6] => [2,5,1,3,6,4] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 1 = 2 - 1
[3,1,5,2,6,4] => [4,2,6,1,5,3] => [2,6,1,5,3,4] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6)
=> 1 = 2 - 1
[3,1,5,4,2,6] => [5,2,4,3,1,6] => [2,4,3,1,6,5] => ([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
[3,1,5,4,6,2] => [6,2,4,3,5,1] => [2,4,3,5,1,6] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
[3,1,5,6,2,4] => [5,2,6,4,1,3] => [2,6,4,1,3,5] => ([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[3,1,5,6,4,2] => [6,2,5,4,3,1] => [2,5,4,3,1,6] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
[3,1,6,4,5,2] => [6,2,5,4,3,1] => [2,5,4,3,1,6] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 1
[3,1,6,5,2,4] => [5,2,6,4,1,3] => [2,6,4,1,3,5] => ([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[3,1,6,5,4,2] => [6,2,5,4,3,1] => [2,5,4,3,1,6] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
[3,2,4,5,1,6] => [5,2,3,4,1,6] => [2,3,4,1,6,5] => ([(0,1),(2,5),(3,5),(4,5)],6)
=> ? = 3 - 1
[3,2,4,5,6,1] => [6,2,3,4,5,1] => [2,3,4,5,1,6] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 3 - 1
[3,2,5,1,4,6] => [4,2,5,1,3,6] => [2,5,1,3,6,4] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 1 = 2 - 1
[3,2,5,1,6,4] => [4,2,6,1,5,3] => [2,6,1,5,3,4] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6)
=> 1 = 2 - 1
[3,2,5,6,1,4] => [5,2,6,4,1,3] => [2,6,4,1,3,5] => ([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[3,2,6,5,1,4] => [5,2,6,4,1,3] => [2,6,4,1,3,5] => ([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[3,6,1,5,2,4] => [5,6,3,4,1,2] => [6,3,4,1,2,5] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 1 = 2 - 1
[3,6,2,5,1,4] => [5,6,3,4,1,2] => [6,3,4,1,2,5] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 1 = 2 - 1
[3,1,4,6,2,5,7] => [5,2,3,6,1,4,7] => [2,3,6,1,4,7,5] => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> 1 = 2 - 1
[3,1,4,6,2,7,5] => [5,2,3,7,1,6,4] => [2,3,7,1,6,4,5] => ([(0,4),(1,4),(2,5),(2,6),(3,5),(3,6),(4,6),(5,6)],7)
=> 1 = 2 - 1
[3,1,4,6,7,2,5] => [6,2,3,7,5,1,4] => [2,3,7,5,1,4,6] => ([(0,6),(1,5),(2,5),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> 1 = 2 - 1
[3,1,4,7,6,2,5] => [6,2,3,7,5,1,4] => [2,3,7,5,1,4,6] => ([(0,6),(1,5),(2,5),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> 1 = 2 - 1
[3,1,5,2,6,4,7] => [4,2,6,1,5,3,7] => [2,6,1,5,3,7,4] => ([(0,4),(1,3),(2,5),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> 1 = 2 - 1
[3,1,5,2,6,7,4] => [4,2,7,1,5,6,3] => [2,7,1,5,6,3,4] => ([(0,1),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> 1 = 2 - 1
[3,1,5,2,7,6,4] => [4,2,7,1,6,5,3] => [2,7,1,6,5,3,4] => ([(0,1),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1 = 2 - 1
[3,1,5,7,2,6,4] => [5,2,7,6,1,4,3] => [2,7,6,1,4,3,5] => ([(0,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1 = 2 - 1
[3,1,6,2,4,5,7] => [4,2,6,1,5,3,7] => [2,6,1,5,3,7,4] => ([(0,4),(1,3),(2,5),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> 1 = 2 - 1
[3,1,6,2,4,7,5] => [4,2,7,1,5,6,3] => [2,7,1,5,6,3,4] => ([(0,1),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> 1 = 2 - 1
[3,1,6,2,5,4,7] => [4,2,6,1,5,3,7] => [2,6,1,5,3,7,4] => ([(0,4),(1,3),(2,5),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> 1 = 2 - 1
[3,1,6,2,5,7,4] => [4,2,7,1,5,6,3] => [2,7,1,5,6,3,4] => ([(0,1),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> 1 = 2 - 1
[3,1,6,2,7,4,5] => [4,2,7,1,6,5,3] => [2,7,1,6,5,3,4] => ([(0,1),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1 = 2 - 1
[3,1,6,2,7,5,4] => [4,2,7,1,6,5,3] => [2,7,1,6,5,3,4] => ([(0,1),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1 = 2 - 1
[3,1,6,4,2,5,7] => [5,2,6,4,1,3,7] => [2,6,4,1,3,7,5] => ([(0,4),(1,3),(2,5),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> 1 = 2 - 1
[3,1,6,4,2,7,5] => [5,2,7,4,1,6,3] => [2,7,4,1,6,3,5] => ([(0,5),(1,4),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,6),(5,6)],7)
=> 1 = 2 - 1
[3,1,6,4,7,2,5] => [6,2,7,4,5,1,3] => [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)
=> 1 = 2 - 1
[3,1,6,5,2,4,7] => [5,2,6,4,1,3,7] => [2,6,4,1,3,7,5] => ([(0,4),(1,3),(2,5),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> 1 = 2 - 1
[3,1,6,5,2,7,4] => [5,2,7,4,1,6,3] => [2,7,4,1,6,3,5] => ([(0,5),(1,4),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,6),(5,6)],7)
=> 1 = 2 - 1
[3,1,6,5,7,2,4] => [6,2,7,4,5,1,3] => [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)
=> 1 = 2 - 1
[3,1,6,7,2,4,5] => [5,2,7,6,1,4,3] => [2,7,6,1,4,3,5] => ([(0,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1 = 2 - 1
[3,1,6,7,2,5,4] => [5,2,7,6,1,4,3] => [2,7,6,1,4,3,5] => ([(0,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1 = 2 - 1
[3,1,6,7,4,2,5] => [6,2,7,5,4,1,3] => [2,7,5,4,1,3,6] => ([(0,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1 = 2 - 1
[3,1,6,7,5,2,4] => [6,2,7,5,4,1,3] => [2,7,5,4,1,3,6] => ([(0,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1 = 2 - 1
[3,1,7,4,6,2,5] => [6,2,7,4,5,1,3] => [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)
=> 1 = 2 - 1
[3,1,7,5,2,6,4] => [5,2,7,6,1,4,3] => [2,7,6,1,4,3,5] => ([(0,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1 = 2 - 1
[3,1,7,6,2,4,5] => [5,2,7,6,1,4,3] => [2,7,6,1,4,3,5] => ([(0,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1 = 2 - 1
[3,1,7,6,2,5,4] => [5,2,7,6,1,4,3] => [2,7,6,1,4,3,5] => ([(0,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1 = 2 - 1
[3,1,7,6,4,2,5] => [6,2,7,5,4,1,3] => [2,7,5,4,1,3,6] => ([(0,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1 = 2 - 1
[3,1,7,6,5,2,4] => [6,2,7,5,4,1,3] => [2,7,5,4,1,3,6] => ([(0,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1 = 2 - 1
[3,2,4,6,1,5,7] => [5,2,3,6,1,4,7] => [2,3,6,1,4,7,5] => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> 1 = 2 - 1
[3,2,4,6,1,7,5] => [5,2,3,7,1,6,4] => [2,3,7,1,6,4,5] => ([(0,4),(1,4),(2,5),(2,6),(3,5),(3,6),(4,6),(5,6)],7)
=> 1 = 2 - 1
[3,2,4,6,7,1,5] => [6,2,3,7,5,1,4] => [2,3,7,5,1,4,6] => ([(0,6),(1,5),(2,5),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> 1 = 2 - 1
[3,2,4,7,6,1,5] => [6,2,3,7,5,1,4] => [2,3,7,5,1,4,6] => ([(0,6),(1,5),(2,5),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> 1 = 2 - 1
[3,2,5,1,6,4,7] => [4,2,6,1,5,3,7] => [2,6,1,5,3,7,4] => ([(0,4),(1,3),(2,5),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> 1 = 2 - 1
[3,2,5,1,6,7,4] => [4,2,7,1,5,6,3] => [2,7,1,5,6,3,4] => ([(0,1),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> 1 = 2 - 1
[3,2,5,1,7,6,4] => [4,2,7,1,6,5,3] => [2,7,1,6,5,3,4] => ([(0,1),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1 = 2 - 1
[3,2,5,7,1,6,4] => [5,2,7,6,1,4,3] => [2,7,6,1,4,3,5] => ([(0,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1 = 2 - 1
[3,2,6,1,4,5,7] => [4,2,6,1,5,3,7] => [2,6,1,5,3,7,4] => ([(0,4),(1,3),(2,5),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> 1 = 2 - 1
[3,2,6,1,4,7,5] => [4,2,7,1,5,6,3] => [2,7,1,5,6,3,4] => ([(0,1),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> 1 = 2 - 1
Description
The multiplicity of the largest distance Laplacian eigenvalue in a connected graph.
The distance Laplacian of a graph is the (symmetric) matrix with row and column sums $0$, which has the negative distances between two vertices as its off-diagonal entries. This statistic is the largest multiplicity of an eigenvalue.
For example, the cycle on four vertices has distance Laplacian
$$
\left(\begin{array}{rrrr}
4 & -1 & -2 & -1 \\
-1 & 4 & -1 & -2 \\
-2 & -1 & 4 & -1 \\
-1 & -2 & -1 & 4
\end{array}\right).
$$
Its eigenvalues are $0,4,4,6$, so the statistic is $1$.
The path on four vertices has eigenvalues $0, 4.7\dots, 6, 9.2\dots$ and therefore also statistic $1$.
The graphs with statistic $n-1$, $n-2$ and $n-3$ have been characterised, see [1].
Matching statistic: St000454
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00011: Binary trees —to graph⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 33%
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00011: Binary trees —to graph⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 33%
Values
[3,1,4,2,5] => [4,1,3,5,2] => [[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2
[3,1,4,5,2] => [5,1,3,4,2] => [[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2
[3,1,5,4,2] => [5,1,4,3,2] => [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2
[3,2,4,1,5] => [2,1,3,5,4] => [[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2
[3,2,4,5,1] => [2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2
[3,2,5,4,1] => [2,1,4,3,5] => [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 2
[3,5,1,4,2] => [5,1,4,2,3] => [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 2
[3,5,2,4,1] => [3,1,4,2,5] => [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 2
[4,1,2,3,5] => [3,4,1,5,2] => [[.,[.,.]],[[.,.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 3
[4,1,2,5,3] => [3,5,1,4,2] => [[.,[.,.]],[[.,.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 3
[4,1,3,2,5] => [4,3,1,5,2] => [[[.,.],.],[[.,.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2
[4,1,3,5,2] => [5,3,1,4,2] => [[[.,.],.],[[.,.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2
[4,1,5,2,3] => [4,5,1,3,2] => [[.,[.,.]],[[.,.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 3
[4,1,5,3,2] => [5,4,1,3,2] => [[[.,.],.],[[.,.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2
[4,2,1,3,5] => [2,4,1,5,3] => [[.,[.,.]],[[.,.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 3
[4,2,1,5,3] => [2,5,1,4,3] => [[.,[.,.]],[[.,.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 3
[4,2,3,1,5] => [2,3,1,5,4] => [[.,[.,.]],[[.,.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2
[4,2,3,5,1] => [2,3,1,4,5] => [[.,[.,.]],[.,[.,.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2
[4,2,5,1,3] => [2,5,1,3,4] => [[.,[.,.]],[.,[.,.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 3
[4,2,5,3,1] => [2,4,1,3,5] => [[.,[.,.]],[.,[.,.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2
[4,3,1,2,5] => [4,2,1,5,3] => [[[.,.],.],[[.,.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2
[4,3,1,5,2] => [5,2,1,4,3] => [[[.,.],.],[[.,.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2
[4,3,2,1,5] => [3,2,1,5,4] => [[[.,.],.],[[.,.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2
[4,3,2,5,1] => [3,2,1,4,5] => [[[.,.],.],[.,[.,.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2
[4,3,5,1,2] => [5,2,1,3,4] => [[[.,.],.],[.,[.,.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2
[4,3,5,2,1] => [4,2,1,3,5] => [[[.,.],.],[.,[.,.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2
[4,5,1,2,3] => [4,5,1,2,3] => [[.,[.,.]],[.,[.,.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 3
[4,5,1,3,2] => [5,4,1,2,3] => [[[.,.],.],[.,[.,.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2
[4,5,2,1,3] => [3,5,1,2,4] => [[.,[.,.]],[.,[.,.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 3
[4,5,2,3,1] => [3,4,1,2,5] => [[.,[.,.]],[.,[.,.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2
[4,5,3,1,2] => [5,3,1,2,4] => [[[.,.],.],[.,[.,.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2
[4,5,3,2,1] => [4,3,1,2,5] => [[[.,.],.],[.,[.,.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2
[5,3,1,4,2] => [5,2,4,1,3] => [[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 2
[5,3,2,4,1] => [3,2,4,1,5] => [[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 2
[5,4,1,2,3] => [4,5,2,1,3] => [[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 3
[5,4,1,3,2] => [5,4,2,1,3] => [[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2
[5,4,2,1,3] => [3,5,2,1,4] => [[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 3
[5,4,2,3,1] => [3,4,2,1,5] => [[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2
[5,4,3,1,2] => [5,3,2,1,4] => [[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2
[5,4,3,2,1] => [4,3,2,1,5] => [[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2
[3,1,4,5,2,6] => [5,1,3,4,6,2] => [[.,.],[[.,[.,[.,.]]],.]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ? = 3
[3,1,4,5,6,2] => [6,1,3,4,5,2] => [[.,.],[[.,[.,[.,.]]],.]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ? = 3
[3,1,4,6,5,2] => [6,1,3,5,4,2] => [[.,.],[[.,[[.,.],.]],.]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ? = 3
[3,1,5,2,4,6] => [4,1,5,3,6,2] => [[.,.],[[[.,.],[.,.]],.]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ? = 2
[3,1,5,2,6,4] => [4,1,6,3,5,2] => [[.,.],[[[.,.],[.,.]],.]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ? = 2
[3,1,5,4,2,6] => [5,1,4,3,6,2] => [[.,.],[[[.,.],[.,.]],.]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ? = 2
[3,1,5,4,6,2] => [6,1,4,3,5,2] => [[.,.],[[[.,.],[.,.]],.]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ? = 2
[3,1,5,6,2,4] => [5,1,6,3,4,2] => [[.,.],[[[.,.],[.,.]],.]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ? = 2
[3,1,5,6,4,2] => [6,1,5,3,4,2] => [[.,.],[[[.,.],[.,.]],.]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ? = 2
[3,1,6,4,5,2] => [6,1,4,5,3,2] => [[.,.],[[[.,[.,.]],.],.]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ? = 3
[3,2,6,1,4,5,7] => [2,1,5,6,3,7,4] => [[.,.],[[.,[.,.]],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
[3,2,6,1,4,7,5] => [2,1,5,7,3,6,4] => [[.,.],[[.,[.,.]],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
[3,2,6,1,5,4,7] => [2,1,6,5,3,7,4] => [[.,.],[[[.,.],.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
[3,2,6,1,5,7,4] => [2,1,7,5,3,6,4] => [[.,.],[[[.,.],.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
[3,2,6,1,7,4,5] => [2,1,6,7,3,5,4] => [[.,.],[[.,[.,.]],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
[3,2,6,1,7,5,4] => [2,1,7,6,3,5,4] => [[.,.],[[[.,.],.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
[3,2,6,4,1,5,7] => [2,1,4,6,3,7,5] => [[.,.],[[.,[.,.]],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
[3,2,6,4,1,7,5] => [2,1,4,7,3,6,5] => [[.,.],[[.,[.,.]],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
[3,2,6,4,5,1,7] => [2,1,4,5,3,7,6] => [[.,.],[[.,[.,.]],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
[3,2,6,4,5,7,1] => [2,1,4,5,3,6,7] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
[3,2,6,4,7,1,5] => [2,1,4,7,3,5,6] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
[3,2,6,4,7,5,1] => [2,1,4,6,3,5,7] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
[3,2,6,5,1,4,7] => [2,1,6,4,3,7,5] => [[.,.],[[[.,.],.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
[3,2,6,5,1,7,4] => [2,1,7,4,3,6,5] => [[.,.],[[[.,.],.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
[3,2,6,5,4,1,7] => [2,1,5,4,3,7,6] => [[.,.],[[[.,.],.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
[3,2,6,5,4,7,1] => [2,1,5,4,3,6,7] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
[3,2,6,5,7,1,4] => [2,1,7,4,3,5,6] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
[3,2,6,5,7,4,1] => [2,1,6,4,3,5,7] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
[3,2,6,7,1,4,5] => [2,1,6,7,3,4,5] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
[3,2,6,7,1,5,4] => [2,1,7,6,3,4,5] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
[3,2,6,7,4,1,5] => [2,1,5,7,3,4,6] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
[3,2,6,7,4,5,1] => [2,1,5,6,3,4,7] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
[3,2,6,7,5,1,4] => [2,1,7,5,3,4,6] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
[3,2,6,7,5,4,1] => [2,1,6,5,3,4,7] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
[3,6,5,1,2,4,7] => [5,1,6,3,2,7,4] => [[.,.],[[[.,.],.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
[3,6,5,1,2,7,4] => [5,1,7,3,2,6,4] => [[.,.],[[[.,.],.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
[3,6,5,1,7,2,4] => [6,1,7,3,2,5,4] => [[.,.],[[[.,.],.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
[3,6,5,2,1,4,7] => [4,1,6,3,2,7,5] => [[.,.],[[[.,.],.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
[3,6,5,2,1,7,4] => [4,1,7,3,2,6,5] => [[.,.],[[[.,.],.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
[3,6,5,2,7,1,4] => [4,1,7,3,2,5,6] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
[3,6,5,4,1,2,7] => [6,1,4,3,2,7,5] => [[.,.],[[[.,.],.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
[3,6,5,4,1,7,2] => [7,1,4,3,2,6,5] => [[.,.],[[[.,.],.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
[3,6,5,4,2,1,7] => [5,1,4,3,2,7,6] => [[.,.],[[[.,.],.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
[3,6,5,4,2,7,1] => [5,1,4,3,2,6,7] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
[3,6,5,4,7,1,2] => [7,1,4,3,2,5,6] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
[3,6,5,4,7,2,1] => [6,1,4,3,2,5,7] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
[3,6,5,7,1,2,4] => [6,1,7,3,2,4,5] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
[3,6,5,7,2,1,4] => [5,1,7,3,2,4,6] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
[3,6,5,7,4,1,2] => [7,1,5,3,2,4,6] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
[3,6,5,7,4,2,1] => [6,1,5,3,2,4,7] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
[3,6,7,5,1,2,4] => [6,1,7,4,2,3,5] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
[3,6,7,5,2,1,4] => [5,1,7,4,2,3,6] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
[3,6,7,5,4,1,2] => [7,1,5,4,2,3,6] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
[3,6,7,5,4,2,1] => [6,1,5,4,2,3,7] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
[5,3,2,4,7,6,1] => [3,2,4,1,6,5,7] => [[[.,.],[.,.]],[[.,.],[.,.]]]
=> ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
=> 2
[5,3,2,7,1,6,4] => [3,2,7,1,6,4,5] => [[[.,.],[.,.]],[[.,.],[.,.]]]
=> ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
=> 2
[5,3,2,7,4,6,1] => [3,2,5,1,6,4,7] => [[[.,.],[.,.]],[[.,.],[.,.]]]
=> ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
=> 2
[5,3,2,7,6,4,1] => [3,2,6,1,5,4,7] => [[[.,.],[.,.]],[[.,.],[.,.]]]
=> ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
=> 2
[5,3,4,2,7,6,1] => [4,2,3,1,6,5,7] => [[[.,.],[.,.]],[[.,.],[.,.]]]
=> ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
=> 2
[5,3,4,7,1,6,2] => [7,2,3,1,6,4,5] => [[[.,.],[.,.]],[[.,.],[.,.]]]
=> ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
=> 2
Description
The largest eigenvalue of a graph if it is integral.
If a graph is $d$-regular, then its largest eigenvalue equals $d$. One can show that the largest eigenvalue always lies between the average degree and the maximal degree.
This statistic is undefined if the largest eigenvalue of the graph is not integral.
Matching statistic: St000422
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00011: Binary trees —to graph⟶ Graphs
St000422: Graphs ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 33%
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00011: Binary trees —to graph⟶ Graphs
St000422: Graphs ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 33%
Values
[3,1,4,2,5] => [4,1,3,5,2] => [[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2 + 6
[3,1,4,5,2] => [5,1,3,4,2] => [[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2 + 6
[3,1,5,4,2] => [5,1,4,3,2] => [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2 + 6
[3,2,4,1,5] => [2,1,3,5,4] => [[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2 + 6
[3,2,4,5,1] => [2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2 + 6
[3,2,5,4,1] => [2,1,4,3,5] => [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 2 + 6
[3,5,1,4,2] => [5,1,4,2,3] => [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 2 + 6
[3,5,2,4,1] => [3,1,4,2,5] => [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 2 + 6
[4,1,2,3,5] => [3,4,1,5,2] => [[.,[.,.]],[[.,.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 3 + 6
[4,1,2,5,3] => [3,5,1,4,2] => [[.,[.,.]],[[.,.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 3 + 6
[4,1,3,2,5] => [4,3,1,5,2] => [[[.,.],.],[[.,.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2 + 6
[4,1,3,5,2] => [5,3,1,4,2] => [[[.,.],.],[[.,.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2 + 6
[4,1,5,2,3] => [4,5,1,3,2] => [[.,[.,.]],[[.,.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 3 + 6
[4,1,5,3,2] => [5,4,1,3,2] => [[[.,.],.],[[.,.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2 + 6
[4,2,1,3,5] => [2,4,1,5,3] => [[.,[.,.]],[[.,.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 3 + 6
[4,2,1,5,3] => [2,5,1,4,3] => [[.,[.,.]],[[.,.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 3 + 6
[4,2,3,1,5] => [2,3,1,5,4] => [[.,[.,.]],[[.,.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2 + 6
[4,2,3,5,1] => [2,3,1,4,5] => [[.,[.,.]],[.,[.,.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2 + 6
[4,2,5,1,3] => [2,5,1,3,4] => [[.,[.,.]],[.,[.,.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 3 + 6
[4,2,5,3,1] => [2,4,1,3,5] => [[.,[.,.]],[.,[.,.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2 + 6
[4,3,1,2,5] => [4,2,1,5,3] => [[[.,.],.],[[.,.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2 + 6
[4,3,1,5,2] => [5,2,1,4,3] => [[[.,.],.],[[.,.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2 + 6
[4,3,2,1,5] => [3,2,1,5,4] => [[[.,.],.],[[.,.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2 + 6
[4,3,2,5,1] => [3,2,1,4,5] => [[[.,.],.],[.,[.,.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2 + 6
[4,3,5,1,2] => [5,2,1,3,4] => [[[.,.],.],[.,[.,.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2 + 6
[4,3,5,2,1] => [4,2,1,3,5] => [[[.,.],.],[.,[.,.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2 + 6
[4,5,1,2,3] => [4,5,1,2,3] => [[.,[.,.]],[.,[.,.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 3 + 6
[4,5,1,3,2] => [5,4,1,2,3] => [[[.,.],.],[.,[.,.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2 + 6
[4,5,2,1,3] => [3,5,1,2,4] => [[.,[.,.]],[.,[.,.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 3 + 6
[4,5,2,3,1] => [3,4,1,2,5] => [[.,[.,.]],[.,[.,.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2 + 6
[4,5,3,1,2] => [5,3,1,2,4] => [[[.,.],.],[.,[.,.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2 + 6
[4,5,3,2,1] => [4,3,1,2,5] => [[[.,.],.],[.,[.,.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2 + 6
[5,3,1,4,2] => [5,2,4,1,3] => [[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 2 + 6
[5,3,2,4,1] => [3,2,4,1,5] => [[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 2 + 6
[5,4,1,2,3] => [4,5,2,1,3] => [[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 3 + 6
[5,4,1,3,2] => [5,4,2,1,3] => [[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2 + 6
[5,4,2,1,3] => [3,5,2,1,4] => [[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 3 + 6
[5,4,2,3,1] => [3,4,2,1,5] => [[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2 + 6
[5,4,3,1,2] => [5,3,2,1,4] => [[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2 + 6
[5,4,3,2,1] => [4,3,2,1,5] => [[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2 + 6
[3,1,4,5,2,6] => [5,1,3,4,6,2] => [[.,.],[[.,[.,[.,.]]],.]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ? = 3 + 6
[3,1,4,5,6,2] => [6,1,3,4,5,2] => [[.,.],[[.,[.,[.,.]]],.]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ? = 3 + 6
[3,1,4,6,5,2] => [6,1,3,5,4,2] => [[.,.],[[.,[[.,.],.]],.]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ? = 3 + 6
[3,1,5,2,4,6] => [4,1,5,3,6,2] => [[.,.],[[[.,.],[.,.]],.]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ? = 2 + 6
[3,1,5,2,6,4] => [4,1,6,3,5,2] => [[.,.],[[[.,.],[.,.]],.]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ? = 2 + 6
[3,1,5,4,2,6] => [5,1,4,3,6,2] => [[.,.],[[[.,.],[.,.]],.]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ? = 2 + 6
[3,1,5,4,6,2] => [6,1,4,3,5,2] => [[.,.],[[[.,.],[.,.]],.]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ? = 2 + 6
[3,1,5,6,2,4] => [5,1,6,3,4,2] => [[.,.],[[[.,.],[.,.]],.]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ? = 2 + 6
[3,1,5,6,4,2] => [6,1,5,3,4,2] => [[.,.],[[[.,.],[.,.]],.]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ? = 2 + 6
[3,1,6,4,5,2] => [6,1,4,5,3,2] => [[.,.],[[[.,[.,.]],.],.]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ? = 3 + 6
[3,2,6,1,4,5,7] => [2,1,5,6,3,7,4] => [[.,.],[[.,[.,.]],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 2 + 6
[3,2,6,1,4,7,5] => [2,1,5,7,3,6,4] => [[.,.],[[.,[.,.]],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 2 + 6
[3,2,6,1,5,4,7] => [2,1,6,5,3,7,4] => [[.,.],[[[.,.],.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 2 + 6
[3,2,6,1,5,7,4] => [2,1,7,5,3,6,4] => [[.,.],[[[.,.],.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 2 + 6
[3,2,6,1,7,4,5] => [2,1,6,7,3,5,4] => [[.,.],[[.,[.,.]],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 2 + 6
[3,2,6,1,7,5,4] => [2,1,7,6,3,5,4] => [[.,.],[[[.,.],.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 2 + 6
[3,2,6,4,1,5,7] => [2,1,4,6,3,7,5] => [[.,.],[[.,[.,.]],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 2 + 6
[3,2,6,4,1,7,5] => [2,1,4,7,3,6,5] => [[.,.],[[.,[.,.]],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 2 + 6
[3,2,6,4,5,1,7] => [2,1,4,5,3,7,6] => [[.,.],[[.,[.,.]],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 2 + 6
[3,2,6,4,5,7,1] => [2,1,4,5,3,6,7] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 2 + 6
[3,2,6,4,7,1,5] => [2,1,4,7,3,5,6] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 2 + 6
[3,2,6,4,7,5,1] => [2,1,4,6,3,5,7] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 2 + 6
[3,2,6,5,1,4,7] => [2,1,6,4,3,7,5] => [[.,.],[[[.,.],.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 2 + 6
[3,2,6,5,1,7,4] => [2,1,7,4,3,6,5] => [[.,.],[[[.,.],.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 2 + 6
[3,2,6,5,4,1,7] => [2,1,5,4,3,7,6] => [[.,.],[[[.,.],.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 2 + 6
[3,2,6,5,4,7,1] => [2,1,5,4,3,6,7] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 2 + 6
[3,2,6,5,7,1,4] => [2,1,7,4,3,5,6] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 2 + 6
[3,2,6,5,7,4,1] => [2,1,6,4,3,5,7] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 2 + 6
[3,2,6,7,1,4,5] => [2,1,6,7,3,4,5] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 2 + 6
[3,2,6,7,1,5,4] => [2,1,7,6,3,4,5] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 2 + 6
[3,2,6,7,4,1,5] => [2,1,5,7,3,4,6] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 2 + 6
[3,2,6,7,4,5,1] => [2,1,5,6,3,4,7] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 2 + 6
[3,2,6,7,5,1,4] => [2,1,7,5,3,4,6] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 2 + 6
[3,2,6,7,5,4,1] => [2,1,6,5,3,4,7] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 2 + 6
[3,6,5,1,2,4,7] => [5,1,6,3,2,7,4] => [[.,.],[[[.,.],.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 2 + 6
[3,6,5,1,2,7,4] => [5,1,7,3,2,6,4] => [[.,.],[[[.,.],.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 2 + 6
[3,6,5,1,7,2,4] => [6,1,7,3,2,5,4] => [[.,.],[[[.,.],.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 2 + 6
[3,6,5,2,1,4,7] => [4,1,6,3,2,7,5] => [[.,.],[[[.,.],.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 2 + 6
[3,6,5,2,1,7,4] => [4,1,7,3,2,6,5] => [[.,.],[[[.,.],.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 2 + 6
[3,6,5,2,7,1,4] => [4,1,7,3,2,5,6] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 2 + 6
[3,6,5,4,1,2,7] => [6,1,4,3,2,7,5] => [[.,.],[[[.,.],.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 2 + 6
[3,6,5,4,1,7,2] => [7,1,4,3,2,6,5] => [[.,.],[[[.,.],.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 2 + 6
[3,6,5,4,2,1,7] => [5,1,4,3,2,7,6] => [[.,.],[[[.,.],.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 2 + 6
[3,6,5,4,2,7,1] => [5,1,4,3,2,6,7] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 2 + 6
[3,6,5,4,7,1,2] => [7,1,4,3,2,5,6] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 2 + 6
[3,6,5,4,7,2,1] => [6,1,4,3,2,5,7] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 2 + 6
[3,6,5,7,1,2,4] => [6,1,7,3,2,4,5] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 2 + 6
[3,6,5,7,2,1,4] => [5,1,7,3,2,4,6] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 2 + 6
[3,6,5,7,4,1,2] => [7,1,5,3,2,4,6] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 2 + 6
[3,6,5,7,4,2,1] => [6,1,5,3,2,4,7] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 2 + 6
[3,6,7,5,1,2,4] => [6,1,7,4,2,3,5] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 2 + 6
[3,6,7,5,2,1,4] => [5,1,7,4,2,3,6] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 2 + 6
[3,6,7,5,4,1,2] => [7,1,5,4,2,3,6] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 2 + 6
[3,6,7,5,4,2,1] => [6,1,5,4,2,3,7] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 2 + 6
[7,4,1,2,5,6,3] => [4,7,2,5,6,1,3] => [[[.,[.,.]],[.,[.,.]]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 2 + 6
[7,4,1,2,6,3,5] => [4,6,2,7,5,1,3] => [[[.,[.,.]],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 2 + 6
[7,4,1,2,6,5,3] => [4,7,2,6,5,1,3] => [[[.,[.,.]],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 2 + 6
[7,4,1,3,5,6,2] => [7,4,2,5,6,1,3] => [[[[.,.],.],[.,[.,.]]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 2 + 6
[7,4,1,3,6,2,5] => [6,4,2,7,5,1,3] => [[[[.,.],.],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 2 + 6
[7,4,1,3,6,5,2] => [7,4,2,6,5,1,3] => [[[[.,.],.],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 2 + 6
Description
The energy of a graph, if it is integral.
The energy of a graph is the sum of the absolute values of its eigenvalues. This statistic is only defined for graphs with integral energy. It is known, that the energy is never an odd integer [2]. In fact, it is never the square root of an odd integer [3].
The energy of a graph is the sum of the energies of the connected components of a graph. The energy of the complete graph $K_n$ equals $2n-2$. For this reason, we do not define the energy of the empty graph.
The following 47 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001171The vector space dimension of $Ext_A^1(I_o,A)$ when $I_o$ is the tilting module corresponding to the permutation $o$ in the Auslander algebra $A$ of $K[x]/(x^n)$. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001768The number of reduced words of a signed permutation. St001823The Stasinski-Voll length of a signed permutation. St001854The size of the left Kazhdan-Lusztig cell, St001168The vector space dimension of the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001616The number of neutral elements in a lattice. St001720The minimal length of a chain of small intervals in a lattice. St001613The binary logarithm of the size of the center of a lattice. St001617The dimension of the space of valuations of a lattice. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001820The size of the image of the pop stack sorting operator. St001881The number of factors of a lattice as a Cartesian product of lattices. St001846The number of elements which do not have a complement in the lattice. St001618The cardinality of the Frattini sublattice of a lattice. St001615The number of join prime elements of a lattice. St000007The number of saliances of the permutation. St000054The first entry of the permutation. St000451The length of the longest pattern of the form k 1 2. St000725The smallest label of a leaf of the increasing binary tree associated to a permutation. St000842The breadth of a permutation. St000891The number of distinct diagonal sums of a permutation matrix. St000028The number of stack-sorts needed to sort a permutation. St000141The maximum drop size of a permutation. St000352The Elizalde-Pak rank of a permutation. St000402Half the size of the symmetry class of a permutation. St000651The maximal size of a rise in a permutation. St000669The number of permutations obtained by switching ascents or descents of size 2. St000696The number of cycles in the breakpoint graph of a permutation. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St000022The number of fixed points of a permutation. St000119The number of occurrences of the pattern 321 in a permutation. St000123The difference in Coxeter length of a permutation and its image under the Simion-Schmidt map. St000223The number of nestings in the permutation. St000359The number of occurrences of the pattern 23-1. St000366The number of double descents of a permutation. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000441The number of successions of a permutation. St000534The number of 2-rises of a permutation. St000546The number of global descents of a permutation. St000648The number of 2-excedences of a permutation. St000665The number of rafts of a permutation. St000731The number of double exceedences of a permutation. St001115The number of even descents of a permutation. St001394The genus of a permutation.
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