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Matching statistic: St000698
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000698: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000698: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> [2,1]
=> 0
[1,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> [1,1]
=> 1
[2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> [2]
=> 1
[1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 0
[1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 2
[1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 2
[1,3,4,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 1
[1,4,2,3] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 1
[1,4,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 1
[2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [3,2]
=> 2
[2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [2,2]
=> 2
[2,3,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [3]
=> 1
[3,1,2,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [3]
=> 1
[3,2,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [3]
=> 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> 0
[1,2,3,5,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> 3
[1,2,4,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> 3
[1,2,4,5,3] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> 2
[1,2,5,3,4] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> 2
[1,2,5,4,3] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> 2
[1,3,2,4,5] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> 3
[1,3,2,5,4] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> 4
[1,3,4,2,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> 2
[1,3,4,5,2] => [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 2
[1,3,5,2,4] => [1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> 1
[1,3,5,4,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 2
[1,4,2,3,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> 2
[1,4,2,5,3] => [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 2
[1,4,3,2,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> 2
[1,4,3,5,2] => [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 2
[1,4,5,2,3] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 2
[1,4,5,3,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 2
[1,5,2,3,4] => [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 2
[1,5,2,4,3] => [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 2
[1,5,3,2,4] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 2
[1,5,3,4,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 2
[1,5,4,2,3] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 2
[1,5,4,3,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 2
[2,1,3,4,5] => [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> 3
[2,1,3,5,4] => [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> 4
[2,1,4,3,5] => [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> 4
[2,1,4,5,3] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> 3
[2,1,5,3,4] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> 3
[2,1,5,4,3] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> 3
[2,3,1,4,5] => [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> 2
[2,3,1,5,4] => [3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> 3
[2,3,4,1,5] => [4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> 2
[2,3,5,1,4] => [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> 1
[2,4,1,3,5] => [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> 1
[2,4,1,5,3] => [3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> 1
Description
The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core.
For any positive integer $k$, one associates a $k$-core to a partition by repeatedly removing all rim hooks of size $k$.
This statistic counts the $2$-rim hooks that are removed in this process to obtain a $2$-core.
Matching statistic: St001645
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001645: Graphs ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 29%
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001645: Graphs ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 29%
Values
[1,2,3] => [1,2,3] => [1,2,3] => ([],3)
=> ? = 0 + 5
[1,3,2] => [1,3,2] => [1,3,2] => ([(1,2)],3)
=> ? = 1 + 5
[2,1,3] => [2,1,3] => [2,1,3] => ([(1,2)],3)
=> ? = 1 + 5
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? = 0 + 5
[1,2,4,3] => [1,2,4,3] => [1,2,4,3] => ([(2,3)],4)
=> ? = 2 + 5
[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
=> ? = 2 + 5
[1,3,4,2] => [1,4,3,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> ? = 1 + 5
[1,4,2,3] => [1,4,3,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> ? = 1 + 5
[1,4,3,2] => [1,4,3,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> ? = 1 + 5
[2,1,3,4] => [2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
=> ? = 2 + 5
[2,1,4,3] => [2,1,4,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> ? = 2 + 5
[2,3,1,4] => [3,2,1,4] => [2,3,1,4] => ([(1,3),(2,3)],4)
=> ? = 1 + 5
[3,1,2,4] => [3,2,1,4] => [2,3,1,4] => ([(1,3),(2,3)],4)
=> ? = 1 + 5
[3,2,1,4] => [3,2,1,4] => [2,3,1,4] => ([(1,3),(2,3)],4)
=> ? = 1 + 5
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> ? = 0 + 5
[1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ? = 3 + 5
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => ([(3,4)],5)
=> ? = 3 + 5
[1,2,4,5,3] => [1,2,5,4,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ? = 2 + 5
[1,2,5,3,4] => [1,2,5,4,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ? = 2 + 5
[1,2,5,4,3] => [1,2,5,4,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ? = 2 + 5
[1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => ([(3,4)],5)
=> ? = 3 + 5
[1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ? = 4 + 5
[1,3,4,2,5] => [1,4,3,2,5] => [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> ? = 2 + 5
[1,3,4,5,2] => [1,5,3,4,2] => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2 + 5
[1,3,5,2,4] => [1,4,5,2,3] => [1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 5
[1,3,5,4,2] => [1,5,4,3,2] => [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 5
[1,4,2,3,5] => [1,4,3,2,5] => [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> ? = 2 + 5
[1,4,2,5,3] => [1,5,3,4,2] => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2 + 5
[1,4,3,2,5] => [1,4,3,2,5] => [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> ? = 2 + 5
[1,4,3,5,2] => [1,5,3,4,2] => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2 + 5
[1,4,5,2,3] => [1,5,4,3,2] => [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 5
[1,4,5,3,2] => [1,5,4,3,2] => [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 5
[1,5,2,3,4] => [1,5,3,4,2] => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2 + 5
[1,5,2,4,3] => [1,5,3,4,2] => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2 + 5
[1,5,3,2,4] => [1,5,4,3,2] => [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 5
[1,5,3,4,2] => [1,5,4,3,2] => [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 5
[1,5,4,2,3] => [1,5,4,3,2] => [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 5
[1,5,4,3,2] => [1,5,4,3,2] => [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 5
[2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ([(3,4)],5)
=> ? = 3 + 5
[2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> ? = 4 + 5
[2,1,4,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => ([(1,4),(2,3)],5)
=> ? = 4 + 5
[2,1,4,5,3] => [2,1,5,4,3] => [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> ? = 3 + 5
[2,1,5,3,4] => [2,1,5,4,3] => [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> ? = 3 + 5
[2,1,5,4,3] => [2,1,5,4,3] => [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> ? = 3 + 5
[2,3,1,4,5] => [3,2,1,4,5] => [2,3,1,4,5] => ([(2,4),(3,4)],5)
=> ? = 2 + 5
[2,3,1,5,4] => [3,2,1,5,4] => [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ? = 3 + 5
[2,3,4,1,5] => [4,2,3,1,5] => [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2 + 5
[2,3,5,1,4] => [4,2,5,1,3] => [2,5,1,4,3] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ? = 1 + 5
[2,4,1,3,5] => [3,4,1,2,5] => [4,1,3,2,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 5
[2,4,1,5,3] => [3,5,1,4,2] => [4,5,1,3,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ? = 1 + 5
[2,4,1,5,6,3] => [3,6,1,4,5,2] => [4,5,6,1,3,2] => ([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 1 + 5
[2,4,1,6,5,3] => [3,6,1,5,4,2] => [5,4,6,1,3,2] => ([(0,1),(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 1 + 5
[2,5,1,3,6,4] => [3,6,1,4,5,2] => [4,5,6,1,3,2] => ([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 1 + 5
[2,5,1,4,6,3] => [3,6,1,4,5,2] => [4,5,6,1,3,2] => ([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 1 + 5
[2,5,1,6,3,4] => [3,6,1,5,4,2] => [5,4,6,1,3,2] => ([(0,1),(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 1 + 5
[2,5,1,6,4,3] => [3,6,1,5,4,2] => [5,4,6,1,3,2] => ([(0,1),(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 1 + 5
[2,6,1,3,4,5] => [3,6,1,4,5,2] => [4,5,6,1,3,2] => ([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 1 + 5
[2,6,1,3,5,4] => [3,6,1,4,5,2] => [4,5,6,1,3,2] => ([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 1 + 5
[2,6,1,4,3,5] => [3,6,1,5,4,2] => [5,4,6,1,3,2] => ([(0,1),(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 1 + 5
[2,6,1,4,5,3] => [3,6,1,5,4,2] => [5,4,6,1,3,2] => ([(0,1),(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 1 + 5
[2,6,1,5,3,4] => [3,6,1,5,4,2] => [5,4,6,1,3,2] => ([(0,1),(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 1 + 5
[2,6,1,5,4,3] => [3,6,1,5,4,2] => [5,4,6,1,3,2] => ([(0,1),(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 1 + 5
[2,4,1,5,6,7,3] => [3,7,1,4,5,6,2] => [4,5,6,7,1,3,2] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(5,6)],7)
=> 7 = 2 + 5
[2,4,1,5,7,6,3] => [3,7,1,4,6,5,2] => [4,6,5,7,1,3,2] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(5,6)],7)
=> 7 = 2 + 5
[2,4,1,6,5,7,3] => [3,7,1,5,4,6,2] => [5,4,6,7,1,3,2] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(5,6)],7)
=> 7 = 2 + 5
[2,4,1,6,7,5,3] => [3,7,1,6,5,4,2] => [5,6,4,7,1,3,2] => ([(0,3),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(4,5),(4,6),(5,6)],7)
=> 7 = 2 + 5
[2,4,1,7,5,6,3] => [3,7,1,6,5,4,2] => [5,6,4,7,1,3,2] => ([(0,3),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(4,5),(4,6),(5,6)],7)
=> 7 = 2 + 5
[2,4,1,7,6,5,3] => [3,7,1,6,5,4,2] => [5,6,4,7,1,3,2] => ([(0,3),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(4,5),(4,6),(5,6)],7)
=> 7 = 2 + 5
[2,4,6,3,1,7,5] => [5,4,7,2,1,6,3] => [6,7,2,4,1,5,3] => ([(0,4),(0,5),(0,6),(1,3),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> 7 = 2 + 5
[2,4,7,3,1,5,6] => [5,4,7,2,1,6,3] => [6,7,2,4,1,5,3] => ([(0,4),(0,5),(0,6),(1,3),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> 7 = 2 + 5
[2,4,7,3,1,6,5] => [5,4,7,2,1,6,3] => [6,7,2,4,1,5,3] => ([(0,4),(0,5),(0,6),(1,3),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> 7 = 2 + 5
[2,5,1,3,6,7,4] => [3,7,1,4,5,6,2] => [4,5,6,7,1,3,2] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(5,6)],7)
=> 7 = 2 + 5
[2,5,1,3,7,6,4] => [3,7,1,4,6,5,2] => [4,6,5,7,1,3,2] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(5,6)],7)
=> 7 = 2 + 5
[2,5,1,4,6,7,3] => [3,7,1,4,5,6,2] => [4,5,6,7,1,3,2] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(5,6)],7)
=> 7 = 2 + 5
[2,5,1,4,7,6,3] => [3,7,1,4,6,5,2] => [4,6,5,7,1,3,2] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(5,6)],7)
=> 7 = 2 + 5
[2,5,1,6,3,7,4] => [3,7,1,5,4,6,2] => [5,4,6,7,1,3,2] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(5,6)],7)
=> 7 = 2 + 5
[2,5,1,6,4,7,3] => [3,7,1,5,4,6,2] => [5,4,6,7,1,3,2] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(5,6)],7)
=> 7 = 2 + 5
[2,5,1,6,7,3,4] => [3,7,1,6,5,4,2] => [5,6,4,7,1,3,2] => ([(0,3),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(4,5),(4,6),(5,6)],7)
=> 7 = 2 + 5
[2,5,1,6,7,4,3] => [3,7,1,6,5,4,2] => [5,6,4,7,1,3,2] => ([(0,3),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(4,5),(4,6),(5,6)],7)
=> 7 = 2 + 5
[2,5,1,7,3,6,4] => [3,7,1,6,5,4,2] => [5,6,4,7,1,3,2] => ([(0,3),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(4,5),(4,6),(5,6)],7)
=> 7 = 2 + 5
[2,5,1,7,4,6,3] => [3,7,1,6,5,4,2] => [5,6,4,7,1,3,2] => ([(0,3),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(4,5),(4,6),(5,6)],7)
=> 7 = 2 + 5
[2,5,1,7,6,3,4] => [3,7,1,6,5,4,2] => [5,6,4,7,1,3,2] => ([(0,3),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(4,5),(4,6),(5,6)],7)
=> 7 = 2 + 5
[2,5,1,7,6,4,3] => [3,7,1,6,5,4,2] => [5,6,4,7,1,3,2] => ([(0,3),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(4,5),(4,6),(5,6)],7)
=> 7 = 2 + 5
[2,6,1,3,4,7,5] => [3,7,1,4,5,6,2] => [4,5,6,7,1,3,2] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(5,6)],7)
=> 7 = 2 + 5
[2,6,1,3,5,7,4] => [3,7,1,4,5,6,2] => [4,5,6,7,1,3,2] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(5,6)],7)
=> 7 = 2 + 5
[2,6,1,3,7,4,5] => [3,7,1,4,6,5,2] => [4,6,5,7,1,3,2] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(5,6)],7)
=> 7 = 2 + 5
[2,6,1,3,7,5,4] => [3,7,1,4,6,5,2] => [4,6,5,7,1,3,2] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(5,6)],7)
=> 7 = 2 + 5
[2,6,1,4,3,7,5] => [3,7,1,5,4,6,2] => [5,4,6,7,1,3,2] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(5,6)],7)
=> 7 = 2 + 5
[2,6,1,4,5,7,3] => [3,7,1,5,4,6,2] => [5,4,6,7,1,3,2] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(5,6)],7)
=> 7 = 2 + 5
[2,6,1,4,7,3,5] => [3,7,1,6,5,4,2] => [5,6,4,7,1,3,2] => ([(0,3),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(4,5),(4,6),(5,6)],7)
=> 7 = 2 + 5
[2,6,1,4,7,5,3] => [3,7,1,6,5,4,2] => [5,6,4,7,1,3,2] => ([(0,3),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(4,5),(4,6),(5,6)],7)
=> 7 = 2 + 5
[2,6,1,5,3,7,4] => [3,7,1,5,4,6,2] => [5,4,6,7,1,3,2] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(5,6)],7)
=> 7 = 2 + 5
[2,6,1,5,4,7,3] => [3,7,1,5,4,6,2] => [5,4,6,7,1,3,2] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(5,6)],7)
=> 7 = 2 + 5
[2,6,1,5,7,3,4] => [3,7,1,6,5,4,2] => [5,6,4,7,1,3,2] => ([(0,3),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(4,5),(4,6),(5,6)],7)
=> 7 = 2 + 5
[2,6,1,5,7,4,3] => [3,7,1,6,5,4,2] => [5,6,4,7,1,3,2] => ([(0,3),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(4,5),(4,6),(5,6)],7)
=> 7 = 2 + 5
[2,6,1,7,3,4,5] => [3,7,1,6,5,4,2] => [5,6,4,7,1,3,2] => ([(0,3),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(4,5),(4,6),(5,6)],7)
=> 7 = 2 + 5
[2,6,1,7,3,5,4] => [3,7,1,6,5,4,2] => [5,6,4,7,1,3,2] => ([(0,3),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(4,5),(4,6),(5,6)],7)
=> 7 = 2 + 5
[2,6,1,7,4,3,5] => [3,7,1,6,5,4,2] => [5,6,4,7,1,3,2] => ([(0,3),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(4,5),(4,6),(5,6)],7)
=> 7 = 2 + 5
[2,6,1,7,4,5,3] => [3,7,1,6,5,4,2] => [5,6,4,7,1,3,2] => ([(0,3),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(4,5),(4,6),(5,6)],7)
=> 7 = 2 + 5
[2,6,1,7,5,3,4] => [3,7,1,6,5,4,2] => [5,6,4,7,1,3,2] => ([(0,3),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(4,5),(4,6),(5,6)],7)
=> 7 = 2 + 5
Description
The pebbling number of a connected graph.
Matching statistic: St001060
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001060: Graphs ⟶ ℤResult quality: 0% ●values known / values provided: 0%●distinct values known / distinct values provided: 14%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001060: Graphs ⟶ ℤResult quality: 0% ●values known / values provided: 0%●distinct values known / distinct values provided: 14%
Values
[1,2,3] => [1,2,3] => [1,2,3] => ([],3)
=> ? = 0 + 1
[1,3,2] => [1,3,2] => [1,2,3] => ([],3)
=> ? = 1 + 1
[2,1,3] => [2,1,3] => [1,2,3] => ([],3)
=> ? = 1 + 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? = 0 + 1
[1,2,4,3] => [1,2,4,3] => [1,2,3,4] => ([],4)
=> ? = 2 + 1
[1,3,2,4] => [1,3,2,4] => [1,2,3,4] => ([],4)
=> ? = 2 + 1
[1,3,4,2] => [1,4,3,2] => [1,2,4,3] => ([(2,3)],4)
=> ? = 1 + 1
[1,4,2,3] => [1,4,3,2] => [1,2,4,3] => ([(2,3)],4)
=> ? = 1 + 1
[1,4,3,2] => [1,4,3,2] => [1,2,4,3] => ([(2,3)],4)
=> ? = 1 + 1
[2,1,3,4] => [2,1,3,4] => [1,2,3,4] => ([],4)
=> ? = 2 + 1
[2,1,4,3] => [2,1,4,3] => [1,2,3,4] => ([],4)
=> ? = 2 + 1
[2,3,1,4] => [3,2,1,4] => [1,3,2,4] => ([(2,3)],4)
=> ? = 1 + 1
[3,1,2,4] => [3,2,1,4] => [1,3,2,4] => ([(2,3)],4)
=> ? = 1 + 1
[3,2,1,4] => [3,2,1,4] => [1,3,2,4] => ([(2,3)],4)
=> ? = 1 + 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> ? = 0 + 1
[1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,4,5] => ([],5)
=> ? = 3 + 1
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,3,4,5] => ([],5)
=> ? = 3 + 1
[1,2,4,5,3] => [1,2,5,4,3] => [1,2,3,5,4] => ([(3,4)],5)
=> ? = 2 + 1
[1,2,5,3,4] => [1,2,5,4,3] => [1,2,3,5,4] => ([(3,4)],5)
=> ? = 2 + 1
[1,2,5,4,3] => [1,2,5,4,3] => [1,2,3,5,4] => ([(3,4)],5)
=> ? = 2 + 1
[1,3,2,4,5] => [1,3,2,4,5] => [1,2,3,4,5] => ([],5)
=> ? = 3 + 1
[1,3,2,5,4] => [1,3,2,5,4] => [1,2,3,4,5] => ([],5)
=> ? = 4 + 1
[1,3,4,2,5] => [1,4,3,2,5] => [1,2,4,3,5] => ([(3,4)],5)
=> ? = 2 + 1
[1,3,4,5,2] => [1,5,3,4,2] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ? = 2 + 1
[1,3,5,2,4] => [1,4,5,2,3] => [1,2,4,3,5] => ([(3,4)],5)
=> ? = 1 + 1
[1,3,5,4,2] => [1,5,4,3,2] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ? = 2 + 1
[1,4,2,3,5] => [1,4,3,2,5] => [1,2,4,3,5] => ([(3,4)],5)
=> ? = 2 + 1
[1,4,2,5,3] => [1,5,3,4,2] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ? = 2 + 1
[1,4,3,2,5] => [1,4,3,2,5] => [1,2,4,3,5] => ([(3,4)],5)
=> ? = 2 + 1
[1,4,3,5,2] => [1,5,3,4,2] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ? = 2 + 1
[1,4,5,2,3] => [1,5,4,3,2] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ? = 2 + 1
[1,4,5,3,2] => [1,5,4,3,2] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ? = 2 + 1
[1,5,2,3,4] => [1,5,3,4,2] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ? = 2 + 1
[1,5,2,4,3] => [1,5,3,4,2] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ? = 2 + 1
[1,5,3,2,4] => [1,5,4,3,2] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ? = 2 + 1
[1,5,3,4,2] => [1,5,4,3,2] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ? = 2 + 1
[1,5,4,2,3] => [1,5,4,3,2] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ? = 2 + 1
[1,5,4,3,2] => [1,5,4,3,2] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ? = 2 + 1
[2,1,3,4,5] => [2,1,3,4,5] => [1,2,3,4,5] => ([],5)
=> ? = 3 + 1
[2,1,3,5,4] => [2,1,3,5,4] => [1,2,3,4,5] => ([],5)
=> ? = 4 + 1
[2,1,4,3,5] => [2,1,4,3,5] => [1,2,3,4,5] => ([],5)
=> ? = 4 + 1
[2,1,4,5,3] => [2,1,5,4,3] => [1,2,3,5,4] => ([(3,4)],5)
=> ? = 3 + 1
[2,1,5,3,4] => [2,1,5,4,3] => [1,2,3,5,4] => ([(3,4)],5)
=> ? = 3 + 1
[2,1,5,4,3] => [2,1,5,4,3] => [1,2,3,5,4] => ([(3,4)],5)
=> ? = 3 + 1
[2,3,1,4,5] => [3,2,1,4,5] => [1,3,2,4,5] => ([(3,4)],5)
=> ? = 2 + 1
[2,3,1,5,4] => [3,2,1,5,4] => [1,3,2,4,5] => ([(3,4)],5)
=> ? = 3 + 1
[2,3,4,1,5] => [4,2,3,1,5] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ? = 2 + 1
[2,3,5,1,4] => [4,2,5,1,3] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ? = 1 + 1
[2,4,1,3,5] => [3,4,1,2,5] => [1,3,2,4,5] => ([(3,4)],5)
=> ? = 1 + 1
[2,4,1,5,3] => [3,5,1,4,2] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ? = 1 + 1
[2,3,5,6,1,4] => [5,2,6,4,1,3] => [1,5,2,3,6,4] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> 2 = 1 + 1
[2,3,6,4,1,5] => [5,2,6,4,1,3] => [1,5,2,3,6,4] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> 2 = 1 + 1
[2,3,6,5,1,4] => [5,2,6,4,1,3] => [1,5,2,3,6,4] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> 2 = 1 + 1
[2,4,5,1,6,3] => [4,6,3,1,5,2] => [1,4,2,6,3,5] => ([(1,5),(2,4),(3,4),(3,5)],6)
=> 2 = 1 + 1
[2,4,6,1,5,3] => [4,6,5,1,3,2] => [1,4,2,6,3,5] => ([(1,5),(2,4),(3,4),(3,5)],6)
=> 2 = 1 + 1
[2,5,3,1,6,4] => [4,6,3,1,5,2] => [1,4,2,6,3,5] => ([(1,5),(2,4),(3,4),(3,5)],6)
=> 2 = 1 + 1
[2,5,4,1,6,3] => [4,6,3,1,5,2] => [1,4,2,6,3,5] => ([(1,5),(2,4),(3,4),(3,5)],6)
=> 2 = 1 + 1
[2,5,6,1,3,4] => [4,6,5,1,3,2] => [1,4,2,6,3,5] => ([(1,5),(2,4),(3,4),(3,5)],6)
=> 2 = 1 + 1
[2,5,6,1,4,3] => [4,6,5,1,3,2] => [1,4,2,6,3,5] => ([(1,5),(2,4),(3,4),(3,5)],6)
=> 2 = 1 + 1
[2,6,3,1,4,5] => [4,6,3,1,5,2] => [1,4,2,6,3,5] => ([(1,5),(2,4),(3,4),(3,5)],6)
=> 2 = 1 + 1
[2,6,3,1,5,4] => [4,6,3,1,5,2] => [1,4,2,6,3,5] => ([(1,5),(2,4),(3,4),(3,5)],6)
=> 2 = 1 + 1
[2,6,4,1,3,5] => [4,6,5,1,3,2] => [1,4,2,6,3,5] => ([(1,5),(2,4),(3,4),(3,5)],6)
=> 2 = 1 + 1
[2,6,4,1,5,3] => [4,6,5,1,3,2] => [1,4,2,6,3,5] => ([(1,5),(2,4),(3,4),(3,5)],6)
=> 2 = 1 + 1
[2,6,5,1,3,4] => [4,6,5,1,3,2] => [1,4,2,6,3,5] => ([(1,5),(2,4),(3,4),(3,5)],6)
=> 2 = 1 + 1
[2,6,5,1,4,3] => [4,6,5,1,3,2] => [1,4,2,6,3,5] => ([(1,5),(2,4),(3,4),(3,5)],6)
=> 2 = 1 + 1
[3,1,5,6,2,4] => [5,2,6,4,1,3] => [1,5,2,3,6,4] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> 2 = 1 + 1
[3,1,6,4,2,5] => [5,2,6,4,1,3] => [1,5,2,3,6,4] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> 2 = 1 + 1
[3,1,6,5,2,4] => [5,2,6,4,1,3] => [1,5,2,3,6,4] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> 2 = 1 + 1
[3,2,5,6,1,4] => [5,2,6,4,1,3] => [1,5,2,3,6,4] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> 2 = 1 + 1
[3,2,6,4,1,5] => [5,2,6,4,1,3] => [1,5,2,3,6,4] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> 2 = 1 + 1
[3,2,6,5,1,4] => [5,2,6,4,1,3] => [1,5,2,3,6,4] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> 2 = 1 + 1
[3,5,1,2,6,4] => [4,6,3,1,5,2] => [1,4,2,6,3,5] => ([(1,5),(2,4),(3,4),(3,5)],6)
=> 2 = 1 + 1
[3,5,2,1,6,4] => [4,6,3,1,5,2] => [1,4,2,6,3,5] => ([(1,5),(2,4),(3,4),(3,5)],6)
=> 2 = 1 + 1
[3,6,1,2,4,5] => [4,6,3,1,5,2] => [1,4,2,6,3,5] => ([(1,5),(2,4),(3,4),(3,5)],6)
=> 2 = 1 + 1
[3,6,1,2,5,4] => [4,6,3,1,5,2] => [1,4,2,6,3,5] => ([(1,5),(2,4),(3,4),(3,5)],6)
=> 2 = 1 + 1
[3,6,2,1,4,5] => [4,6,3,1,5,2] => [1,4,2,6,3,5] => ([(1,5),(2,4),(3,4),(3,5)],6)
=> 2 = 1 + 1
[3,6,2,1,5,4] => [4,6,3,1,5,2] => [1,4,2,6,3,5] => ([(1,5),(2,4),(3,4),(3,5)],6)
=> 2 = 1 + 1
[4,1,6,2,3,5] => [5,2,6,4,1,3] => [1,5,2,3,6,4] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> 2 = 1 + 1
[4,1,6,3,2,5] => [5,2,6,4,1,3] => [1,5,2,3,6,4] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> 2 = 1 + 1
Description
The distinguishing index of a graph.
This is the smallest number of colours such that there is a colouring of the edges which is not preserved by any automorphism.
If the graph has a connected component which is a single edge, or at least two isolated vertices, this statistic is undefined.
Matching statistic: St000454
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00011: Binary trees —to graph⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 0% ●values known / values provided: 0%●distinct values known / distinct values provided: 14%
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00011: Binary trees —to graph⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 0% ●values known / values provided: 0%●distinct values known / distinct values provided: 14%
Values
[1,2,3] => [1,3,2] => [.,[[.,.],.]]
=> ([(0,2),(1,2)],3)
=> ? = 0 - 1
[1,3,2] => [1,3,2] => [.,[[.,.],.]]
=> ([(0,2),(1,2)],3)
=> ? = 1 - 1
[2,1,3] => [2,1,3] => [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ? = 1 - 1
[1,2,3,4] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 0 - 1
[1,2,4,3] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2 - 1
[1,3,2,4] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2 - 1
[1,3,4,2] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 1 - 1
[1,4,2,3] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 1 - 1
[1,4,3,2] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 1 - 1
[2,1,3,4] => [2,1,4,3] => [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2 - 1
[2,1,4,3] => [2,1,4,3] => [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2 - 1
[2,3,1,4] => [2,4,1,3] => [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 1 - 1
[3,1,2,4] => [3,1,4,2] => [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 1 - 1
[3,2,1,4] => [3,2,1,4] => [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 1 - 1
[1,2,3,4,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 0 - 1
[1,2,3,5,4] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 3 - 1
[1,2,4,3,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 3 - 1
[1,2,4,5,3] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2 - 1
[1,2,5,3,4] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2 - 1
[1,2,5,4,3] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2 - 1
[1,3,2,4,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 3 - 1
[1,3,2,5,4] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 4 - 1
[1,3,4,2,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2 - 1
[1,3,4,5,2] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2 - 1
[1,3,5,2,4] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 1 - 1
[1,3,5,4,2] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2 - 1
[1,4,2,3,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2 - 1
[1,4,2,5,3] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2 - 1
[1,4,3,2,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2 - 1
[1,4,3,5,2] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2 - 1
[1,4,5,2,3] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2 - 1
[1,4,5,3,2] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2 - 1
[1,5,2,3,4] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2 - 1
[1,5,2,4,3] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2 - 1
[1,5,3,2,4] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2 - 1
[1,5,3,4,2] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2 - 1
[1,5,4,2,3] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2 - 1
[1,5,4,3,2] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2 - 1
[2,1,3,4,5] => [2,1,5,4,3] => [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 3 - 1
[2,1,3,5,4] => [2,1,5,4,3] => [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 4 - 1
[2,1,4,3,5] => [2,1,5,4,3] => [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 4 - 1
[2,1,4,5,3] => [2,1,5,4,3] => [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 3 - 1
[2,1,5,3,4] => [2,1,5,4,3] => [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 3 - 1
[2,1,5,4,3] => [2,1,5,4,3] => [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 3 - 1
[2,3,1,4,5] => [2,5,1,4,3] => [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2 - 1
[2,3,1,5,4] => [2,5,1,4,3] => [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 3 - 1
[2,3,4,1,5] => [2,5,4,1,3] => [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2 - 1
[2,3,5,1,4] => [2,5,4,1,3] => [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 1 - 1
[2,4,1,3,5] => [2,5,1,4,3] => [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 1 - 1
[2,4,1,5,3] => [2,5,1,4,3] => [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 1 - 1
[6,3,1,2,4,5,7] => [6,3,1,7,5,4,2] => [[[.,[.,.]],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2 = 3 - 1
[6,3,1,2,5,4,7] => [6,3,1,7,5,4,2] => [[[.,[.,.]],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2 = 3 - 1
[6,3,1,4,2,5,7] => [6,3,1,7,5,4,2] => [[[.,[.,.]],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2 = 3 - 1
[6,3,1,4,5,2,7] => [6,3,1,7,5,4,2] => [[[.,[.,.]],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2 = 3 - 1
[6,3,1,5,2,4,7] => [6,3,1,7,5,4,2] => [[[.,[.,.]],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2 = 3 - 1
[6,3,1,5,4,2,7] => [6,3,1,7,5,4,2] => [[[.,[.,.]],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2 = 3 - 1
[6,3,2,1,4,5,7] => [6,3,2,1,7,5,4] => [[[[.,.],.],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2 = 3 - 1
[6,3,2,1,5,4,7] => [6,3,2,1,7,5,4] => [[[[.,.],.],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2 = 3 - 1
[6,3,2,4,1,5,7] => [6,3,2,7,1,5,4] => [[[[.,.],.],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2 = 3 - 1
[6,3,2,4,5,1,7] => [6,3,2,7,5,1,4] => [[[[.,.],.],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2 = 3 - 1
[6,3,2,5,1,4,7] => [6,3,2,7,1,5,4] => [[[[.,.],.],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2 = 3 - 1
[6,3,2,5,4,1,7] => [6,3,2,7,5,1,4] => [[[[.,.],.],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2 = 3 - 1
[6,3,4,1,2,5,7] => [6,3,7,1,5,4,2] => [[[.,[.,.]],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2 = 3 - 1
[6,3,4,1,5,2,7] => [6,3,7,1,5,4,2] => [[[.,[.,.]],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2 = 3 - 1
[6,3,4,2,1,5,7] => [6,3,7,2,1,5,4] => [[[[.,.],.],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2 = 3 - 1
[6,3,4,2,5,1,7] => [6,3,7,2,5,1,4] => [[[[.,.],.],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2 = 3 - 1
[6,3,4,5,1,2,7] => [6,3,7,5,1,4,2] => [[[.,[.,.]],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2 = 3 - 1
[6,3,4,5,2,1,7] => [6,3,7,5,2,1,4] => [[[[.,.],.],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2 = 3 - 1
[6,3,5,1,2,4,7] => [6,3,7,1,5,4,2] => [[[.,[.,.]],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2 = 3 - 1
[6,3,5,1,4,2,7] => [6,3,7,1,5,4,2] => [[[.,[.,.]],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2 = 3 - 1
[6,3,5,2,1,4,7] => [6,3,7,2,1,5,4] => [[[[.,.],.],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2 = 3 - 1
[6,3,5,2,4,1,7] => [6,3,7,2,5,1,4] => [[[[.,.],.],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2 = 3 - 1
[6,3,5,4,1,2,7] => [6,3,7,5,1,4,2] => [[[.,[.,.]],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2 = 3 - 1
[6,3,5,4,2,1,7] => [6,3,7,5,2,1,4] => [[[[.,.],.],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2 = 3 - 1
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
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00011: Binary trees —to graph⟶ Graphs
St000422: Graphs ⟶ ℤResult quality: 0% ●values known / values provided: 0%●distinct values known / distinct values provided: 14%
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00011: Binary trees —to graph⟶ Graphs
St000422: Graphs ⟶ ℤResult quality: 0% ●values known / values provided: 0%●distinct values known / distinct values provided: 14%
Values
[1,2,3] => [1,3,2] => [.,[[.,.],.]]
=> ([(0,2),(1,2)],3)
=> ? = 0 + 5
[1,3,2] => [1,3,2] => [.,[[.,.],.]]
=> ([(0,2),(1,2)],3)
=> ? = 1 + 5
[2,1,3] => [2,1,3] => [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ? = 1 + 5
[1,2,3,4] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 0 + 5
[1,2,4,3] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2 + 5
[1,3,2,4] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2 + 5
[1,3,4,2] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 1 + 5
[1,4,2,3] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 1 + 5
[1,4,3,2] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 1 + 5
[2,1,3,4] => [2,1,4,3] => [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2 + 5
[2,1,4,3] => [2,1,4,3] => [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2 + 5
[2,3,1,4] => [2,4,1,3] => [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 1 + 5
[3,1,2,4] => [3,1,4,2] => [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 1 + 5
[3,2,1,4] => [3,2,1,4] => [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 1 + 5
[1,2,3,4,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 0 + 5
[1,2,3,5,4] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 3 + 5
[1,2,4,3,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 3 + 5
[1,2,4,5,3] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2 + 5
[1,2,5,3,4] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2 + 5
[1,2,5,4,3] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2 + 5
[1,3,2,4,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 3 + 5
[1,3,2,5,4] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 4 + 5
[1,3,4,2,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2 + 5
[1,3,4,5,2] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2 + 5
[1,3,5,2,4] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 1 + 5
[1,3,5,4,2] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2 + 5
[1,4,2,3,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2 + 5
[1,4,2,5,3] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2 + 5
[1,4,3,2,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2 + 5
[1,4,3,5,2] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2 + 5
[1,4,5,2,3] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2 + 5
[1,4,5,3,2] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2 + 5
[1,5,2,3,4] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2 + 5
[1,5,2,4,3] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2 + 5
[1,5,3,2,4] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2 + 5
[1,5,3,4,2] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2 + 5
[1,5,4,2,3] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2 + 5
[1,5,4,3,2] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2 + 5
[2,1,3,4,5] => [2,1,5,4,3] => [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 3 + 5
[2,1,3,5,4] => [2,1,5,4,3] => [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 4 + 5
[2,1,4,3,5] => [2,1,5,4,3] => [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 4 + 5
[2,1,4,5,3] => [2,1,5,4,3] => [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 3 + 5
[2,1,5,3,4] => [2,1,5,4,3] => [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 3 + 5
[2,1,5,4,3] => [2,1,5,4,3] => [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 3 + 5
[2,3,1,4,5] => [2,5,1,4,3] => [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2 + 5
[2,3,1,5,4] => [2,5,1,4,3] => [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 3 + 5
[2,3,4,1,5] => [2,5,4,1,3] => [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2 + 5
[2,3,5,1,4] => [2,5,4,1,3] => [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 1 + 5
[2,4,1,3,5] => [2,5,1,4,3] => [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 1 + 5
[2,4,1,5,3] => [2,5,1,4,3] => [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 1 + 5
[6,3,1,2,4,5,7] => [6,3,1,7,5,4,2] => [[[.,[.,.]],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 3 + 5
[6,3,1,2,5,4,7] => [6,3,1,7,5,4,2] => [[[.,[.,.]],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 3 + 5
[6,3,1,4,2,5,7] => [6,3,1,7,5,4,2] => [[[.,[.,.]],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 3 + 5
[6,3,1,4,5,2,7] => [6,3,1,7,5,4,2] => [[[.,[.,.]],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 3 + 5
[6,3,1,5,2,4,7] => [6,3,1,7,5,4,2] => [[[.,[.,.]],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 3 + 5
[6,3,1,5,4,2,7] => [6,3,1,7,5,4,2] => [[[.,[.,.]],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 3 + 5
[6,3,2,1,4,5,7] => [6,3,2,1,7,5,4] => [[[[.,.],.],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 3 + 5
[6,3,2,1,5,4,7] => [6,3,2,1,7,5,4] => [[[[.,.],.],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 3 + 5
[6,3,2,4,1,5,7] => [6,3,2,7,1,5,4] => [[[[.,.],.],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 3 + 5
[6,3,2,4,5,1,7] => [6,3,2,7,5,1,4] => [[[[.,.],.],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 3 + 5
[6,3,2,5,1,4,7] => [6,3,2,7,1,5,4] => [[[[.,.],.],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 3 + 5
[6,3,2,5,4,1,7] => [6,3,2,7,5,1,4] => [[[[.,.],.],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 3 + 5
[6,3,4,1,2,5,7] => [6,3,7,1,5,4,2] => [[[.,[.,.]],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 3 + 5
[6,3,4,1,5,2,7] => [6,3,7,1,5,4,2] => [[[.,[.,.]],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 3 + 5
[6,3,4,2,1,5,7] => [6,3,7,2,1,5,4] => [[[[.,.],.],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 3 + 5
[6,3,4,2,5,1,7] => [6,3,7,2,5,1,4] => [[[[.,.],.],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 3 + 5
[6,3,4,5,1,2,7] => [6,3,7,5,1,4,2] => [[[.,[.,.]],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 3 + 5
[6,3,4,5,2,1,7] => [6,3,7,5,2,1,4] => [[[[.,.],.],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 3 + 5
[6,3,5,1,2,4,7] => [6,3,7,1,5,4,2] => [[[.,[.,.]],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 3 + 5
[6,3,5,1,4,2,7] => [6,3,7,1,5,4,2] => [[[.,[.,.]],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 3 + 5
[6,3,5,2,1,4,7] => [6,3,7,2,1,5,4] => [[[[.,.],.],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 3 + 5
[6,3,5,2,4,1,7] => [6,3,7,2,5,1,4] => [[[[.,.],.],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 3 + 5
[6,3,5,4,1,2,7] => [6,3,7,5,1,4,2] => [[[.,[.,.]],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 3 + 5
[6,3,5,4,2,1,7] => [6,3,7,5,2,1,4] => [[[[.,.],.],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 3 + 5
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.
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