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Your data matches 13 different statistics following compositions of up to 3 maps.
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Matching statistic: St000441
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000441: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00069: Permutations —complement⟶ Permutations
St000441: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => 0
[1,2] => [1,2] => [2,1] => 0
[2,1] => [1,2] => [2,1] => 0
[1,2,3] => [1,2,3] => [3,2,1] => 0
[1,3,2] => [1,2,3] => [3,2,1] => 0
[2,1,3] => [1,2,3] => [3,2,1] => 0
[2,3,1] => [1,2,3] => [3,2,1] => 0
[3,1,2] => [1,3,2] => [3,1,2] => 1
[3,2,1] => [1,3,2] => [3,1,2] => 1
[1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 0
[1,2,4,3] => [1,2,3,4] => [4,3,2,1] => 0
[1,3,2,4] => [1,2,3,4] => [4,3,2,1] => 0
[1,3,4,2] => [1,2,3,4] => [4,3,2,1] => 0
[1,4,2,3] => [1,2,4,3] => [4,3,1,2] => 1
[1,4,3,2] => [1,2,4,3] => [4,3,1,2] => 1
[2,1,3,4] => [1,2,3,4] => [4,3,2,1] => 0
[2,1,4,3] => [1,2,3,4] => [4,3,2,1] => 0
[2,3,1,4] => [1,2,3,4] => [4,3,2,1] => 0
[2,3,4,1] => [1,2,3,4] => [4,3,2,1] => 0
[2,4,1,3] => [1,2,4,3] => [4,3,1,2] => 1
[2,4,3,1] => [1,2,4,3] => [4,3,1,2] => 1
[3,1,2,4] => [1,3,2,4] => [4,2,3,1] => 1
[3,1,4,2] => [1,3,4,2] => [4,2,1,3] => 0
[3,2,1,4] => [1,3,2,4] => [4,2,3,1] => 1
[3,2,4,1] => [1,3,4,2] => [4,2,1,3] => 0
[3,4,1,2] => [1,3,2,4] => [4,2,3,1] => 1
[3,4,2,1] => [1,3,2,4] => [4,2,3,1] => 1
[4,1,2,3] => [1,4,3,2] => [4,1,2,3] => 2
[4,1,3,2] => [1,4,2,3] => [4,1,3,2] => 0
[4,2,1,3] => [1,4,3,2] => [4,1,2,3] => 2
[4,2,3,1] => [1,4,2,3] => [4,1,3,2] => 0
[4,3,1,2] => [1,4,2,3] => [4,1,3,2] => 0
[4,3,2,1] => [1,4,2,3] => [4,1,3,2] => 0
[1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 0
[1,2,3,5,4] => [1,2,3,4,5] => [5,4,3,2,1] => 0
[1,2,4,3,5] => [1,2,3,4,5] => [5,4,3,2,1] => 0
[1,2,4,5,3] => [1,2,3,4,5] => [5,4,3,2,1] => 0
[1,2,5,3,4] => [1,2,3,5,4] => [5,4,3,1,2] => 1
[1,2,5,4,3] => [1,2,3,5,4] => [5,4,3,1,2] => 1
[1,3,2,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 0
[1,3,2,5,4] => [1,2,3,4,5] => [5,4,3,2,1] => 0
[1,3,4,2,5] => [1,2,3,4,5] => [5,4,3,2,1] => 0
[1,3,4,5,2] => [1,2,3,4,5] => [5,4,3,2,1] => 0
[1,3,5,2,4] => [1,2,3,5,4] => [5,4,3,1,2] => 1
[1,3,5,4,2] => [1,2,3,5,4] => [5,4,3,1,2] => 1
[1,4,2,3,5] => [1,2,4,3,5] => [5,4,2,3,1] => 1
[1,4,2,5,3] => [1,2,4,5,3] => [5,4,2,1,3] => 0
[1,4,3,2,5] => [1,2,4,3,5] => [5,4,2,3,1] => 1
[1,4,3,5,2] => [1,2,4,5,3] => [5,4,2,1,3] => 0
[1,4,5,2,3] => [1,2,4,3,5] => [5,4,2,3,1] => 1
Description
The number of successions of a permutation.
A succession of a permutation $\pi$ is an index $i$ such that $\pi(i)+1 = \pi(i+1)$. Successions are also known as ''small ascents'' or ''1-rises''.
Matching statistic: St000214
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(load all 8 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000214: Permutations ⟶ ℤResult quality: 59% ●values known / values provided: 59%●distinct values known / distinct values provided: 100%
St000214: Permutations ⟶ ℤResult quality: 59% ●values known / values provided: 59%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => 0
[1,2] => [1,2] => 0
[2,1] => [1,2] => 0
[1,2,3] => [1,2,3] => 0
[1,3,2] => [1,2,3] => 0
[2,1,3] => [1,2,3] => 0
[2,3,1] => [1,2,3] => 0
[3,1,2] => [1,3,2] => 1
[3,2,1] => [1,3,2] => 1
[1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,3,4] => 0
[1,3,2,4] => [1,2,3,4] => 0
[1,3,4,2] => [1,2,3,4] => 0
[1,4,2,3] => [1,2,4,3] => 1
[1,4,3,2] => [1,2,4,3] => 1
[2,1,3,4] => [1,2,3,4] => 0
[2,1,4,3] => [1,2,3,4] => 0
[2,3,1,4] => [1,2,3,4] => 0
[2,3,4,1] => [1,2,3,4] => 0
[2,4,1,3] => [1,2,4,3] => 1
[2,4,3,1] => [1,2,4,3] => 1
[3,1,2,4] => [1,3,2,4] => 1
[3,1,4,2] => [1,3,4,2] => 0
[3,2,1,4] => [1,3,2,4] => 1
[3,2,4,1] => [1,3,4,2] => 0
[3,4,1,2] => [1,3,2,4] => 1
[3,4,2,1] => [1,3,2,4] => 1
[4,1,2,3] => [1,4,3,2] => 2
[4,1,3,2] => [1,4,2,3] => 0
[4,2,1,3] => [1,4,3,2] => 2
[4,2,3,1] => [1,4,2,3] => 0
[4,3,1,2] => [1,4,2,3] => 0
[4,3,2,1] => [1,4,2,3] => 0
[1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,2,3,4,5] => 0
[1,2,4,3,5] => [1,2,3,4,5] => 0
[1,2,4,5,3] => [1,2,3,4,5] => 0
[1,2,5,3,4] => [1,2,3,5,4] => 1
[1,2,5,4,3] => [1,2,3,5,4] => 1
[1,3,2,4,5] => [1,2,3,4,5] => 0
[1,3,2,5,4] => [1,2,3,4,5] => 0
[1,3,4,2,5] => [1,2,3,4,5] => 0
[1,3,4,5,2] => [1,2,3,4,5] => 0
[1,3,5,2,4] => [1,2,3,5,4] => 1
[1,3,5,4,2] => [1,2,3,5,4] => 1
[1,4,2,3,5] => [1,2,4,3,5] => 1
[1,4,2,5,3] => [1,2,4,5,3] => 0
[1,4,3,2,5] => [1,2,4,3,5] => 1
[1,4,3,5,2] => [1,2,4,5,3] => 0
[1,4,5,2,3] => [1,2,4,3,5] => 1
[1,2,3,7,4,5,6] => [1,2,3,4,7,6,5] => ? = 2
[1,2,3,7,5,4,6] => [1,2,3,4,7,6,5] => ? = 2
[1,2,4,7,3,5,6] => [1,2,3,4,7,6,5] => ? = 2
[1,2,4,7,5,3,6] => [1,2,3,4,7,6,5] => ? = 2
[1,2,5,7,3,4,6] => [1,2,3,5,4,7,6] => ? = 2
[1,2,5,7,3,6,4] => [1,2,3,5,4,7,6] => ? = 2
[1,2,5,7,4,3,6] => [1,2,3,5,4,7,6] => ? = 2
[1,2,5,7,4,6,3] => [1,2,3,5,4,7,6] => ? = 2
[1,2,7,3,4,5,6] => [1,2,3,7,6,5,4] => ? = 3
[1,2,7,4,3,5,6] => [1,2,3,7,6,5,4] => ? = 3
[1,3,2,7,4,5,6] => [1,2,3,4,7,6,5] => ? = 2
[1,3,2,7,5,4,6] => [1,2,3,4,7,6,5] => ? = 2
[1,3,4,7,2,5,6] => [1,2,3,4,7,6,5] => ? = 2
[1,3,4,7,5,2,6] => [1,2,3,4,7,6,5] => ? = 2
[1,3,5,7,2,4,6] => [1,2,3,5,4,7,6] => ? = 2
[1,3,5,7,2,6,4] => [1,2,3,5,4,7,6] => ? = 2
[1,3,5,7,4,2,6] => [1,2,3,5,4,7,6] => ? = 2
[1,3,5,7,4,6,2] => [1,2,3,5,4,7,6] => ? = 2
[1,3,7,2,4,5,6] => [1,2,3,7,6,5,4] => ? = 3
[1,3,7,4,2,5,6] => [1,2,3,7,6,5,4] => ? = 3
[1,4,7,2,3,5,6] => [1,2,4,3,7,6,5] => ? = 3
[1,4,7,2,5,3,6] => [1,2,4,3,7,6,5] => ? = 3
[1,4,7,3,2,5,6] => [1,2,4,3,7,6,5] => ? = 3
[1,4,7,3,5,2,6] => [1,2,4,3,7,6,5] => ? = 3
[1,7,2,3,4,5,6] => [1,2,7,6,5,4,3] => ? = 4
[1,7,3,2,4,5,6] => [1,2,7,6,5,4,3] => ? = 4
[2,1,3,7,4,5,6] => [1,2,3,4,7,6,5] => ? = 2
[2,1,3,7,5,4,6] => [1,2,3,4,7,6,5] => ? = 2
[2,1,4,7,3,5,6] => [1,2,3,4,7,6,5] => ? = 2
[2,1,4,7,5,3,6] => [1,2,3,4,7,6,5] => ? = 2
[2,1,5,7,3,4,6] => [1,2,3,5,4,7,6] => ? = 2
[2,1,5,7,3,6,4] => [1,2,3,5,4,7,6] => ? = 2
[2,1,5,7,4,3,6] => [1,2,3,5,4,7,6] => ? = 2
[2,1,5,7,4,6,3] => [1,2,3,5,4,7,6] => ? = 2
[2,1,7,3,4,5,6] => [1,2,3,7,6,5,4] => ? = 3
[2,1,7,4,3,5,6] => [1,2,3,7,6,5,4] => ? = 3
[2,3,1,7,4,5,6] => [1,2,3,4,7,6,5] => ? = 2
[2,3,1,7,5,4,6] => [1,2,3,4,7,6,5] => ? = 2
[2,3,4,7,1,5,6] => [1,2,3,4,7,6,5] => ? = 2
[2,3,4,7,5,1,6] => [1,2,3,4,7,6,5] => ? = 2
[2,3,5,7,1,4,6] => [1,2,3,5,4,7,6] => ? = 2
[2,3,5,7,1,6,4] => [1,2,3,5,4,7,6] => ? = 2
[2,3,5,7,4,1,6] => [1,2,3,5,4,7,6] => ? = 2
[2,3,5,7,4,6,1] => [1,2,3,5,4,7,6] => ? = 2
[2,3,7,1,4,5,6] => [1,2,3,7,6,5,4] => ? = 3
[2,3,7,4,1,5,6] => [1,2,3,7,6,5,4] => ? = 3
[2,4,7,1,3,5,6] => [1,2,4,3,7,6,5] => ? = 3
[2,4,7,1,5,3,6] => [1,2,4,3,7,6,5] => ? = 3
[2,4,7,3,1,5,6] => [1,2,4,3,7,6,5] => ? = 3
[2,4,7,3,5,1,6] => [1,2,4,3,7,6,5] => ? = 3
Description
The number of adjacencies of a permutation.
An adjacency of a permutation $\pi$ is an index $i$ such that $\pi(i)-1 = \pi(i+1)$. Adjacencies are also known as ''small descents''.
This can be also described as an occurrence of the bivincular pattern ([2,1], {((0,1),(1,0),(1,1),(1,2),(2,1)}), i.e., the middle row and the middle column are shaded, see [3].
Matching statistic: St000237
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000237: Permutations ⟶ ℤResult quality: 50% ●values known / values provided: 53%●distinct values known / distinct values provided: 50%
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000237: Permutations ⟶ ℤResult quality: 50% ●values known / values provided: 53%●distinct values known / distinct values provided: 50%
Values
[1] => [1] => [1] => [1] => 0
[1,2] => [1,2] => [1,2] => [1,2] => 0
[2,1] => [1,2] => [1,2] => [1,2] => 0
[1,2,3] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,2] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[2,1,3] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[2,3,1] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[3,1,2] => [1,3,2] => [1,3,2] => [1,3,2] => 1
[3,2,1] => [1,3,2] => [1,3,2] => [1,3,2] => 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,3,2,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,3,4,2] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,4,2,3] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 1
[1,4,3,2] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 1
[2,1,3,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[2,1,4,3] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[2,3,1,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[2,4,1,3] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 1
[2,4,3,1] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 1
[3,1,2,4] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 1
[3,1,4,2] => [1,3,4,2] => [1,4,3,2] => [1,4,3,2] => 0
[3,2,1,4] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 1
[3,2,4,1] => [1,3,4,2] => [1,4,3,2] => [1,4,3,2] => 0
[3,4,1,2] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 1
[3,4,2,1] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 1
[4,1,2,3] => [1,4,3,2] => [1,4,2,3] => [1,3,4,2] => 2
[4,1,3,2] => [1,4,2,3] => [1,3,4,2] => [1,4,2,3] => 0
[4,2,1,3] => [1,4,3,2] => [1,4,2,3] => [1,3,4,2] => 2
[4,2,3,1] => [1,4,2,3] => [1,3,4,2] => [1,4,2,3] => 0
[4,3,1,2] => [1,4,2,3] => [1,3,4,2] => [1,4,2,3] => 0
[4,3,2,1] => [1,4,2,3] => [1,3,4,2] => [1,4,2,3] => 0
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,5,3,4] => [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 1
[1,2,5,4,3] => [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 1
[1,3,2,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,2,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,4,2,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,4,5,2] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,5,2,4] => [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 1
[1,3,5,4,2] => [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 1
[1,4,2,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 1
[1,4,2,5,3] => [1,2,4,5,3] => [1,2,5,4,3] => [1,2,5,4,3] => 0
[1,4,3,2,5] => [1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 1
[1,4,3,5,2] => [1,2,4,5,3] => [1,2,5,4,3] => [1,2,5,4,3] => 0
[1,4,5,2,3] => [1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 1
[1,2,3,7,4,5,6] => [1,2,3,4,7,6,5] => [1,2,3,4,7,5,6] => [1,2,3,4,6,7,5] => ? = 2
[1,2,3,7,5,4,6] => [1,2,3,4,7,6,5] => [1,2,3,4,7,5,6] => [1,2,3,4,6,7,5] => ? = 2
[1,2,4,7,3,5,6] => [1,2,3,4,7,6,5] => [1,2,3,4,7,5,6] => [1,2,3,4,6,7,5] => ? = 2
[1,2,4,7,5,3,6] => [1,2,3,4,7,6,5] => [1,2,3,4,7,5,6] => [1,2,3,4,6,7,5] => ? = 2
[1,2,5,7,3,4,6] => [1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => ? = 2
[1,2,5,7,3,6,4] => [1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => ? = 2
[1,2,5,7,4,3,6] => [1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => ? = 2
[1,2,5,7,4,6,3] => [1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => ? = 2
[1,2,7,3,4,5,6] => [1,2,3,7,6,5,4] => [1,2,3,7,4,5,6] => [1,2,3,5,6,7,4] => ? = 3
[1,2,7,4,3,5,6] => [1,2,3,7,6,5,4] => [1,2,3,7,4,5,6] => [1,2,3,5,6,7,4] => ? = 3
[1,3,2,7,4,5,6] => [1,2,3,4,7,6,5] => [1,2,3,4,7,5,6] => [1,2,3,4,6,7,5] => ? = 2
[1,3,2,7,5,4,6] => [1,2,3,4,7,6,5] => [1,2,3,4,7,5,6] => [1,2,3,4,6,7,5] => ? = 2
[1,3,4,7,2,5,6] => [1,2,3,4,7,6,5] => [1,2,3,4,7,5,6] => [1,2,3,4,6,7,5] => ? = 2
[1,3,4,7,5,2,6] => [1,2,3,4,7,6,5] => [1,2,3,4,7,5,6] => [1,2,3,4,6,7,5] => ? = 2
[1,3,5,7,2,4,6] => [1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => ? = 2
[1,3,5,7,2,6,4] => [1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => ? = 2
[1,3,5,7,4,2,6] => [1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => ? = 2
[1,3,5,7,4,6,2] => [1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => ? = 2
[1,3,7,2,4,5,6] => [1,2,3,7,6,5,4] => [1,2,3,7,4,5,6] => [1,2,3,5,6,7,4] => ? = 3
[1,3,7,4,2,5,6] => [1,2,3,7,6,5,4] => [1,2,3,7,4,5,6] => [1,2,3,5,6,7,4] => ? = 3
[1,4,7,2,3,5,6] => [1,2,4,3,7,6,5] => [1,2,4,3,7,5,6] => [1,2,4,3,6,7,5] => ? = 3
[1,4,7,2,5,3,6] => [1,2,4,3,7,6,5] => [1,2,4,3,7,5,6] => [1,2,4,3,6,7,5] => ? = 3
[1,4,7,3,2,5,6] => [1,2,4,3,7,6,5] => [1,2,4,3,7,5,6] => [1,2,4,3,6,7,5] => ? = 3
[1,4,7,3,5,2,6] => [1,2,4,3,7,6,5] => [1,2,4,3,7,5,6] => [1,2,4,3,6,7,5] => ? = 3
[1,7,2,3,4,5,6] => [1,2,7,6,5,4,3] => [1,2,7,3,4,5,6] => [1,2,4,5,6,7,3] => ? = 4
[1,7,3,2,4,5,6] => [1,2,7,6,5,4,3] => [1,2,7,3,4,5,6] => [1,2,4,5,6,7,3] => ? = 4
[2,1,3,7,4,5,6] => [1,2,3,4,7,6,5] => [1,2,3,4,7,5,6] => [1,2,3,4,6,7,5] => ? = 2
[2,1,3,7,5,4,6] => [1,2,3,4,7,6,5] => [1,2,3,4,7,5,6] => [1,2,3,4,6,7,5] => ? = 2
[2,1,4,7,3,5,6] => [1,2,3,4,7,6,5] => [1,2,3,4,7,5,6] => [1,2,3,4,6,7,5] => ? = 2
[2,1,4,7,5,3,6] => [1,2,3,4,7,6,5] => [1,2,3,4,7,5,6] => [1,2,3,4,6,7,5] => ? = 2
[2,1,5,7,3,4,6] => [1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => ? = 2
[2,1,5,7,3,6,4] => [1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => ? = 2
[2,1,5,7,4,3,6] => [1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => ? = 2
[2,1,5,7,4,6,3] => [1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => ? = 2
[2,1,7,3,4,5,6] => [1,2,3,7,6,5,4] => [1,2,3,7,4,5,6] => [1,2,3,5,6,7,4] => ? = 3
[2,1,7,4,3,5,6] => [1,2,3,7,6,5,4] => [1,2,3,7,4,5,6] => [1,2,3,5,6,7,4] => ? = 3
[2,3,1,7,4,5,6] => [1,2,3,4,7,6,5] => [1,2,3,4,7,5,6] => [1,2,3,4,6,7,5] => ? = 2
[2,3,1,7,5,4,6] => [1,2,3,4,7,6,5] => [1,2,3,4,7,5,6] => [1,2,3,4,6,7,5] => ? = 2
[2,3,4,7,1,5,6] => [1,2,3,4,7,6,5] => [1,2,3,4,7,5,6] => [1,2,3,4,6,7,5] => ? = 2
[2,3,4,7,5,1,6] => [1,2,3,4,7,6,5] => [1,2,3,4,7,5,6] => [1,2,3,4,6,7,5] => ? = 2
[2,3,5,7,1,4,6] => [1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => ? = 2
[2,3,5,7,1,6,4] => [1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => ? = 2
[2,3,5,7,4,1,6] => [1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => ? = 2
[2,3,5,7,4,6,1] => [1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => ? = 2
[2,3,7,1,4,5,6] => [1,2,3,7,6,5,4] => [1,2,3,7,4,5,6] => [1,2,3,5,6,7,4] => ? = 3
[2,3,7,4,1,5,6] => [1,2,3,7,6,5,4] => [1,2,3,7,4,5,6] => [1,2,3,5,6,7,4] => ? = 3
[2,4,7,1,3,5,6] => [1,2,4,3,7,6,5] => [1,2,4,3,7,5,6] => [1,2,4,3,6,7,5] => ? = 3
[2,4,7,1,5,3,6] => [1,2,4,3,7,6,5] => [1,2,4,3,7,5,6] => [1,2,4,3,6,7,5] => ? = 3
[2,4,7,3,1,5,6] => [1,2,4,3,7,6,5] => [1,2,4,3,7,5,6] => [1,2,4,3,6,7,5] => ? = 3
[2,4,7,3,5,1,6] => [1,2,4,3,7,6,5] => [1,2,4,3,7,5,6] => [1,2,4,3,6,7,5] => ? = 3
Description
The number of small exceedances.
This is the number of indices $i$ such that $\pi_i=i+1$.
Matching statistic: St000648
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
St000648: Permutations ⟶ ℤResult quality: 40% ●values known / values provided: 40%●distinct values known / distinct values provided: 50%
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
St000648: Permutations ⟶ ℤResult quality: 40% ●values known / values provided: 40%●distinct values known / distinct values provided: 50%
Values
[1] => [1] => [1] => [1] => 0
[1,2] => [1,2] => [1,2] => [2,1] => 0
[2,1] => [1,2] => [1,2] => [2,1] => 0
[1,2,3] => [1,2,3] => [1,2,3] => [2,3,1] => 0
[1,3,2] => [1,2,3] => [1,2,3] => [2,3,1] => 0
[2,1,3] => [1,2,3] => [1,2,3] => [2,3,1] => 0
[2,3,1] => [1,2,3] => [1,2,3] => [2,3,1] => 0
[3,1,2] => [1,3,2] => [1,3,2] => [3,2,1] => 1
[3,2,1] => [1,3,2] => [1,3,2] => [3,2,1] => 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [2,3,4,1] => 0
[1,2,4,3] => [1,2,3,4] => [1,2,3,4] => [2,3,4,1] => 0
[1,3,2,4] => [1,2,3,4] => [1,2,3,4] => [2,3,4,1] => 0
[1,3,4,2] => [1,2,3,4] => [1,2,3,4] => [2,3,4,1] => 0
[1,4,2,3] => [1,2,4,3] => [1,2,4,3] => [2,4,3,1] => 1
[1,4,3,2] => [1,2,4,3] => [1,2,4,3] => [2,4,3,1] => 1
[2,1,3,4] => [1,2,3,4] => [1,2,3,4] => [2,3,4,1] => 0
[2,1,4,3] => [1,2,3,4] => [1,2,3,4] => [2,3,4,1] => 0
[2,3,1,4] => [1,2,3,4] => [1,2,3,4] => [2,3,4,1] => 0
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => [2,3,4,1] => 0
[2,4,1,3] => [1,2,4,3] => [1,2,4,3] => [2,4,3,1] => 1
[2,4,3,1] => [1,2,4,3] => [1,2,4,3] => [2,4,3,1] => 1
[3,1,2,4] => [1,3,2,4] => [1,3,2,4] => [3,2,4,1] => 1
[3,1,4,2] => [1,3,4,2] => [1,4,3,2] => [4,3,2,1] => 0
[3,2,1,4] => [1,3,2,4] => [1,3,2,4] => [3,2,4,1] => 1
[3,2,4,1] => [1,3,4,2] => [1,4,3,2] => [4,3,2,1] => 0
[3,4,1,2] => [1,3,2,4] => [1,3,2,4] => [3,2,4,1] => 1
[3,4,2,1] => [1,3,2,4] => [1,3,2,4] => [3,2,4,1] => 1
[4,1,2,3] => [1,4,3,2] => [1,4,2,3] => [3,4,2,1] => 2
[4,1,3,2] => [1,4,2,3] => [1,3,4,2] => [4,2,3,1] => 0
[4,2,1,3] => [1,4,3,2] => [1,4,2,3] => [3,4,2,1] => 2
[4,2,3,1] => [1,4,2,3] => [1,3,4,2] => [4,2,3,1] => 0
[4,3,1,2] => [1,4,2,3] => [1,3,4,2] => [4,2,3,1] => 0
[4,3,2,1] => [1,4,2,3] => [1,3,4,2] => [4,2,3,1] => 0
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => 0
[1,2,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => 0
[1,2,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => 0
[1,2,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => 0
[1,2,5,3,4] => [1,2,3,5,4] => [1,2,3,5,4] => [2,3,5,4,1] => 1
[1,2,5,4,3] => [1,2,3,5,4] => [1,2,3,5,4] => [2,3,5,4,1] => 1
[1,3,2,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => 0
[1,3,2,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => 0
[1,3,4,2,5] => [1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => 0
[1,3,4,5,2] => [1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => 0
[1,3,5,2,4] => [1,2,3,5,4] => [1,2,3,5,4] => [2,3,5,4,1] => 1
[1,3,5,4,2] => [1,2,3,5,4] => [1,2,3,5,4] => [2,3,5,4,1] => 1
[1,4,2,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => [2,4,3,5,1] => 1
[1,4,2,5,3] => [1,2,4,5,3] => [1,2,5,4,3] => [2,5,4,3,1] => 0
[1,4,3,2,5] => [1,2,4,3,5] => [1,2,4,3,5] => [2,4,3,5,1] => 1
[1,4,3,5,2] => [1,2,4,5,3] => [1,2,5,4,3] => [2,5,4,3,1] => 0
[1,4,5,2,3] => [1,2,4,3,5] => [1,2,4,3,5] => [2,4,3,5,1] => 1
[1,2,3,4,7,5,6] => [1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => [2,3,4,5,7,6,1] => ? = 1
[1,2,3,4,7,6,5] => [1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => [2,3,4,5,7,6,1] => ? = 1
[1,2,3,5,7,4,6] => [1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => [2,3,4,5,7,6,1] => ? = 1
[1,2,3,5,7,6,4] => [1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => [2,3,4,5,7,6,1] => ? = 1
[1,2,3,7,4,5,6] => [1,2,3,4,7,6,5] => [1,2,3,4,7,5,6] => [2,3,4,6,7,5,1] => ? = 2
[1,2,3,7,5,4,6] => [1,2,3,4,7,6,5] => [1,2,3,4,7,5,6] => [2,3,4,6,7,5,1] => ? = 2
[1,2,4,3,7,5,6] => [1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => [2,3,4,5,7,6,1] => ? = 1
[1,2,4,3,7,6,5] => [1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => [2,3,4,5,7,6,1] => ? = 1
[1,2,4,5,7,3,6] => [1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => [2,3,4,5,7,6,1] => ? = 1
[1,2,4,5,7,6,3] => [1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => [2,3,4,5,7,6,1] => ? = 1
[1,2,4,7,3,5,6] => [1,2,3,4,7,6,5] => [1,2,3,4,7,5,6] => [2,3,4,6,7,5,1] => ? = 2
[1,2,4,7,5,3,6] => [1,2,3,4,7,6,5] => [1,2,3,4,7,5,6] => [2,3,4,6,7,5,1] => ? = 2
[1,2,5,7,3,4,6] => [1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => [2,3,5,4,7,6,1] => ? = 2
[1,2,5,7,3,6,4] => [1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => [2,3,5,4,7,6,1] => ? = 2
[1,2,5,7,4,3,6] => [1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => [2,3,5,4,7,6,1] => ? = 2
[1,2,5,7,4,6,3] => [1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => [2,3,5,4,7,6,1] => ? = 2
[1,2,7,3,4,5,6] => [1,2,3,7,6,5,4] => [1,2,3,7,4,5,6] => [2,3,5,6,7,4,1] => ? = 3
[1,2,7,4,3,5,6] => [1,2,3,7,6,5,4] => [1,2,3,7,4,5,6] => [2,3,5,6,7,4,1] => ? = 3
[1,3,2,4,7,5,6] => [1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => [2,3,4,5,7,6,1] => ? = 1
[1,3,2,4,7,6,5] => [1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => [2,3,4,5,7,6,1] => ? = 1
[1,3,2,5,7,4,6] => [1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => [2,3,4,5,7,6,1] => ? = 1
[1,3,2,5,7,6,4] => [1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => [2,3,4,5,7,6,1] => ? = 1
[1,3,2,7,4,5,6] => [1,2,3,4,7,6,5] => [1,2,3,4,7,5,6] => [2,3,4,6,7,5,1] => ? = 2
[1,3,2,7,5,4,6] => [1,2,3,4,7,6,5] => [1,2,3,4,7,5,6] => [2,3,4,6,7,5,1] => ? = 2
[1,3,4,2,7,5,6] => [1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => [2,3,4,5,7,6,1] => ? = 1
[1,3,4,2,7,6,5] => [1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => [2,3,4,5,7,6,1] => ? = 1
[1,3,4,5,7,2,6] => [1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => [2,3,4,5,7,6,1] => ? = 1
[1,3,4,5,7,6,2] => [1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => [2,3,4,5,7,6,1] => ? = 1
[1,3,4,7,2,5,6] => [1,2,3,4,7,6,5] => [1,2,3,4,7,5,6] => [2,3,4,6,7,5,1] => ? = 2
[1,3,4,7,5,2,6] => [1,2,3,4,7,6,5] => [1,2,3,4,7,5,6] => [2,3,4,6,7,5,1] => ? = 2
[1,3,5,7,2,4,6] => [1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => [2,3,5,4,7,6,1] => ? = 2
[1,3,5,7,2,6,4] => [1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => [2,3,5,4,7,6,1] => ? = 2
[1,3,5,7,4,2,6] => [1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => [2,3,5,4,7,6,1] => ? = 2
[1,3,5,7,4,6,2] => [1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => [2,3,5,4,7,6,1] => ? = 2
[1,3,7,2,4,5,6] => [1,2,3,7,6,5,4] => [1,2,3,7,4,5,6] => [2,3,5,6,7,4,1] => ? = 3
[1,3,7,4,2,5,6] => [1,2,3,7,6,5,4] => [1,2,3,7,4,5,6] => [2,3,5,6,7,4,1] => ? = 3
[1,4,7,2,3,5,6] => [1,2,4,3,7,6,5] => [1,2,4,3,7,5,6] => [2,4,3,6,7,5,1] => ? = 3
[1,4,7,2,5,3,6] => [1,2,4,3,7,6,5] => [1,2,4,3,7,5,6] => [2,4,3,6,7,5,1] => ? = 3
[1,4,7,3,2,5,6] => [1,2,4,3,7,6,5] => [1,2,4,3,7,5,6] => [2,4,3,6,7,5,1] => ? = 3
[1,4,7,3,5,2,6] => [1,2,4,3,7,6,5] => [1,2,4,3,7,5,6] => [2,4,3,6,7,5,1] => ? = 3
[1,7,2,3,4,5,6] => [1,2,7,6,5,4,3] => [1,2,7,3,4,5,6] => [2,4,5,6,7,3,1] => ? = 4
[1,7,3,2,4,5,6] => [1,2,7,6,5,4,3] => [1,2,7,3,4,5,6] => [2,4,5,6,7,3,1] => ? = 4
[2,1,3,4,7,5,6] => [1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => [2,3,4,5,7,6,1] => ? = 1
[2,1,3,4,7,6,5] => [1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => [2,3,4,5,7,6,1] => ? = 1
[2,1,3,5,7,4,6] => [1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => [2,3,4,5,7,6,1] => ? = 1
[2,1,3,5,7,6,4] => [1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => [2,3,4,5,7,6,1] => ? = 1
[2,1,3,7,4,5,6] => [1,2,3,4,7,6,5] => [1,2,3,4,7,5,6] => [2,3,4,6,7,5,1] => ? = 2
[2,1,3,7,5,4,6] => [1,2,3,4,7,6,5] => [1,2,3,4,7,5,6] => [2,3,4,6,7,5,1] => ? = 2
[2,1,4,3,7,5,6] => [1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => [2,3,4,5,7,6,1] => ? = 1
[2,1,4,3,7,6,5] => [1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => [2,3,4,5,7,6,1] => ? = 1
Description
The number of 2-excedences of a permutation.
This is the number of positions $1\leq i\leq n$ such that $\sigma(i)=i+2$.
Matching statistic: St000259
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St000259: Graphs ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 10%
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St000259: Graphs ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 10%
Values
[1] => [1] => ([],1)
=> ([],1)
=> 0
[1,2] => [1,2] => ([],2)
=> ([],1)
=> 0
[2,1] => [1,2] => ([],2)
=> ([],1)
=> 0
[1,2,3] => [1,2,3] => ([],3)
=> ([],1)
=> 0
[1,3,2] => [1,2,3] => ([],3)
=> ([],1)
=> 0
[2,1,3] => [1,2,3] => ([],3)
=> ([],1)
=> 0
[2,3,1] => [1,2,3] => ([],3)
=> ([],1)
=> 0
[3,1,2] => [1,3,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> ? = 1
[3,2,1] => [1,3,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> ? = 1
[1,2,3,4] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[1,2,4,3] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[1,3,2,4] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[1,3,4,2] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[1,4,2,3] => [1,2,4,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 1
[1,4,3,2] => [1,2,4,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 1
[2,1,3,4] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[2,1,4,3] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[2,3,1,4] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[2,3,4,1] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[2,4,1,3] => [1,2,4,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 1
[2,4,3,1] => [1,2,4,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 1
[3,1,2,4] => [1,3,2,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 1
[3,1,4,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 0
[3,2,1,4] => [1,3,2,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 1
[3,2,4,1] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 0
[3,4,1,2] => [1,3,2,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 1
[3,4,2,1] => [1,3,2,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 1
[4,1,2,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 2
[4,1,3,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 0
[4,2,1,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 2
[4,2,3,1] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 0
[4,3,1,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 0
[4,3,2,1] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 0
[1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,2,3,5,4] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,2,4,3,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,2,4,5,3] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,2,5,3,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[1,2,5,4,3] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[1,3,2,4,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,3,2,5,4] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,3,4,2,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,3,4,5,2] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,3,5,2,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[1,3,5,4,2] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[1,4,2,3,5] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[1,4,2,5,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[1,4,3,2,5] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[1,4,3,5,2] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[1,4,5,2,3] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[1,4,5,3,2] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[1,5,2,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 2
[1,5,2,4,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[1,5,3,2,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 2
[1,5,3,4,2] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[1,5,4,2,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[1,5,4,3,2] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[2,1,3,4,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,1,3,5,4] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,1,4,3,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,1,4,5,3] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,1,5,3,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[2,1,5,4,3] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[2,3,1,4,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,3,1,5,4] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,3,4,1,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,3,4,5,1] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,3,5,1,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[2,3,5,4,1] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[2,4,1,3,5] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[2,4,1,5,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[2,4,3,1,5] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[2,4,3,5,1] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[2,4,5,1,3] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[2,4,5,3,1] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[2,5,1,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 2
[2,5,1,4,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[2,5,3,1,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 2
[2,5,3,4,1] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[2,5,4,1,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[2,5,4,3,1] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[1,2,3,4,5,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,2,3,4,6,5] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,2,3,5,4,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,2,3,5,6,4] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,2,4,3,5,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,2,4,3,6,5] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,2,4,5,3,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,2,4,5,6,3] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,2,4,5,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,2,4,6,5] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,2,5,4,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,2,5,6,4] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,4,2,5,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,4,2,6,5] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,4,5,2,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,4,5,6,2] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[2,1,3,4,5,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[2,1,3,4,6,5] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[2,1,3,5,4,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
Description
The diameter of a connected graph.
This is the greatest distance between any pair of vertices.
Matching statistic: St000260
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 10%
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 10%
Values
[1] => [1] => ([],1)
=> ([],1)
=> 0
[1,2] => [1,2] => ([],2)
=> ([],1)
=> 0
[2,1] => [1,2] => ([],2)
=> ([],1)
=> 0
[1,2,3] => [1,2,3] => ([],3)
=> ([],1)
=> 0
[1,3,2] => [1,2,3] => ([],3)
=> ([],1)
=> 0
[2,1,3] => [1,2,3] => ([],3)
=> ([],1)
=> 0
[2,3,1] => [1,2,3] => ([],3)
=> ([],1)
=> 0
[3,1,2] => [1,3,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> ? = 1
[3,2,1] => [1,3,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> ? = 1
[1,2,3,4] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[1,2,4,3] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[1,3,2,4] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[1,3,4,2] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[1,4,2,3] => [1,2,4,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 1
[1,4,3,2] => [1,2,4,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 1
[2,1,3,4] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[2,1,4,3] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[2,3,1,4] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[2,3,4,1] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[2,4,1,3] => [1,2,4,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 1
[2,4,3,1] => [1,2,4,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 1
[3,1,2,4] => [1,3,2,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 1
[3,1,4,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 0
[3,2,1,4] => [1,3,2,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 1
[3,2,4,1] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 0
[3,4,1,2] => [1,3,2,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 1
[3,4,2,1] => [1,3,2,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 1
[4,1,2,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 2
[4,1,3,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 0
[4,2,1,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 2
[4,2,3,1] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 0
[4,3,1,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 0
[4,3,2,1] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 0
[1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,2,3,5,4] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,2,4,3,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,2,4,5,3] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,2,5,3,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[1,2,5,4,3] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[1,3,2,4,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,3,2,5,4] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,3,4,2,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,3,4,5,2] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,3,5,2,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[1,3,5,4,2] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[1,4,2,3,5] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[1,4,2,5,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[1,4,3,2,5] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[1,4,3,5,2] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[1,4,5,2,3] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[1,4,5,3,2] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[1,5,2,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 2
[1,5,2,4,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[1,5,3,2,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 2
[1,5,3,4,2] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[1,5,4,2,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[1,5,4,3,2] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[2,1,3,4,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,1,3,5,4] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,1,4,3,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,1,4,5,3] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,1,5,3,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[2,1,5,4,3] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[2,3,1,4,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,3,1,5,4] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,3,4,1,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,3,4,5,1] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,3,5,1,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[2,3,5,4,1] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[2,4,1,3,5] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[2,4,1,5,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[2,4,3,1,5] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[2,4,3,5,1] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[2,4,5,1,3] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[2,4,5,3,1] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[2,5,1,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 2
[2,5,1,4,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[2,5,3,1,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 2
[2,5,3,4,1] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[2,5,4,1,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[2,5,4,3,1] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[1,2,3,4,5,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,2,3,4,6,5] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,2,3,5,4,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,2,3,5,6,4] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,2,4,3,5,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,2,4,3,6,5] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,2,4,5,3,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,2,4,5,6,3] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,2,4,5,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,2,4,6,5] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,2,5,4,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,2,5,6,4] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,4,2,5,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,4,2,6,5] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,4,5,2,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,4,5,6,2] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[2,1,3,4,5,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[2,1,3,4,6,5] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[2,1,3,5,4,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
Description
The radius of a connected graph.
This is the minimum eccentricity of any vertex.
Matching statistic: St000302
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St000302: Graphs ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 10%
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St000302: Graphs ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 10%
Values
[1] => [1] => ([],1)
=> ([],1)
=> 0
[1,2] => [1,2] => ([],2)
=> ([],1)
=> 0
[2,1] => [1,2] => ([],2)
=> ([],1)
=> 0
[1,2,3] => [1,2,3] => ([],3)
=> ([],1)
=> 0
[1,3,2] => [1,2,3] => ([],3)
=> ([],1)
=> 0
[2,1,3] => [1,2,3] => ([],3)
=> ([],1)
=> 0
[2,3,1] => [1,2,3] => ([],3)
=> ([],1)
=> 0
[3,1,2] => [1,3,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> ? = 1
[3,2,1] => [1,3,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> ? = 1
[1,2,3,4] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[1,2,4,3] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[1,3,2,4] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[1,3,4,2] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[1,4,2,3] => [1,2,4,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 1
[1,4,3,2] => [1,2,4,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 1
[2,1,3,4] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[2,1,4,3] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[2,3,1,4] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[2,3,4,1] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[2,4,1,3] => [1,2,4,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 1
[2,4,3,1] => [1,2,4,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 1
[3,1,2,4] => [1,3,2,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 1
[3,1,4,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 0
[3,2,1,4] => [1,3,2,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 1
[3,2,4,1] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 0
[3,4,1,2] => [1,3,2,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 1
[3,4,2,1] => [1,3,2,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 1
[4,1,2,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 2
[4,1,3,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 0
[4,2,1,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 2
[4,2,3,1] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 0
[4,3,1,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 0
[4,3,2,1] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 0
[1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,2,3,5,4] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,2,4,3,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,2,4,5,3] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,2,5,3,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[1,2,5,4,3] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[1,3,2,4,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,3,2,5,4] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,3,4,2,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,3,4,5,2] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,3,5,2,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[1,3,5,4,2] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[1,4,2,3,5] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[1,4,2,5,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[1,4,3,2,5] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[1,4,3,5,2] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[1,4,5,2,3] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[1,4,5,3,2] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[1,5,2,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 2
[1,5,2,4,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[1,5,3,2,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 2
[1,5,3,4,2] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[1,5,4,2,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[1,5,4,3,2] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[2,1,3,4,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,1,3,5,4] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,1,4,3,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,1,4,5,3] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,1,5,3,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[2,1,5,4,3] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[2,3,1,4,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,3,1,5,4] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,3,4,1,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,3,4,5,1] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,3,5,1,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[2,3,5,4,1] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[2,4,1,3,5] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[2,4,1,5,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[2,4,3,1,5] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[2,4,3,5,1] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[2,4,5,1,3] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[2,4,5,3,1] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[2,5,1,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 2
[2,5,1,4,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[2,5,3,1,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 2
[2,5,3,4,1] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[2,5,4,1,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[2,5,4,3,1] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[1,2,3,4,5,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,2,3,4,6,5] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,2,3,5,4,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,2,3,5,6,4] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,2,4,3,5,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,2,4,3,6,5] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,2,4,5,3,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,2,4,5,6,3] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,2,4,5,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,2,4,6,5] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,2,5,4,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,2,5,6,4] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,4,2,5,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,4,2,6,5] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,4,5,2,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,4,5,6,2] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[2,1,3,4,5,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[2,1,3,4,6,5] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[2,1,3,5,4,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
Description
The determinant of the distance matrix of a connected graph.
Matching statistic: St000466
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St000466: Graphs ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 10%
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St000466: Graphs ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 10%
Values
[1] => [1] => ([],1)
=> ([],1)
=> 0
[1,2] => [1,2] => ([],2)
=> ([],1)
=> 0
[2,1] => [1,2] => ([],2)
=> ([],1)
=> 0
[1,2,3] => [1,2,3] => ([],3)
=> ([],1)
=> 0
[1,3,2] => [1,2,3] => ([],3)
=> ([],1)
=> 0
[2,1,3] => [1,2,3] => ([],3)
=> ([],1)
=> 0
[2,3,1] => [1,2,3] => ([],3)
=> ([],1)
=> 0
[3,1,2] => [1,3,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> ? = 1
[3,2,1] => [1,3,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> ? = 1
[1,2,3,4] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[1,2,4,3] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[1,3,2,4] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[1,3,4,2] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[1,4,2,3] => [1,2,4,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 1
[1,4,3,2] => [1,2,4,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 1
[2,1,3,4] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[2,1,4,3] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[2,3,1,4] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[2,3,4,1] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[2,4,1,3] => [1,2,4,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 1
[2,4,3,1] => [1,2,4,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 1
[3,1,2,4] => [1,3,2,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 1
[3,1,4,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 0
[3,2,1,4] => [1,3,2,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 1
[3,2,4,1] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 0
[3,4,1,2] => [1,3,2,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 1
[3,4,2,1] => [1,3,2,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 1
[4,1,2,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 2
[4,1,3,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 0
[4,2,1,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 2
[4,2,3,1] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 0
[4,3,1,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 0
[4,3,2,1] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 0
[1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,2,3,5,4] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,2,4,3,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,2,4,5,3] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,2,5,3,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[1,2,5,4,3] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[1,3,2,4,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,3,2,5,4] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,3,4,2,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,3,4,5,2] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,3,5,2,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[1,3,5,4,2] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[1,4,2,3,5] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[1,4,2,5,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[1,4,3,2,5] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[1,4,3,5,2] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[1,4,5,2,3] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[1,4,5,3,2] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[1,5,2,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 2
[1,5,2,4,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[1,5,3,2,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 2
[1,5,3,4,2] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[1,5,4,2,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[1,5,4,3,2] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[2,1,3,4,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,1,3,5,4] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,1,4,3,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,1,4,5,3] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,1,5,3,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[2,1,5,4,3] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[2,3,1,4,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,3,1,5,4] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,3,4,1,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,3,4,5,1] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,3,5,1,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[2,3,5,4,1] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[2,4,1,3,5] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[2,4,1,5,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[2,4,3,1,5] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[2,4,3,5,1] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[2,4,5,1,3] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[2,4,5,3,1] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[2,5,1,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 2
[2,5,1,4,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[2,5,3,1,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 2
[2,5,3,4,1] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[2,5,4,1,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[2,5,4,3,1] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[1,2,3,4,5,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,2,3,4,6,5] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,2,3,5,4,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,2,3,5,6,4] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,2,4,3,5,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,2,4,3,6,5] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,2,4,5,3,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,2,4,5,6,3] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,2,4,5,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,2,4,6,5] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,2,5,4,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,2,5,6,4] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,4,2,5,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,4,2,6,5] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,4,5,2,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,4,5,6,2] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[2,1,3,4,5,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[2,1,3,4,6,5] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[2,1,3,5,4,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
Description
The Gutman (or modified Schultz) index of a connected graph.
This is
$$\sum_{\{u,v\}\subseteq V} d(u)d(v)d(u,v)$$
where $d(u)$ is the degree of vertex $u$ and $d(u,v)$ is the distance between vertices $u$ and $v$.
For trees on $n$ vertices, the modified Schultz index is related to the Wiener index via $S^\ast(T)=4W(T)-(n-1)(2n-1)$ [1].
Matching statistic: St000467
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St000467: Graphs ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 10%
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St000467: Graphs ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 10%
Values
[1] => [1] => ([],1)
=> ([],1)
=> 0
[1,2] => [1,2] => ([],2)
=> ([],1)
=> 0
[2,1] => [1,2] => ([],2)
=> ([],1)
=> 0
[1,2,3] => [1,2,3] => ([],3)
=> ([],1)
=> 0
[1,3,2] => [1,2,3] => ([],3)
=> ([],1)
=> 0
[2,1,3] => [1,2,3] => ([],3)
=> ([],1)
=> 0
[2,3,1] => [1,2,3] => ([],3)
=> ([],1)
=> 0
[3,1,2] => [1,3,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> ? = 1
[3,2,1] => [1,3,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> ? = 1
[1,2,3,4] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[1,2,4,3] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[1,3,2,4] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[1,3,4,2] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[1,4,2,3] => [1,2,4,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 1
[1,4,3,2] => [1,2,4,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 1
[2,1,3,4] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[2,1,4,3] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[2,3,1,4] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[2,3,4,1] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[2,4,1,3] => [1,2,4,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 1
[2,4,3,1] => [1,2,4,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 1
[3,1,2,4] => [1,3,2,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 1
[3,1,4,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 0
[3,2,1,4] => [1,3,2,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 1
[3,2,4,1] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 0
[3,4,1,2] => [1,3,2,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 1
[3,4,2,1] => [1,3,2,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 1
[4,1,2,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 2
[4,1,3,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 0
[4,2,1,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 2
[4,2,3,1] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 0
[4,3,1,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 0
[4,3,2,1] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 0
[1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,2,3,5,4] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,2,4,3,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,2,4,5,3] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,2,5,3,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[1,2,5,4,3] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[1,3,2,4,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,3,2,5,4] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,3,4,2,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,3,4,5,2] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,3,5,2,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[1,3,5,4,2] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[1,4,2,3,5] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[1,4,2,5,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[1,4,3,2,5] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[1,4,3,5,2] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[1,4,5,2,3] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[1,4,5,3,2] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[1,5,2,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 2
[1,5,2,4,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[1,5,3,2,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 2
[1,5,3,4,2] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[1,5,4,2,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[1,5,4,3,2] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[2,1,3,4,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,1,3,5,4] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,1,4,3,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,1,4,5,3] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,1,5,3,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[2,1,5,4,3] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[2,3,1,4,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,3,1,5,4] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,3,4,1,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,3,4,5,1] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,3,5,1,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[2,3,5,4,1] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[2,4,1,3,5] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[2,4,1,5,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[2,4,3,1,5] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[2,4,3,5,1] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[2,4,5,1,3] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[2,4,5,3,1] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[2,5,1,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 2
[2,5,1,4,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[2,5,3,1,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 2
[2,5,3,4,1] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[2,5,4,1,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[2,5,4,3,1] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[1,2,3,4,5,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,2,3,4,6,5] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,2,3,5,4,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,2,3,5,6,4] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,2,4,3,5,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,2,4,3,6,5] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,2,4,5,3,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,2,4,5,6,3] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,2,4,5,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,2,4,6,5] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,2,5,4,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,2,5,6,4] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,4,2,5,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,4,2,6,5] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,4,5,2,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,4,5,6,2] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[2,1,3,4,5,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[2,1,3,4,6,5] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[2,1,3,5,4,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
Description
The hyper-Wiener index of a connected graph.
This is
$$
\sum_{\{u,v\}\subseteq V} d(u,v)+d(u,v)^2.
$$
Matching statistic: St000771
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St000771: Graphs ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 10%
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St000771: Graphs ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 10%
Values
[1] => [1] => ([],1)
=> ([],1)
=> 1 = 0 + 1
[1,2] => [1,2] => ([],2)
=> ([],1)
=> 1 = 0 + 1
[2,1] => [1,2] => ([],2)
=> ([],1)
=> 1 = 0 + 1
[1,2,3] => [1,2,3] => ([],3)
=> ([],1)
=> 1 = 0 + 1
[1,3,2] => [1,2,3] => ([],3)
=> ([],1)
=> 1 = 0 + 1
[2,1,3] => [1,2,3] => ([],3)
=> ([],1)
=> 1 = 0 + 1
[2,3,1] => [1,2,3] => ([],3)
=> ([],1)
=> 1 = 0 + 1
[3,1,2] => [1,3,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> ? = 1 + 1
[3,2,1] => [1,3,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> ? = 1 + 1
[1,2,3,4] => [1,2,3,4] => ([],4)
=> ([],1)
=> 1 = 0 + 1
[1,2,4,3] => [1,2,3,4] => ([],4)
=> ([],1)
=> 1 = 0 + 1
[1,3,2,4] => [1,2,3,4] => ([],4)
=> ([],1)
=> 1 = 0 + 1
[1,3,4,2] => [1,2,3,4] => ([],4)
=> ([],1)
=> 1 = 0 + 1
[1,4,2,3] => [1,2,4,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 1 + 1
[1,4,3,2] => [1,2,4,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 1 + 1
[2,1,3,4] => [1,2,3,4] => ([],4)
=> ([],1)
=> 1 = 0 + 1
[2,1,4,3] => [1,2,3,4] => ([],4)
=> ([],1)
=> 1 = 0 + 1
[2,3,1,4] => [1,2,3,4] => ([],4)
=> ([],1)
=> 1 = 0 + 1
[2,3,4,1] => [1,2,3,4] => ([],4)
=> ([],1)
=> 1 = 0 + 1
[2,4,1,3] => [1,2,4,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 1 + 1
[2,4,3,1] => [1,2,4,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 1 + 1
[3,1,2,4] => [1,3,2,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 1 + 1
[3,1,4,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 0 + 1
[3,2,1,4] => [1,3,2,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 1 + 1
[3,2,4,1] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 0 + 1
[3,4,1,2] => [1,3,2,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 1 + 1
[3,4,2,1] => [1,3,2,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 1 + 1
[4,1,2,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 2 + 1
[4,1,3,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 0 + 1
[4,2,1,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 2 + 1
[4,2,3,1] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 0 + 1
[4,3,1,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 0 + 1
[4,3,2,1] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 0 + 1
[1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 1 = 0 + 1
[1,2,3,5,4] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 1 = 0 + 1
[1,2,4,3,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 1 = 0 + 1
[1,2,4,5,3] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 1 = 0 + 1
[1,2,5,3,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 1 + 1
[1,2,5,4,3] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 1 + 1
[1,3,2,4,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 1 = 0 + 1
[1,3,2,5,4] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 1 = 0 + 1
[1,3,4,2,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 1 = 0 + 1
[1,3,4,5,2] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 1 = 0 + 1
[1,3,5,2,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 1 + 1
[1,3,5,4,2] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 1 + 1
[1,4,2,3,5] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 1 + 1
[1,4,2,5,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0 + 1
[1,4,3,2,5] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 1 + 1
[1,4,3,5,2] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0 + 1
[1,4,5,2,3] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 1 + 1
[1,4,5,3,2] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 1 + 1
[1,5,2,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 2 + 1
[1,5,2,4,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0 + 1
[1,5,3,2,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 2 + 1
[1,5,3,4,2] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0 + 1
[1,5,4,2,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0 + 1
[1,5,4,3,2] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0 + 1
[2,1,3,4,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 1 = 0 + 1
[2,1,3,5,4] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 1 = 0 + 1
[2,1,4,3,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 1 = 0 + 1
[2,1,4,5,3] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 1 = 0 + 1
[2,1,5,3,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 1 + 1
[2,1,5,4,3] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 1 + 1
[2,3,1,4,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 1 = 0 + 1
[2,3,1,5,4] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 1 = 0 + 1
[2,3,4,1,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 1 = 0 + 1
[2,3,4,5,1] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 1 = 0 + 1
[2,3,5,1,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 1 + 1
[2,3,5,4,1] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 1 + 1
[2,4,1,3,5] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 1 + 1
[2,4,1,5,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0 + 1
[2,4,3,1,5] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 1 + 1
[2,4,3,5,1] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0 + 1
[2,4,5,1,3] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 1 + 1
[2,4,5,3,1] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 1 + 1
[2,5,1,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 2 + 1
[2,5,1,4,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0 + 1
[2,5,3,1,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 2 + 1
[2,5,3,4,1] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0 + 1
[2,5,4,1,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0 + 1
[2,5,4,3,1] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0 + 1
[1,2,3,4,5,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 1 = 0 + 1
[1,2,3,4,6,5] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 1 = 0 + 1
[1,2,3,5,4,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 1 = 0 + 1
[1,2,3,5,6,4] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 1 = 0 + 1
[1,2,4,3,5,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 1 = 0 + 1
[1,2,4,3,6,5] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 1 = 0 + 1
[1,2,4,5,3,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 1 = 0 + 1
[1,2,4,5,6,3] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 1 = 0 + 1
[1,3,2,4,5,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 1 = 0 + 1
[1,3,2,4,6,5] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 1 = 0 + 1
[1,3,2,5,4,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 1 = 0 + 1
[1,3,2,5,6,4] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 1 = 0 + 1
[1,3,4,2,5,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 1 = 0 + 1
[1,3,4,2,6,5] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 1 = 0 + 1
[1,3,4,5,2,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 1 = 0 + 1
[1,3,4,5,6,2] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 1 = 0 + 1
[2,1,3,4,5,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 1 = 0 + 1
[2,1,3,4,6,5] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 1 = 0 + 1
[2,1,3,5,4,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 1 = 0 + 1
Description
The largest multiplicity of a 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 $2$.
The path on four vertices has eigenvalues $0, 4.7\dots, 6, 9.2\dots$ and therefore statistic $1$.
The following 3 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
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