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Your data matches 19 different statistics following compositions of up to 3 maps.
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Matching statistic: St000237
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Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000237: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000237: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●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] => 2
[3,2,1,4] => [1,3,2,4] => 1
[3,2,4,1] => [1,3,4,2] => 2
[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] => 0
[4,1,3,2] => [1,4,2,3] => 0
[4,2,1,3] => [1,4,3,2] => 0
[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] => 2
[1,4,3,2,5] => [1,2,4,3,5] => 1
[1,4,3,5,2] => [1,2,4,5,3] => 2
[1,4,5,2,3] => [1,2,4,3,5] => 1
Description
The number of small exceedances.
This is the number of indices $i$ such that $\pi_i=i+1$.
Matching statistic: St001810
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Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
St001810: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
St001810: 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] => [2,3,1] => 0
[1,3,2] => [1,2,3] => [2,3,1] => 0
[2,1,3] => [1,2,3] => [2,3,1] => 0
[2,3,1] => [1,2,3] => [2,3,1] => 0
[3,1,2] => [1,3,2] => [3,2,1] => 1
[3,2,1] => [1,3,2] => [3,2,1] => 1
[1,2,3,4] => [1,2,3,4] => [2,3,4,1] => 0
[1,2,4,3] => [1,2,3,4] => [2,3,4,1] => 0
[1,3,2,4] => [1,2,3,4] => [2,3,4,1] => 0
[1,3,4,2] => [1,2,3,4] => [2,3,4,1] => 0
[1,4,2,3] => [1,2,4,3] => [2,4,3,1] => 1
[1,4,3,2] => [1,2,4,3] => [2,4,3,1] => 1
[2,1,3,4] => [1,2,3,4] => [2,3,4,1] => 0
[2,1,4,3] => [1,2,3,4] => [2,3,4,1] => 0
[2,3,1,4] => [1,2,3,4] => [2,3,4,1] => 0
[2,3,4,1] => [1,2,3,4] => [2,3,4,1] => 0
[2,4,1,3] => [1,2,4,3] => [2,4,3,1] => 1
[2,4,3,1] => [1,2,4,3] => [2,4,3,1] => 1
[3,1,2,4] => [1,3,2,4] => [3,2,4,1] => 1
[3,1,4,2] => [1,3,4,2] => [4,2,3,1] => 2
[3,2,1,4] => [1,3,2,4] => [3,2,4,1] => 1
[3,2,4,1] => [1,3,4,2] => [4,2,3,1] => 2
[3,4,1,2] => [1,3,2,4] => [3,2,4,1] => 1
[3,4,2,1] => [1,3,2,4] => [3,2,4,1] => 1
[4,1,2,3] => [1,4,3,2] => [4,3,2,1] => 0
[4,1,3,2] => [1,4,2,3] => [3,4,2,1] => 0
[4,2,1,3] => [1,4,3,2] => [4,3,2,1] => 0
[4,2,3,1] => [1,4,2,3] => [3,4,2,1] => 0
[4,3,1,2] => [1,4,2,3] => [3,4,2,1] => 0
[4,3,2,1] => [1,4,2,3] => [3,4,2,1] => 0
[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] => [2,3,4,5,1] => 0
[1,2,4,3,5] => [1,2,3,4,5] => [2,3,4,5,1] => 0
[1,2,4,5,3] => [1,2,3,4,5] => [2,3,4,5,1] => 0
[1,2,5,3,4] => [1,2,3,5,4] => [2,3,5,4,1] => 1
[1,2,5,4,3] => [1,2,3,5,4] => [2,3,5,4,1] => 1
[1,3,2,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => 0
[1,3,2,5,4] => [1,2,3,4,5] => [2,3,4,5,1] => 0
[1,3,4,2,5] => [1,2,3,4,5] => [2,3,4,5,1] => 0
[1,3,4,5,2] => [1,2,3,4,5] => [2,3,4,5,1] => 0
[1,3,5,2,4] => [1,2,3,5,4] => [2,3,5,4,1] => 1
[1,3,5,4,2] => [1,2,3,5,4] => [2,3,5,4,1] => 1
[1,4,2,3,5] => [1,2,4,3,5] => [2,4,3,5,1] => 1
[1,4,2,5,3] => [1,2,4,5,3] => [2,5,3,4,1] => 2
[1,4,3,2,5] => [1,2,4,3,5] => [2,4,3,5,1] => 1
[1,4,3,5,2] => [1,2,4,5,3] => [2,5,3,4,1] => 2
[1,4,5,2,3] => [1,2,4,3,5] => [2,4,3,5,1] => 1
Description
The number of fixed points of a permutation smaller than its largest moved point.
Matching statistic: St000214
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000214: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00066: Permutations —inverse⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000214: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1] => 0
[1,2] => [1,2] => [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] => [1,4,2,3] => [1,4,3,2] => 2
[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] => [1,4,2,3] => [1,4,3,2] => 2
[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] => [1,4,3,2] => [1,3,4,2] => 0
[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] => [1,4,3,2] => [1,3,4,2] => 0
[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] => [1,2,5,3,4] => [1,2,5,4,3] => 2
[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] => [1,2,5,3,4] => [1,2,5,4,3] => 2
[1,4,5,2,3] => [1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 1
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: St000648
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
St000648: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00066: Permutations —inverse⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
St000648: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1] => 0
[1,2] => [1,2] => [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] => [1,4,2,3] => [3,4,2,1] => 2
[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] => [1,4,2,3] => [3,4,2,1] => 2
[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] => [1,4,3,2] => [4,3,2,1] => 0
[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] => [1,4,3,2] => [4,3,2,1] => 0
[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] => [1,2,5,3,4] => [2,4,5,3,1] => 2
[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] => [1,2,5,3,4] => [2,4,5,3,1] => 2
[1,4,5,2,3] => [1,2,4,3,5] => [1,2,4,3,5] => [2,4,3,5,1] => 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: St000248
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000248: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000248: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => {{1}}
=> ? = 0
[1,2] => [1,2] => {{1},{2}}
=> 0
[2,1] => [1,2] => {{1},{2}}
=> 0
[1,2,3] => [1,2,3] => {{1},{2},{3}}
=> 0
[1,3,2] => [1,2,3] => {{1},{2},{3}}
=> 0
[2,1,3] => [1,2,3] => {{1},{2},{3}}
=> 0
[2,3,1] => [1,2,3] => {{1},{2},{3}}
=> 0
[3,1,2] => [1,3,2] => {{1},{2,3}}
=> 1
[3,2,1] => [1,3,2] => {{1},{2,3}}
=> 1
[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}}
=> 0
[1,3,2,4] => [1,2,3,4] => {{1},{2},{3},{4}}
=> 0
[1,3,4,2] => [1,2,3,4] => {{1},{2},{3},{4}}
=> 0
[1,4,2,3] => [1,2,4,3] => {{1},{2},{3,4}}
=> 1
[1,4,3,2] => [1,2,4,3] => {{1},{2},{3,4}}
=> 1
[2,1,3,4] => [1,2,3,4] => {{1},{2},{3},{4}}
=> 0
[2,1,4,3] => [1,2,3,4] => {{1},{2},{3},{4}}
=> 0
[2,3,1,4] => [1,2,3,4] => {{1},{2},{3},{4}}
=> 0
[2,3,4,1] => [1,2,3,4] => {{1},{2},{3},{4}}
=> 0
[2,4,1,3] => [1,2,4,3] => {{1},{2},{3,4}}
=> 1
[2,4,3,1] => [1,2,4,3] => {{1},{2},{3,4}}
=> 1
[3,1,2,4] => [1,3,2,4] => {{1},{2,3},{4}}
=> 1
[3,1,4,2] => [1,3,4,2] => {{1},{2,3,4}}
=> 2
[3,2,1,4] => [1,3,2,4] => {{1},{2,3},{4}}
=> 1
[3,2,4,1] => [1,3,4,2] => {{1},{2,3,4}}
=> 2
[3,4,1,2] => [1,3,2,4] => {{1},{2,3},{4}}
=> 1
[3,4,2,1] => [1,3,2,4] => {{1},{2,3},{4}}
=> 1
[4,1,2,3] => [1,4,3,2] => {{1},{2,4},{3}}
=> 0
[4,1,3,2] => [1,4,2,3] => {{1},{2,4},{3}}
=> 0
[4,2,1,3] => [1,4,3,2] => {{1},{2,4},{3}}
=> 0
[4,2,3,1] => [1,4,2,3] => {{1},{2,4},{3}}
=> 0
[4,3,1,2] => [1,4,2,3] => {{1},{2,4},{3}}
=> 0
[4,3,2,1] => [1,4,2,3] => {{1},{2,4},{3}}
=> 0
[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}}
=> 0
[1,2,4,3,5] => [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}}
=> 0
[1,2,5,3,4] => [1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> 1
[1,2,5,4,3] => [1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> 1
[1,3,2,4,5] => [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}}
=> 0
[1,3,4,2,5] => [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}}
=> 0
[1,3,5,2,4] => [1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> 1
[1,3,5,4,2] => [1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> 1
[1,4,2,3,5] => [1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> 1
[1,4,2,5,3] => [1,2,4,5,3] => {{1},{2},{3,4,5}}
=> 2
[1,4,3,2,5] => [1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> 1
[1,4,3,5,2] => [1,2,4,5,3] => {{1},{2},{3,4,5}}
=> 2
[1,4,5,2,3] => [1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> 1
[1,4,5,3,2] => [1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> 1
Description
The number of anti-singletons of a set partition.
An anti-singleton of a set partition $S$ is an index $i$ such that $i$ and $i+1$ (considered cyclically) are both in the same block of $S$.
For noncrossing set partitions, this is also the number of singletons of the image of $S$ under the Kreweras complement.
Matching statistic: St000502
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000502: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000502: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => {{1}}
=> ? = 0
[1,2] => [1,2] => {{1},{2}}
=> 0
[2,1] => [1,2] => {{1},{2}}
=> 0
[1,2,3] => [1,2,3] => {{1},{2},{3}}
=> 0
[1,3,2] => [1,2,3] => {{1},{2},{3}}
=> 0
[2,1,3] => [1,2,3] => {{1},{2},{3}}
=> 0
[2,3,1] => [1,2,3] => {{1},{2},{3}}
=> 0
[3,1,2] => [1,3,2] => {{1},{2,3}}
=> 1
[3,2,1] => [1,3,2] => {{1},{2,3}}
=> 1
[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}}
=> 0
[1,3,2,4] => [1,2,3,4] => {{1},{2},{3},{4}}
=> 0
[1,3,4,2] => [1,2,3,4] => {{1},{2},{3},{4}}
=> 0
[1,4,2,3] => [1,2,4,3] => {{1},{2},{3,4}}
=> 1
[1,4,3,2] => [1,2,4,3] => {{1},{2},{3,4}}
=> 1
[2,1,3,4] => [1,2,3,4] => {{1},{2},{3},{4}}
=> 0
[2,1,4,3] => [1,2,3,4] => {{1},{2},{3},{4}}
=> 0
[2,3,1,4] => [1,2,3,4] => {{1},{2},{3},{4}}
=> 0
[2,3,4,1] => [1,2,3,4] => {{1},{2},{3},{4}}
=> 0
[2,4,1,3] => [1,2,4,3] => {{1},{2},{3,4}}
=> 1
[2,4,3,1] => [1,2,4,3] => {{1},{2},{3,4}}
=> 1
[3,1,2,4] => [1,3,2,4] => {{1},{2,3},{4}}
=> 1
[3,1,4,2] => [1,3,4,2] => {{1},{2,3,4}}
=> 2
[3,2,1,4] => [1,3,2,4] => {{1},{2,3},{4}}
=> 1
[3,2,4,1] => [1,3,4,2] => {{1},{2,3,4}}
=> 2
[3,4,1,2] => [1,3,2,4] => {{1},{2,3},{4}}
=> 1
[3,4,2,1] => [1,3,2,4] => {{1},{2,3},{4}}
=> 1
[4,1,2,3] => [1,4,3,2] => {{1},{2,4},{3}}
=> 0
[4,1,3,2] => [1,4,2,3] => {{1},{2,4},{3}}
=> 0
[4,2,1,3] => [1,4,3,2] => {{1},{2,4},{3}}
=> 0
[4,2,3,1] => [1,4,2,3] => {{1},{2,4},{3}}
=> 0
[4,3,1,2] => [1,4,2,3] => {{1},{2,4},{3}}
=> 0
[4,3,2,1] => [1,4,2,3] => {{1},{2,4},{3}}
=> 0
[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}}
=> 0
[1,2,4,3,5] => [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}}
=> 0
[1,2,5,3,4] => [1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> 1
[1,2,5,4,3] => [1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> 1
[1,3,2,4,5] => [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}}
=> 0
[1,3,4,2,5] => [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}}
=> 0
[1,3,5,2,4] => [1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> 1
[1,3,5,4,2] => [1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> 1
[1,4,2,3,5] => [1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> 1
[1,4,2,5,3] => [1,2,4,5,3] => {{1},{2},{3,4,5}}
=> 2
[1,4,3,2,5] => [1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> 1
[1,4,3,5,2] => [1,2,4,5,3] => {{1},{2},{3,4,5}}
=> 2
[1,4,5,2,3] => [1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> 1
[1,4,5,3,2] => [1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> 1
Description
The number of successions of a set partitions.
This is the number of indices $i$ such that $i$ and $i+1$ belonging to the same block.
Matching statistic: St000247
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
St000247: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
St000247: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => {{1}}
=> ? = 0
[1,2] => [1,2] => [2,1] => {{1,2}}
=> 0
[2,1] => [1,2] => [2,1] => {{1,2}}
=> 0
[1,2,3] => [1,2,3] => [2,3,1] => {{1,2,3}}
=> 0
[1,3,2] => [1,2,3] => [2,3,1] => {{1,2,3}}
=> 0
[2,1,3] => [1,2,3] => [2,3,1] => {{1,2,3}}
=> 0
[2,3,1] => [1,2,3] => [2,3,1] => {{1,2,3}}
=> 0
[3,1,2] => [1,3,2] => [3,2,1] => {{1,3},{2}}
=> 1
[3,2,1] => [1,3,2] => [3,2,1] => {{1,3},{2}}
=> 1
[1,2,3,4] => [1,2,3,4] => [2,3,4,1] => {{1,2,3,4}}
=> 0
[1,2,4,3] => [1,2,3,4] => [2,3,4,1] => {{1,2,3,4}}
=> 0
[1,3,2,4] => [1,2,3,4] => [2,3,4,1] => {{1,2,3,4}}
=> 0
[1,3,4,2] => [1,2,3,4] => [2,3,4,1] => {{1,2,3,4}}
=> 0
[1,4,2,3] => [1,2,4,3] => [2,4,3,1] => {{1,2,4},{3}}
=> 1
[1,4,3,2] => [1,2,4,3] => [2,4,3,1] => {{1,2,4},{3}}
=> 1
[2,1,3,4] => [1,2,3,4] => [2,3,4,1] => {{1,2,3,4}}
=> 0
[2,1,4,3] => [1,2,3,4] => [2,3,4,1] => {{1,2,3,4}}
=> 0
[2,3,1,4] => [1,2,3,4] => [2,3,4,1] => {{1,2,3,4}}
=> 0
[2,3,4,1] => [1,2,3,4] => [2,3,4,1] => {{1,2,3,4}}
=> 0
[2,4,1,3] => [1,2,4,3] => [2,4,3,1] => {{1,2,4},{3}}
=> 1
[2,4,3,1] => [1,2,4,3] => [2,4,3,1] => {{1,2,4},{3}}
=> 1
[3,1,2,4] => [1,3,2,4] => [3,2,4,1] => {{1,3,4},{2}}
=> 1
[3,1,4,2] => [1,3,4,2] => [4,2,3,1] => {{1,4},{2},{3}}
=> 2
[3,2,1,4] => [1,3,2,4] => [3,2,4,1] => {{1,3,4},{2}}
=> 1
[3,2,4,1] => [1,3,4,2] => [4,2,3,1] => {{1,4},{2},{3}}
=> 2
[3,4,1,2] => [1,3,2,4] => [3,2,4,1] => {{1,3,4},{2}}
=> 1
[3,4,2,1] => [1,3,2,4] => [3,2,4,1] => {{1,3,4},{2}}
=> 1
[4,1,2,3] => [1,4,3,2] => [4,3,2,1] => {{1,4},{2,3}}
=> 0
[4,1,3,2] => [1,4,2,3] => [3,4,2,1] => {{1,2,3,4}}
=> 0
[4,2,1,3] => [1,4,3,2] => [4,3,2,1] => {{1,4},{2,3}}
=> 0
[4,2,3,1] => [1,4,2,3] => [3,4,2,1] => {{1,2,3,4}}
=> 0
[4,3,1,2] => [1,4,2,3] => [3,4,2,1] => {{1,2,3,4}}
=> 0
[4,3,2,1] => [1,4,2,3] => [3,4,2,1] => {{1,2,3,4}}
=> 0
[1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => {{1,2,3,4,5}}
=> 0
[1,2,3,5,4] => [1,2,3,4,5] => [2,3,4,5,1] => {{1,2,3,4,5}}
=> 0
[1,2,4,3,5] => [1,2,3,4,5] => [2,3,4,5,1] => {{1,2,3,4,5}}
=> 0
[1,2,4,5,3] => [1,2,3,4,5] => [2,3,4,5,1] => {{1,2,3,4,5}}
=> 0
[1,2,5,3,4] => [1,2,3,5,4] => [2,3,5,4,1] => {{1,2,3,5},{4}}
=> 1
[1,2,5,4,3] => [1,2,3,5,4] => [2,3,5,4,1] => {{1,2,3,5},{4}}
=> 1
[1,3,2,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => {{1,2,3,4,5}}
=> 0
[1,3,2,5,4] => [1,2,3,4,5] => [2,3,4,5,1] => {{1,2,3,4,5}}
=> 0
[1,3,4,2,5] => [1,2,3,4,5] => [2,3,4,5,1] => {{1,2,3,4,5}}
=> 0
[1,3,4,5,2] => [1,2,3,4,5] => [2,3,4,5,1] => {{1,2,3,4,5}}
=> 0
[1,3,5,2,4] => [1,2,3,5,4] => [2,3,5,4,1] => {{1,2,3,5},{4}}
=> 1
[1,3,5,4,2] => [1,2,3,5,4] => [2,3,5,4,1] => {{1,2,3,5},{4}}
=> 1
[1,4,2,3,5] => [1,2,4,3,5] => [2,4,3,5,1] => {{1,2,4,5},{3}}
=> 1
[1,4,2,5,3] => [1,2,4,5,3] => [2,5,3,4,1] => {{1,2,5},{3},{4}}
=> 2
[1,4,3,2,5] => [1,2,4,3,5] => [2,4,3,5,1] => {{1,2,4,5},{3}}
=> 1
[1,4,3,5,2] => [1,2,4,5,3] => [2,5,3,4,1] => {{1,2,5},{3},{4}}
=> 2
[1,4,5,2,3] => [1,2,4,3,5] => [2,4,3,5,1] => {{1,2,4,5},{3}}
=> 1
[1,4,5,3,2] => [1,2,4,3,5] => [2,4,3,5,1] => {{1,2,4,5},{3}}
=> 1
Description
The number of singleton blocks of a set partition.
Matching statistic: St000504
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
Mp00171: Set partitions —intertwining number to dual major index⟶ Set partitions
St000504: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
Mp00171: Set partitions —intertwining number to dual major index⟶ Set partitions
St000504: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => {{1}}
=> {{1}}
=> ? = 0 + 1
[1,2] => [1,2] => {{1},{2}}
=> {{1},{2}}
=> 1 = 0 + 1
[2,1] => [1,2] => {{1},{2}}
=> {{1},{2}}
=> 1 = 0 + 1
[1,2,3] => [1,2,3] => {{1},{2},{3}}
=> {{1},{2},{3}}
=> 1 = 0 + 1
[1,3,2] => [1,2,3] => {{1},{2},{3}}
=> {{1},{2},{3}}
=> 1 = 0 + 1
[2,1,3] => [1,2,3] => {{1},{2},{3}}
=> {{1},{2},{3}}
=> 1 = 0 + 1
[2,3,1] => [1,2,3] => {{1},{2},{3}}
=> {{1},{2},{3}}
=> 1 = 0 + 1
[3,1,2] => [1,3,2] => {{1},{2,3}}
=> {{1,3},{2}}
=> 2 = 1 + 1
[3,2,1] => [1,3,2] => {{1},{2,3}}
=> {{1,3},{2}}
=> 2 = 1 + 1
[1,2,3,4] => [1,2,3,4] => {{1},{2},{3},{4}}
=> {{1},{2},{3},{4}}
=> 1 = 0 + 1
[1,2,4,3] => [1,2,3,4] => {{1},{2},{3},{4}}
=> {{1},{2},{3},{4}}
=> 1 = 0 + 1
[1,3,2,4] => [1,2,3,4] => {{1},{2},{3},{4}}
=> {{1},{2},{3},{4}}
=> 1 = 0 + 1
[1,3,4,2] => [1,2,3,4] => {{1},{2},{3},{4}}
=> {{1},{2},{3},{4}}
=> 1 = 0 + 1
[1,4,2,3] => [1,2,4,3] => {{1},{2},{3,4}}
=> {{1,4},{2},{3}}
=> 2 = 1 + 1
[1,4,3,2] => [1,2,4,3] => {{1},{2},{3,4}}
=> {{1,4},{2},{3}}
=> 2 = 1 + 1
[2,1,3,4] => [1,2,3,4] => {{1},{2},{3},{4}}
=> {{1},{2},{3},{4}}
=> 1 = 0 + 1
[2,1,4,3] => [1,2,3,4] => {{1},{2},{3},{4}}
=> {{1},{2},{3},{4}}
=> 1 = 0 + 1
[2,3,1,4] => [1,2,3,4] => {{1},{2},{3},{4}}
=> {{1},{2},{3},{4}}
=> 1 = 0 + 1
[2,3,4,1] => [1,2,3,4] => {{1},{2},{3},{4}}
=> {{1},{2},{3},{4}}
=> 1 = 0 + 1
[2,4,1,3] => [1,2,4,3] => {{1},{2},{3,4}}
=> {{1,4},{2},{3}}
=> 2 = 1 + 1
[2,4,3,1] => [1,2,4,3] => {{1},{2},{3,4}}
=> {{1,4},{2},{3}}
=> 2 = 1 + 1
[3,1,2,4] => [1,3,2,4] => {{1},{2,3},{4}}
=> {{1,3},{2},{4}}
=> 2 = 1 + 1
[3,1,4,2] => [1,3,4,2] => {{1},{2,3,4}}
=> {{1,3,4},{2}}
=> 3 = 2 + 1
[3,2,1,4] => [1,3,2,4] => {{1},{2,3},{4}}
=> {{1,3},{2},{4}}
=> 2 = 1 + 1
[3,2,4,1] => [1,3,4,2] => {{1},{2,3,4}}
=> {{1,3,4},{2}}
=> 3 = 2 + 1
[3,4,1,2] => [1,3,2,4] => {{1},{2,3},{4}}
=> {{1,3},{2},{4}}
=> 2 = 1 + 1
[3,4,2,1] => [1,3,2,4] => {{1},{2,3},{4}}
=> {{1,3},{2},{4}}
=> 2 = 1 + 1
[4,1,2,3] => [1,4,3,2] => {{1},{2,4},{3}}
=> {{1},{2,4},{3}}
=> 1 = 0 + 1
[4,1,3,2] => [1,4,2,3] => {{1},{2,4},{3}}
=> {{1},{2,4},{3}}
=> 1 = 0 + 1
[4,2,1,3] => [1,4,3,2] => {{1},{2,4},{3}}
=> {{1},{2,4},{3}}
=> 1 = 0 + 1
[4,2,3,1] => [1,4,2,3] => {{1},{2,4},{3}}
=> {{1},{2,4},{3}}
=> 1 = 0 + 1
[4,3,1,2] => [1,4,2,3] => {{1},{2,4},{3}}
=> {{1},{2,4},{3}}
=> 1 = 0 + 1
[4,3,2,1] => [1,4,2,3] => {{1},{2,4},{3}}
=> {{1},{2,4},{3}}
=> 1 = 0 + 1
[1,2,3,4,5] => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> {{1},{2},{3},{4},{5}}
=> 1 = 0 + 1
[1,2,3,5,4] => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> {{1},{2},{3},{4},{5}}
=> 1 = 0 + 1
[1,2,4,3,5] => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> {{1},{2},{3},{4},{5}}
=> 1 = 0 + 1
[1,2,4,5,3] => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> {{1},{2},{3},{4},{5}}
=> 1 = 0 + 1
[1,2,5,3,4] => [1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> {{1,5},{2},{3},{4}}
=> 2 = 1 + 1
[1,2,5,4,3] => [1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> {{1,5},{2},{3},{4}}
=> 2 = 1 + 1
[1,3,2,4,5] => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> {{1},{2},{3},{4},{5}}
=> 1 = 0 + 1
[1,3,2,5,4] => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> {{1},{2},{3},{4},{5}}
=> 1 = 0 + 1
[1,3,4,2,5] => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> {{1},{2},{3},{4},{5}}
=> 1 = 0 + 1
[1,3,4,5,2] => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> {{1},{2},{3},{4},{5}}
=> 1 = 0 + 1
[1,3,5,2,4] => [1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> {{1,5},{2},{3},{4}}
=> 2 = 1 + 1
[1,3,5,4,2] => [1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> {{1,5},{2},{3},{4}}
=> 2 = 1 + 1
[1,4,2,3,5] => [1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> {{1,4},{2},{3},{5}}
=> 2 = 1 + 1
[1,4,2,5,3] => [1,2,4,5,3] => {{1},{2},{3,4,5}}
=> {{1,4,5},{2},{3}}
=> 3 = 2 + 1
[1,4,3,2,5] => [1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> {{1,4},{2},{3},{5}}
=> 2 = 1 + 1
[1,4,3,5,2] => [1,2,4,5,3] => {{1},{2},{3,4,5}}
=> {{1,4,5},{2},{3}}
=> 3 = 2 + 1
[1,4,5,2,3] => [1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> {{1,4},{2},{3},{5}}
=> 2 = 1 + 1
[1,4,5,3,2] => [1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> {{1,4},{2},{3},{5}}
=> 2 = 1 + 1
Description
The cardinality of the first block of a set partition.
The number of partitions of $\{1,\ldots,n\}$ into $k$ blocks in which the first block has cardinality $j+1$ is given by $\binom{n-1}{j}S(n-j-1,k-1)$, see [1, Theorem 1.1] and the references therein. Here, $S(n,k)$ are the ''Stirling numbers of the second kind'' counting all set partitions of $\{1,\ldots,n\}$ into $k$ blocks [2].
Matching statistic: St000894
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00088: Permutations —Kreweras complement⟶ Permutations
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
St000894: Alternating sign matrices ⟶ ℤResult quality: 34% ●values known / values provided: 34%●distinct values known / distinct values provided: 100%
Mp00088: Permutations —Kreweras complement⟶ Permutations
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
St000894: Alternating sign matrices ⟶ ℤResult quality: 34% ●values known / values provided: 34%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [[1]]
=> ? = 0
[1,2] => [1,2] => [2,1] => [[0,1],[1,0]]
=> 0
[2,1] => [1,2] => [2,1] => [[0,1],[1,0]]
=> 0
[1,2,3] => [1,2,3] => [2,3,1] => [[0,0,1],[1,0,0],[0,1,0]]
=> 0
[1,3,2] => [1,2,3] => [2,3,1] => [[0,0,1],[1,0,0],[0,1,0]]
=> 0
[2,1,3] => [1,2,3] => [2,3,1] => [[0,0,1],[1,0,0],[0,1,0]]
=> 0
[2,3,1] => [1,2,3] => [2,3,1] => [[0,0,1],[1,0,0],[0,1,0]]
=> 0
[3,1,2] => [1,3,2] => [2,1,3] => [[0,1,0],[1,0,0],[0,0,1]]
=> 1
[3,2,1] => [1,3,2] => [2,1,3] => [[0,1,0],[1,0,0],[0,0,1]]
=> 1
[1,2,3,4] => [1,2,3,4] => [2,3,4,1] => [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> 0
[1,2,4,3] => [1,2,3,4] => [2,3,4,1] => [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> 0
[1,3,2,4] => [1,2,3,4] => [2,3,4,1] => [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> 0
[1,3,4,2] => [1,2,3,4] => [2,3,4,1] => [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> 0
[1,4,2,3] => [1,2,4,3] => [2,3,1,4] => [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> 1
[1,4,3,2] => [1,2,4,3] => [2,3,1,4] => [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> 1
[2,1,3,4] => [1,2,3,4] => [2,3,4,1] => [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> 0
[2,1,4,3] => [1,2,3,4] => [2,3,4,1] => [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> 0
[2,3,1,4] => [1,2,3,4] => [2,3,4,1] => [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> 0
[2,3,4,1] => [1,2,3,4] => [2,3,4,1] => [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> 0
[2,4,1,3] => [1,2,4,3] => [2,3,1,4] => [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> 1
[2,4,3,1] => [1,2,4,3] => [2,3,1,4] => [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> 1
[3,1,2,4] => [1,3,2,4] => [2,4,3,1] => [[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
=> 1
[3,1,4,2] => [1,3,4,2] => [2,1,3,4] => [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> 2
[3,2,1,4] => [1,3,2,4] => [2,4,3,1] => [[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
=> 1
[3,2,4,1] => [1,3,4,2] => [2,1,3,4] => [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> 2
[3,4,1,2] => [1,3,2,4] => [2,4,3,1] => [[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
=> 1
[3,4,2,1] => [1,3,2,4] => [2,4,3,1] => [[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
=> 1
[4,1,2,3] => [1,4,3,2] => [2,1,4,3] => [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> 0
[4,1,3,2] => [1,4,2,3] => [2,4,1,3] => [[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
=> 0
[4,2,1,3] => [1,4,3,2] => [2,1,4,3] => [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> 0
[4,2,3,1] => [1,4,2,3] => [2,4,1,3] => [[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
=> 0
[4,3,1,2] => [1,4,2,3] => [2,4,1,3] => [[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
=> 0
[4,3,2,1] => [1,4,2,3] => [2,4,1,3] => [[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
=> 0
[1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> 0
[1,2,3,5,4] => [1,2,3,4,5] => [2,3,4,5,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> 0
[1,2,4,3,5] => [1,2,3,4,5] => [2,3,4,5,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> 0
[1,2,4,5,3] => [1,2,3,4,5] => [2,3,4,5,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> 0
[1,2,5,3,4] => [1,2,3,5,4] => [2,3,4,1,5] => [[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> 1
[1,2,5,4,3] => [1,2,3,5,4] => [2,3,4,1,5] => [[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> 1
[1,3,2,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> 0
[1,3,2,5,4] => [1,2,3,4,5] => [2,3,4,5,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> 0
[1,3,4,2,5] => [1,2,3,4,5] => [2,3,4,5,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> 0
[1,3,4,5,2] => [1,2,3,4,5] => [2,3,4,5,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> 0
[1,3,5,2,4] => [1,2,3,5,4] => [2,3,4,1,5] => [[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> 1
[1,3,5,4,2] => [1,2,3,5,4] => [2,3,4,1,5] => [[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> 1
[1,4,2,3,5] => [1,2,4,3,5] => [2,3,5,4,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0]]
=> 1
[1,4,2,5,3] => [1,2,4,5,3] => [2,3,1,4,5] => [[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> 2
[1,4,3,2,5] => [1,2,4,3,5] => [2,3,5,4,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0]]
=> 1
[1,4,3,5,2] => [1,2,4,5,3] => [2,3,1,4,5] => [[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> 2
[1,4,5,2,3] => [1,2,4,3,5] => [2,3,5,4,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0]]
=> 1
[1,4,5,3,2] => [1,2,4,3,5] => [2,3,5,4,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0]]
=> 1
[1,2,3,4,5,6] => [1,2,3,4,5,6] => [2,3,4,5,6,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> ? = 0
[1,2,3,4,6,5] => [1,2,3,4,5,6] => [2,3,4,5,6,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> ? = 0
[1,2,3,5,4,6] => [1,2,3,4,5,6] => [2,3,4,5,6,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> ? = 0
[1,2,3,5,6,4] => [1,2,3,4,5,6] => [2,3,4,5,6,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> ? = 0
[1,2,4,3,5,6] => [1,2,3,4,5,6] => [2,3,4,5,6,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> ? = 0
[1,2,4,3,6,5] => [1,2,3,4,5,6] => [2,3,4,5,6,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> ? = 0
[1,2,4,5,3,6] => [1,2,3,4,5,6] => [2,3,4,5,6,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> ? = 0
[1,2,4,5,6,3] => [1,2,3,4,5,6] => [2,3,4,5,6,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> ? = 0
[1,2,5,3,4,6] => [1,2,3,5,4,6] => [2,3,4,6,5,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0]]
=> ? = 1
[1,2,5,4,3,6] => [1,2,3,5,4,6] => [2,3,4,6,5,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0]]
=> ? = 1
[1,2,5,6,3,4] => [1,2,3,5,4,6] => [2,3,4,6,5,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0]]
=> ? = 1
[1,2,5,6,4,3] => [1,2,3,5,4,6] => [2,3,4,6,5,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0]]
=> ? = 1
[1,2,6,3,5,4] => [1,2,3,6,4,5] => [2,3,4,6,1,5] => [[0,0,0,0,1,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0]]
=> ? = 0
[1,2,6,4,5,3] => [1,2,3,6,4,5] => [2,3,4,6,1,5] => [[0,0,0,0,1,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0]]
=> ? = 0
[1,2,6,5,3,4] => [1,2,3,6,4,5] => [2,3,4,6,1,5] => [[0,0,0,0,1,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0]]
=> ? = 0
[1,2,6,5,4,3] => [1,2,3,6,4,5] => [2,3,4,6,1,5] => [[0,0,0,0,1,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0]]
=> ? = 0
[1,3,2,4,5,6] => [1,2,3,4,5,6] => [2,3,4,5,6,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> ? = 0
[1,3,2,4,6,5] => [1,2,3,4,5,6] => [2,3,4,5,6,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> ? = 0
[1,3,2,5,4,6] => [1,2,3,4,5,6] => [2,3,4,5,6,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> ? = 0
[1,3,2,5,6,4] => [1,2,3,4,5,6] => [2,3,4,5,6,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> ? = 0
[1,3,4,2,5,6] => [1,2,3,4,5,6] => [2,3,4,5,6,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> ? = 0
[1,3,4,2,6,5] => [1,2,3,4,5,6] => [2,3,4,5,6,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> ? = 0
[1,3,4,5,2,6] => [1,2,3,4,5,6] => [2,3,4,5,6,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> ? = 0
[1,3,4,5,6,2] => [1,2,3,4,5,6] => [2,3,4,5,6,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> ? = 0
[1,3,5,2,4,6] => [1,2,3,5,4,6] => [2,3,4,6,5,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0]]
=> ? = 1
[1,3,5,4,2,6] => [1,2,3,5,4,6] => [2,3,4,6,5,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0]]
=> ? = 1
[1,3,5,6,2,4] => [1,2,3,5,4,6] => [2,3,4,6,5,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0]]
=> ? = 1
[1,3,5,6,4,2] => [1,2,3,5,4,6] => [2,3,4,6,5,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0]]
=> ? = 1
[1,3,6,2,5,4] => [1,2,3,6,4,5] => [2,3,4,6,1,5] => [[0,0,0,0,1,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0]]
=> ? = 0
[1,3,6,4,5,2] => [1,2,3,6,4,5] => [2,3,4,6,1,5] => [[0,0,0,0,1,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0]]
=> ? = 0
[1,3,6,5,2,4] => [1,2,3,6,4,5] => [2,3,4,6,1,5] => [[0,0,0,0,1,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0]]
=> ? = 0
[1,3,6,5,4,2] => [1,2,3,6,4,5] => [2,3,4,6,1,5] => [[0,0,0,0,1,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0]]
=> ? = 0
[1,4,2,3,5,6] => [1,2,4,3,5,6] => [2,3,5,4,6,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> ? = 1
[1,4,2,3,6,5] => [1,2,4,3,5,6] => [2,3,5,4,6,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> ? = 1
[1,4,2,5,3,6] => [1,2,4,5,3,6] => [2,3,6,4,5,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0]]
=> ? = 2
[1,4,2,6,5,3] => [1,2,4,6,3,5] => [2,3,6,4,1,5] => [[0,0,0,0,1,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0]]
=> ? = 1
[1,4,3,2,5,6] => [1,2,4,3,5,6] => [2,3,5,4,6,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> ? = 1
[1,4,3,2,6,5] => [1,2,4,3,5,6] => [2,3,5,4,6,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> ? = 1
[1,4,3,5,2,6] => [1,2,4,5,3,6] => [2,3,6,4,5,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0]]
=> ? = 2
[1,4,3,6,5,2] => [1,2,4,6,3,5] => [2,3,6,4,1,5] => [[0,0,0,0,1,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0]]
=> ? = 1
[1,4,5,2,3,6] => [1,2,4,3,5,6] => [2,3,5,4,6,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> ? = 1
[1,4,5,2,6,3] => [1,2,4,3,5,6] => [2,3,5,4,6,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> ? = 1
[1,4,5,3,2,6] => [1,2,4,3,5,6] => [2,3,5,4,6,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> ? = 1
[1,4,5,3,6,2] => [1,2,4,3,5,6] => [2,3,5,4,6,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> ? = 1
[1,4,5,6,2,3] => [1,2,4,6,3,5] => [2,3,6,4,1,5] => [[0,0,0,0,1,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0]]
=> ? = 1
[1,4,5,6,3,2] => [1,2,4,6,3,5] => [2,3,6,4,1,5] => [[0,0,0,0,1,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0]]
=> ? = 1
[1,4,6,5,2,3] => [1,2,4,5,3,6] => [2,3,6,4,5,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0]]
=> ? = 2
[1,4,6,5,3,2] => [1,2,4,5,3,6] => [2,3,6,4,5,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0]]
=> ? = 2
[1,5,2,3,4,6] => [1,2,5,4,3,6] => [2,3,6,5,4,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]
=> ? = 0
Description
The trace of an alternating sign matrix.
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: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 20%
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St000259: Graphs ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 20%
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)
=> ? = 2
[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)
=> ? = 2
[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)
=> ? = 0
[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)
=> ? = 0
[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)
=> ? = 2
[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)
=> ? = 2
[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)
=> ? = 0
[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)
=> ? = 0
[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)
=> ? = 2
[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)
=> ? = 2
[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)
=> ? = 0
[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)
=> ? = 0
[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.
The following 9 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000260The radius of a connected graph. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001645The pebbling number of a connected graph. St001330The hat guessing number of a graph.
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