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Your data matches 192 different statistics following compositions of up to 3 maps.
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Matching statistic: St000058
(load all 50 compositions to match this statistic)
(load all 50 compositions to match this statistic)
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
St000058: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000058: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => 1
[1,2] => [1,2] => 1
[2,1] => [2,1] => 2
[1,2,3] => [1,2,3] => 1
[1,3,2] => [1,3,2] => 2
[2,1,3] => [2,1,3] => 2
[2,3,1] => [3,2,1] => 2
[3,1,2] => [3,2,1] => 2
[3,2,1] => [3,2,1] => 2
[1,2,3,4] => [1,2,3,4] => 1
[1,2,4,3] => [1,2,4,3] => 2
[1,3,2,4] => [1,3,2,4] => 2
[1,3,4,2] => [1,4,3,2] => 2
[1,4,2,3] => [1,4,3,2] => 2
[1,4,3,2] => [1,4,3,2] => 2
[2,1,3,4] => [2,1,3,4] => 2
[2,1,4,3] => [2,1,4,3] => 2
[2,3,1,4] => [3,2,1,4] => 2
[2,3,4,1] => [4,2,3,1] => 2
[2,4,1,3] => [3,4,1,2] => 2
[2,4,3,1] => [4,3,2,1] => 2
[3,1,2,4] => [3,2,1,4] => 2
[3,1,4,2] => [4,2,3,1] => 2
[3,2,1,4] => [3,2,1,4] => 2
[3,2,4,1] => [4,2,3,1] => 2
[3,4,1,2] => [4,3,2,1] => 2
[3,4,2,1] => [4,3,2,1] => 2
[4,1,2,3] => [4,2,3,1] => 2
[4,1,3,2] => [4,2,3,1] => 2
[4,2,1,3] => [4,3,2,1] => 2
[4,2,3,1] => [4,3,2,1] => 2
[4,3,1,2] => [4,3,2,1] => 2
[4,3,2,1] => [4,3,2,1] => 2
[1,2,3,4,5] => [1,2,3,4,5] => 1
[1,2,3,5,4] => [1,2,3,5,4] => 2
[1,2,4,3,5] => [1,2,4,3,5] => 2
[1,2,4,5,3] => [1,2,5,4,3] => 2
[1,2,5,3,4] => [1,2,5,4,3] => 2
[1,2,5,4,3] => [1,2,5,4,3] => 2
[1,3,2,4,5] => [1,3,2,4,5] => 2
[1,3,2,5,4] => [1,3,2,5,4] => 2
[1,3,4,2,5] => [1,4,3,2,5] => 2
[1,3,4,5,2] => [1,5,3,4,2] => 2
[1,3,5,2,4] => [1,4,5,2,3] => 2
[1,3,5,4,2] => [1,5,4,3,2] => 2
[1,4,2,3,5] => [1,4,3,2,5] => 2
[1,4,2,5,3] => [1,5,3,4,2] => 2
[1,4,3,2,5] => [1,4,3,2,5] => 2
[1,4,3,5,2] => [1,5,3,4,2] => 2
[1,4,5,2,3] => [1,5,4,3,2] => 2
Description
The order of a permutation.
$\operatorname{ord}(\pi)$ is given by the minimial $k$ for which $\pi^k$ is the identity permutation.
Matching statistic: St000062
(load all 31 compositions to match this statistic)
(load all 31 compositions to match this statistic)
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000062: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000062: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => 1
[1,2] => [1,2] => 2
[2,1] => [2,1] => 1
[1,2,3] => [1,3,2] => 2
[1,3,2] => [1,3,2] => 2
[2,1,3] => [2,1,3] => 2
[2,3,1] => [2,3,1] => 2
[3,1,2] => [3,1,2] => 2
[3,2,1] => [3,2,1] => 1
[1,2,3,4] => [1,4,3,2] => 2
[1,2,4,3] => [1,4,3,2] => 2
[1,3,2,4] => [1,4,3,2] => 2
[1,3,4,2] => [1,4,3,2] => 2
[1,4,2,3] => [1,4,3,2] => 2
[1,4,3,2] => [1,4,3,2] => 2
[2,1,3,4] => [2,1,4,3] => 2
[2,1,4,3] => [2,1,4,3] => 2
[2,3,1,4] => [2,4,1,3] => 2
[2,3,4,1] => [2,4,3,1] => 2
[2,4,1,3] => [2,4,1,3] => 2
[2,4,3,1] => [2,4,3,1] => 2
[3,1,2,4] => [3,1,4,2] => 2
[3,1,4,2] => [3,1,4,2] => 2
[3,2,1,4] => [3,2,1,4] => 2
[3,2,4,1] => [3,2,4,1] => 2
[3,4,1,2] => [3,4,1,2] => 2
[3,4,2,1] => [3,4,2,1] => 2
[4,1,2,3] => [4,1,3,2] => 2
[4,1,3,2] => [4,1,3,2] => 2
[4,2,1,3] => [4,2,1,3] => 2
[4,2,3,1] => [4,2,3,1] => 2
[4,3,1,2] => [4,3,1,2] => 2
[4,3,2,1] => [4,3,2,1] => 1
[1,2,3,4,5] => [1,5,4,3,2] => 2
[1,2,3,5,4] => [1,5,4,3,2] => 2
[1,2,4,3,5] => [1,5,4,3,2] => 2
[1,2,4,5,3] => [1,5,4,3,2] => 2
[1,2,5,3,4] => [1,5,4,3,2] => 2
[1,2,5,4,3] => [1,5,4,3,2] => 2
[1,3,2,4,5] => [1,5,4,3,2] => 2
[1,3,2,5,4] => [1,5,4,3,2] => 2
[1,3,4,2,5] => [1,5,4,3,2] => 2
[1,3,4,5,2] => [1,5,4,3,2] => 2
[1,3,5,2,4] => [1,5,4,3,2] => 2
[1,3,5,4,2] => [1,5,4,3,2] => 2
[1,4,2,3,5] => [1,5,4,3,2] => 2
[1,4,2,5,3] => [1,5,4,3,2] => 2
[1,4,3,2,5] => [1,5,4,3,2] => 2
[1,4,3,5,2] => [1,5,4,3,2] => 2
[1,4,5,2,3] => [1,5,4,3,2] => 2
Description
The length of the longest increasing subsequence of the permutation.
Matching statistic: St000298
(load all 49 compositions to match this statistic)
(load all 49 compositions to match this statistic)
Mp00065: Permutations —permutation poset⟶ Posets
St000298: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000298: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
=> 1
[1,2] => ([(0,1)],2)
=> 1
[2,1] => ([],2)
=> 2
[1,2,3] => ([(0,2),(2,1)],3)
=> 1
[1,3,2] => ([(0,1),(0,2)],3)
=> 2
[2,1,3] => ([(0,2),(1,2)],3)
=> 2
[2,3,1] => ([(1,2)],3)
=> 2
[3,1,2] => ([(1,2)],3)
=> 2
[3,2,1] => ([],3)
=> 2
[1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> 2
[1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> 2
[1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> 2
[1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> 2
[2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> 2
[2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> 2
[2,3,4,1] => ([(1,2),(2,3)],4)
=> 2
[2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> 2
[2,4,3,1] => ([(1,2),(1,3)],4)
=> 2
[3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> 2
[3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> 2
[3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> 2
[3,2,4,1] => ([(1,3),(2,3)],4)
=> 2
[3,4,1,2] => ([(0,3),(1,2)],4)
=> 2
[3,4,2,1] => ([(2,3)],4)
=> 2
[4,1,2,3] => ([(1,2),(2,3)],4)
=> 2
[4,1,3,2] => ([(1,2),(1,3)],4)
=> 2
[4,2,1,3] => ([(1,3),(2,3)],4)
=> 2
[4,2,3,1] => ([(2,3)],4)
=> 2
[4,3,1,2] => ([(2,3)],4)
=> 2
[4,3,2,1] => ([],4)
=> 2
[1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
=> 2
[1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2
[1,2,4,5,3] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> 2
[1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> 2
[1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
=> 2
[1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2
[1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 2
[1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
=> 2
[1,3,5,2,4] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> 2
[1,3,5,4,2] => ([(0,3),(0,4),(4,1),(4,2)],5)
=> 2
[1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 2
[1,4,2,5,3] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> 2
[1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 2
[1,4,3,5,2] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> 2
[1,4,5,2,3] => ([(0,3),(0,4),(3,2),(4,1)],5)
=> 2
Description
The order dimension or Dushnik-Miller dimension of a poset.
This is the minimal number of linear orderings whose intersection is the given poset.
Matching statistic: St000308
(load all 32 compositions to match this statistic)
(load all 32 compositions to match this statistic)
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000308: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000308: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => 1
[1,2] => [1,2] => 2
[2,1] => [2,1] => 1
[1,2,3] => [1,3,2] => 2
[1,3,2] => [1,3,2] => 2
[2,1,3] => [2,1,3] => 2
[2,3,1] => [2,3,1] => 2
[3,1,2] => [3,1,2] => 2
[3,2,1] => [3,2,1] => 1
[1,2,3,4] => [1,4,3,2] => 2
[1,2,4,3] => [1,4,3,2] => 2
[1,3,2,4] => [1,4,3,2] => 2
[1,3,4,2] => [1,4,3,2] => 2
[1,4,2,3] => [1,4,3,2] => 2
[1,4,3,2] => [1,4,3,2] => 2
[2,1,3,4] => [2,1,4,3] => 2
[2,1,4,3] => [2,1,4,3] => 2
[2,3,1,4] => [2,4,1,3] => 2
[2,3,4,1] => [2,4,3,1] => 2
[2,4,1,3] => [2,4,1,3] => 2
[2,4,3,1] => [2,4,3,1] => 2
[3,1,2,4] => [3,1,4,2] => 2
[3,1,4,2] => [3,1,4,2] => 2
[3,2,1,4] => [3,2,1,4] => 2
[3,2,4,1] => [3,2,4,1] => 2
[3,4,1,2] => [3,4,1,2] => 2
[3,4,2,1] => [3,4,2,1] => 2
[4,1,2,3] => [4,1,3,2] => 2
[4,1,3,2] => [4,1,3,2] => 2
[4,2,1,3] => [4,2,1,3] => 2
[4,2,3,1] => [4,2,3,1] => 2
[4,3,1,2] => [4,3,1,2] => 2
[4,3,2,1] => [4,3,2,1] => 1
[1,2,3,4,5] => [1,5,4,3,2] => 2
[1,2,3,5,4] => [1,5,4,3,2] => 2
[1,2,4,3,5] => [1,5,4,3,2] => 2
[1,2,4,5,3] => [1,5,4,3,2] => 2
[1,2,5,3,4] => [1,5,4,3,2] => 2
[1,2,5,4,3] => [1,5,4,3,2] => 2
[1,3,2,4,5] => [1,5,4,3,2] => 2
[1,3,2,5,4] => [1,5,4,3,2] => 2
[1,3,4,2,5] => [1,5,4,3,2] => 2
[1,3,4,5,2] => [1,5,4,3,2] => 2
[1,3,5,2,4] => [1,5,4,3,2] => 2
[1,3,5,4,2] => [1,5,4,3,2] => 2
[1,4,2,3,5] => [1,5,4,3,2] => 2
[1,4,2,5,3] => [1,5,4,3,2] => 2
[1,4,3,2,5] => [1,5,4,3,2] => 2
[1,4,3,5,2] => [1,5,4,3,2] => 2
[1,4,5,2,3] => [1,5,4,3,2] => 2
Description
The height of the tree associated to a permutation.
A permutation can be mapped to a rooted tree with vertices $\{0,1,2,\ldots,n\}$ and root $0$ in the following way. Entries of the permutations are inserted one after the other, each child is larger than its parent and the children are in strict order from left to right. Details of the construction are found in [1].
The statistic is given by the height of this tree.
See also [[St000325]] for the width of this tree.
Matching statistic: St000093
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000093: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00160: Permutations —graph of inversions⟶ Graphs
St000093: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => ([],1)
=> 1
[1,2] => [1,2] => ([],2)
=> 2
[2,1] => [2,1] => ([(0,1)],2)
=> 1
[1,2,3] => [1,3,2] => ([(1,2)],3)
=> 2
[1,3,2] => [1,3,2] => ([(1,2)],3)
=> 2
[2,1,3] => [2,1,3] => ([(1,2)],3)
=> 2
[2,3,1] => [2,3,1] => ([(0,2),(1,2)],3)
=> 2
[3,1,2] => [3,1,2] => ([(0,2),(1,2)],3)
=> 2
[3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[1,2,3,4] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
[1,2,4,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
[1,3,2,4] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
[1,3,4,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
[1,4,2,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
[1,4,3,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
[2,1,3,4] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> 2
[2,1,4,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> 2
[2,3,1,4] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> 2
[2,3,4,1] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[2,4,1,3] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> 2
[2,4,3,1] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[3,1,2,4] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 2
[3,1,4,2] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 2
[3,2,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 2
[3,2,4,1] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[3,4,1,2] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[3,4,2,1] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[4,1,2,3] => [4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[4,1,3,2] => [4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[4,2,1,3] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[4,2,3,1] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[4,3,1,2] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[4,3,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,2,3,4,5] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,2,3,5,4] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,2,4,3,5] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,2,4,5,3] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,2,5,3,4] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,2,5,4,3] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,3,2,4,5] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,3,2,5,4] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,3,4,2,5] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,3,4,5,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,3,5,2,4] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,3,5,4,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,4,2,3,5] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,4,2,5,3] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,4,3,2,5] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,4,3,5,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,4,5,2,3] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
Description
The cardinality of a maximal independent set of vertices of a graph.
An independent set of a graph is a set of pairwise non-adjacent vertices. A maximum independent set is an independent set of maximum cardinality. This statistic is also called the independence number or stability number $\alpha(G)$ of $G$.
Matching statistic: St000097
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Values
[1] => ([],1)
=> ([],1)
=> 1
[1,2] => ([(0,1)],2)
=> ([(0,1)],2)
=> 2
[2,1] => ([],2)
=> ([],2)
=> 1
[1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 2
[1,3,2] => ([(0,1),(0,2)],3)
=> ([(0,2),(1,2)],3)
=> 2
[2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 2
[2,3,1] => ([(1,2)],3)
=> ([(1,2)],3)
=> 2
[3,1,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> 2
[3,2,1] => ([],3)
=> ([],3)
=> 1
[1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 2
[1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 2
[2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 2
[2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[2,3,4,1] => ([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2
[2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[2,4,3,1] => ([(1,2),(1,3)],4)
=> ([(1,3),(2,3)],4)
=> 2
[3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 2
[3,2,4,1] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2
[3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> 2
[3,4,2,1] => ([(2,3)],4)
=> ([(2,3)],4)
=> 2
[4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2
[4,1,3,2] => ([(1,2),(1,3)],4)
=> ([(1,3),(2,3)],4)
=> 2
[4,2,1,3] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2
[4,2,3,1] => ([(2,3)],4)
=> ([(2,3)],4)
=> 2
[4,3,1,2] => ([(2,3)],4)
=> ([(2,3)],4)
=> 2
[4,3,2,1] => ([],4)
=> ([],4)
=> 1
[1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2
[1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2
[1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 2
[1,2,4,5,3] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2
[1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2
[1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 2
[1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 2
[1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2
[1,3,5,2,4] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 2
[1,3,5,4,2] => ([(0,3),(0,4),(4,1),(4,2)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2
[1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 2
[1,4,2,5,3] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 2
[1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[1,4,3,5,2] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 2
[1,4,5,2,3] => ([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2
Description
The order of the largest clique of the graph.
A clique in a graph $G$ is a subset $U \subseteq V(G)$ such that any pair of vertices in $U$ are adjacent. I.e. the subgraph induced by $U$ is a complete graph.
Matching statistic: St000147
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1]
=> 1
[1,2] => [1,2] => [2]
=> 2
[2,1] => [2,1] => [1,1]
=> 1
[1,2,3] => [1,3,2] => [2,1]
=> 2
[1,3,2] => [1,3,2] => [2,1]
=> 2
[2,1,3] => [2,1,3] => [2,1]
=> 2
[2,3,1] => [2,3,1] => [2,1]
=> 2
[3,1,2] => [3,1,2] => [2,1]
=> 2
[3,2,1] => [3,2,1] => [1,1,1]
=> 1
[1,2,3,4] => [1,4,3,2] => [2,1,1]
=> 2
[1,2,4,3] => [1,4,3,2] => [2,1,1]
=> 2
[1,3,2,4] => [1,4,3,2] => [2,1,1]
=> 2
[1,3,4,2] => [1,4,3,2] => [2,1,1]
=> 2
[1,4,2,3] => [1,4,3,2] => [2,1,1]
=> 2
[1,4,3,2] => [1,4,3,2] => [2,1,1]
=> 2
[2,1,3,4] => [2,1,4,3] => [2,2]
=> 2
[2,1,4,3] => [2,1,4,3] => [2,2]
=> 2
[2,3,1,4] => [2,4,1,3] => [2,2]
=> 2
[2,3,4,1] => [2,4,3,1] => [2,1,1]
=> 2
[2,4,1,3] => [2,4,1,3] => [2,2]
=> 2
[2,4,3,1] => [2,4,3,1] => [2,1,1]
=> 2
[3,1,2,4] => [3,1,4,2] => [2,2]
=> 2
[3,1,4,2] => [3,1,4,2] => [2,2]
=> 2
[3,2,1,4] => [3,2,1,4] => [2,1,1]
=> 2
[3,2,4,1] => [3,2,4,1] => [2,1,1]
=> 2
[3,4,1,2] => [3,4,1,2] => [2,2]
=> 2
[3,4,2,1] => [3,4,2,1] => [2,1,1]
=> 2
[4,1,2,3] => [4,1,3,2] => [2,1,1]
=> 2
[4,1,3,2] => [4,1,3,2] => [2,1,1]
=> 2
[4,2,1,3] => [4,2,1,3] => [2,1,1]
=> 2
[4,2,3,1] => [4,2,3,1] => [2,1,1]
=> 2
[4,3,1,2] => [4,3,1,2] => [2,1,1]
=> 2
[4,3,2,1] => [4,3,2,1] => [1,1,1,1]
=> 1
[1,2,3,4,5] => [1,5,4,3,2] => [2,1,1,1]
=> 2
[1,2,3,5,4] => [1,5,4,3,2] => [2,1,1,1]
=> 2
[1,2,4,3,5] => [1,5,4,3,2] => [2,1,1,1]
=> 2
[1,2,4,5,3] => [1,5,4,3,2] => [2,1,1,1]
=> 2
[1,2,5,3,4] => [1,5,4,3,2] => [2,1,1,1]
=> 2
[1,2,5,4,3] => [1,5,4,3,2] => [2,1,1,1]
=> 2
[1,3,2,4,5] => [1,5,4,3,2] => [2,1,1,1]
=> 2
[1,3,2,5,4] => [1,5,4,3,2] => [2,1,1,1]
=> 2
[1,3,4,2,5] => [1,5,4,3,2] => [2,1,1,1]
=> 2
[1,3,4,5,2] => [1,5,4,3,2] => [2,1,1,1]
=> 2
[1,3,5,2,4] => [1,5,4,3,2] => [2,1,1,1]
=> 2
[1,3,5,4,2] => [1,5,4,3,2] => [2,1,1,1]
=> 2
[1,4,2,3,5] => [1,5,4,3,2] => [2,1,1,1]
=> 2
[1,4,2,5,3] => [1,5,4,3,2] => [2,1,1,1]
=> 2
[1,4,3,2,5] => [1,5,4,3,2] => [2,1,1,1]
=> 2
[1,4,3,5,2] => [1,5,4,3,2] => [2,1,1,1]
=> 2
[1,4,5,2,3] => [1,5,4,3,2] => [2,1,1,1]
=> 2
Description
The largest part of an integer partition.
Matching statistic: St000258
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Values
[1] => ([],1)
=> ([],1)
=> 1
[1,2] => ([],2)
=> ([],1)
=> 1
[2,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> 2
[1,2,3] => ([],3)
=> ([],1)
=> 1
[1,3,2] => ([(1,2)],3)
=> ([(0,1)],2)
=> 2
[2,1,3] => ([(1,2)],3)
=> ([(0,1)],2)
=> 2
[2,3,1] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2
[3,1,2] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
[1,2,3,4] => ([],4)
=> ([],1)
=> 1
[1,2,4,3] => ([(2,3)],4)
=> ([(0,1)],2)
=> 2
[1,3,2,4] => ([(2,3)],4)
=> ([(0,1)],2)
=> 2
[1,3,4,2] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2
[1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
[2,1,3,4] => ([(2,3)],4)
=> ([(0,1)],2)
=> 2
[2,1,4,3] => ([(0,3),(1,2)],4)
=> ([(0,1)],2)
=> 2
[2,3,1,4] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> 2
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
[3,1,2,4] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> 2
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> 2
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,2,3,4,5] => ([],5)
=> ([],1)
=> 1
[1,2,3,5,4] => ([(3,4)],5)
=> ([(0,1)],2)
=> 2
[1,2,4,3,5] => ([(3,4)],5)
=> ([(0,1)],2)
=> 2
[1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 2
[1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 2
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
[1,3,2,4,5] => ([(3,4)],5)
=> ([(0,1)],2)
=> 2
[1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ([(0,1)],2)
=> 2
[1,3,4,2,5] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 2
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 2
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> 2
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
[1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 2
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> 2
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> 2
Description
The burning number of a graph.
This is the minimum number of rounds needed to burn all vertices of a graph. In each round, the neighbours of all burned vertices are burnt. Additionally, an unburned vertex may be chosen to be burned.
Matching statistic: St000259
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Values
[1] => ([],1)
=> ([(0,1)],2)
=> 1
[1,2] => ([],2)
=> ([(0,2),(1,2)],3)
=> 2
[2,1] => ([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
[1,2,3] => ([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 2
[1,3,2] => ([(1,2)],3)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[2,1,3] => ([(1,2)],3)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[2,3,1] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[3,1,2] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,2,3,4] => ([],4)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,2,4,3] => ([(2,3)],4)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,3,2,4] => ([(2,3)],4)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,3,4,2] => ([(1,3),(2,3)],4)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[2,1,3,4] => ([(2,3)],4)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[2,1,4,3] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 2
[2,3,1,4] => ([(1,3),(2,3)],4)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[3,1,2,4] => ([(1,3),(2,3)],4)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 2
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,2,3,4,5] => ([],5)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[1,2,3,5,4] => ([(3,4)],5)
=> ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,2,4,3,5] => ([(3,4)],5)
=> ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,3,2,4,5] => ([(3,4)],5)
=> ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> 2
[1,3,4,2,5] => ([(2,4),(3,4)],5)
=> ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 2
Description
The diameter of a connected graph.
This is the greatest distance between any pair of vertices.
Matching statistic: St000299
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Values
[1] => ([],1)
=> ([],1)
=> 1
[1,2] => ([],2)
=> ([],1)
=> 1
[2,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> 2
[1,2,3] => ([],3)
=> ([],1)
=> 1
[1,3,2] => ([(1,2)],3)
=> ([(0,1)],2)
=> 2
[2,1,3] => ([(1,2)],3)
=> ([(0,1)],2)
=> 2
[2,3,1] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2
[3,1,2] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
[1,2,3,4] => ([],4)
=> ([],1)
=> 1
[1,2,4,3] => ([(2,3)],4)
=> ([(0,1)],2)
=> 2
[1,3,2,4] => ([(2,3)],4)
=> ([(0,1)],2)
=> 2
[1,3,4,2] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2
[1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
[2,1,3,4] => ([(2,3)],4)
=> ([(0,1)],2)
=> 2
[2,1,4,3] => ([(0,3),(1,2)],4)
=> ([(0,1)],2)
=> 2
[2,3,1,4] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> 2
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
[3,1,2,4] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> 2
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> 2
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,2,3,4,5] => ([],5)
=> ([],1)
=> 1
[1,2,3,5,4] => ([(3,4)],5)
=> ([(0,1)],2)
=> 2
[1,2,4,3,5] => ([(3,4)],5)
=> ([(0,1)],2)
=> 2
[1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 2
[1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 2
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
[1,3,2,4,5] => ([(3,4)],5)
=> ([(0,1)],2)
=> 2
[1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ([(0,1)],2)
=> 2
[1,3,4,2,5] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 2
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 2
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> 2
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
[1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 2
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> 2
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> 2
Description
The number of nonisomorphic vertex-induced subtrees.
The following 182 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000381The largest part of an integer composition. St000451The length of the longest pattern of the form k 1 2. St000452The number of distinct eigenvalues of a graph. St000453The number of distinct Laplacian eigenvalues of a graph. St000528The height of a poset. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000918The 2-limited packing number of a graph. St000991The number of right-to-left minima of a permutation. St001093The detour number of a graph. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001261The Castelnuovo-Mumford regularity of a graph. St001315The dissociation number of a graph. St001318The number of vertices of the largest induced subforest with the same number of connected components of a graph. St001321The number of vertices of the largest induced subforest of a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001343The dimension of the reduced incidence algebra of a poset. St001674The number of vertices of the largest induced star graph in the graph. St001717The largest size of an interval in a poset. St001720The minimal length of a chain of small intervals in a lattice. St000080The rank of the poset. St000260The radius of a connected graph. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000480The number of lower covers of a partition in dominance order. St000535The rank-width of a graph. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St001092The number of distinct even parts of a partition. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001271The competition number of a graph. St001333The cardinality of a minimal edge-isolating set of a graph. St001335The cardinality of a minimal cycle-isolating set of a graph. St001340The cardinality of a minimal non-edge isolating set of a graph. St001349The number of different graphs obtained from the given graph by removing an edge. St001352The number of internal nodes in the modular decomposition of a graph. St001354The number of series nodes in the modular decomposition of a graph. St001393The induced matching number of a graph. St001512The minimum rank of a graph. St001587Half of the largest even part of an integer partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St000007The number of saliances of the permutation. St000010The length of the partition. St000013The height of a Dyck path. St000098The chromatic number of a graph. St000166The depth minus 1 of an ordered tree. St000314The number of left-to-right-maxima of a permutation. St000328The maximum number of child nodes in a tree. St000346The number of coarsenings of a partition. St000378The diagonal inversion number of an integer partition. St000382The first part of an integer composition. St000397The Strahler number of a rooted tree. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000527The width of the poset. St000542The number of left-to-right-minima of a permutation. St000676The number of odd rises of a Dyck path. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St000734The last entry in the first row of a standard tableau. St000808The number of up steps of the associated bargraph. St000917The open packing number of a graph. St000920The logarithmic height of a Dyck path. St001029The size of the core of a graph. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001316The domatic number of a graph. St001330The hat guessing number of a graph. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001486The number of corners of the ribbon associated with an integer composition. St001530The depth of a Dyck path. St001642The Prague dimension of a graph. St001654The monophonic hull number of a graph. St001656The monophonic position number of a graph. St001746The coalition number of a graph. St001784The minimum of the smallest closer and the second element of the block containing 1 in a set partition. St000028The number of stack-sorts needed to sort a permutation. St000035The number of left outer peaks of a permutation. St000094The depth of an ordered tree. St000133The "bounce" of a permutation. St000141The maximum drop size of a permutation. St000143The largest repeated part of a partition. St000183The side length of the Durfee square of an integer partition. St000209Maximum difference of elements in cycles. St000245The number of ascents of a permutation. St000256The number of parts from which one can substract 2 and still get an integer partition. St000257The number of distinct parts of a partition that occur at least twice. St000374The number of exclusive right-to-left minima of a permutation. St000473The number of parts of a partition that are strictly bigger than the number of ones. St000481The number of upper covers of a partition in dominance order. St000651The maximal size of a rise in a permutation. St000662The staircase size of the code of a permutation. St000741The Colin de Verdière graph invariant. St000742The number of big ascents of a permutation after prepending zero. St000834The number of right outer peaks of a permutation. St000845The maximal number of elements covered by an element in a poset. St000864The number of circled entries of the shifted recording tableau of a permutation. St000897The number of different multiplicities of parts of an integer partition. St000996The number of exclusive left-to-right maxima of a permutation. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001056The Grundy value for the game of deleting vertices of a graph until it has no edges. St001071The beta invariant of the graph. St001090The number of pop-stack-sorts needed to sort a permutation. St001188The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001280The number of parts of an integer partition that are at least two. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001484The number of singletons of an integer partition. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001665The number of pure excedances of a permutation. St001737The number of descents of type 2 in a permutation. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St001777The number of weak descents in an integer composition. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001931The weak major index of an integer composition regarded as a word. St000485The length of the longest cycle of a permutation. St000668The least common multiple of the parts of the partition. St001062The maximal size of a block of a set partition. St000253The crossing number of a set partition. St000640The rank of the largest boolean interval in a poset. St000659The number of rises of length at least 2 of a Dyck path. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001859The number of factors of the Stanley symmetric function associated with a permutation. St000444The length of the maximal rise of a Dyck path. St000504The cardinality of the first block of a set partition. St000844The size of the largest block in the direct sum decomposition of a permutation. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001568The smallest positive integer that does not appear twice in the partition. St000254The nesting number of a set partition. St000354The number of recoils of a permutation. St000390The number of runs of ones in a binary word. St000392The length of the longest run of ones in a binary word. St000442The maximal area to the right of an up step of a Dyck path. St000472The sum of the ascent bottoms of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000730The maximal arc length of a set partition. St000919The number of maximal left branches of a binary tree. St000956The maximal displacement of a permutation. St000989The number of final rises of a permutation. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St000092The number of outer peaks of a permutation. St000353The number of inner valleys of a permutation. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St000630The length of the shortest palindromic decomposition of a binary word. St000642The size of the smallest orbit of antichains under Panyushev complementation. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001471The magnitude of a Dyck path. St000511The number of invariant subsets when acting with a permutation of given cycle type. St000159The number of distinct parts of the integer partition. St000160The multiplicity of the smallest part of a partition. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000548The number of different non-empty partial sums of an integer partition. St001057The Grundy value of the game of creating an independent set in a graph. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000783The side length of the largest staircase partition fitting into a partition. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St000862The number of parts of the shifted shape of a permutation. St001323The independence gap of a graph. St001624The breadth of a lattice. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001734The lettericity of a graph. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001569The maximal modular displacement of a permutation. St001555The order of a signed permutation. St000454The largest eigenvalue of a graph if it is integral. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St000455The second largest eigenvalue of a graph if it is integral. St001870The number of positive entries followed by a negative entry in a signed permutation. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$.
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