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Your data matches 16 different statistics following compositions of up to 3 maps.
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Matching statistic: St000002
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Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000002: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000002: 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] => 1
[1,3,2] => [1,2,3] => 1
[2,1,3] => [1,2,3] => 1
[2,3,1] => [1,2,3] => 1
[3,1,2] => [1,3,2] => 0
[3,2,1] => [1,3,2] => 0
[1,2,3,4] => [1,2,3,4] => 4
[1,2,4,3] => [1,2,3,4] => 4
[1,3,2,4] => [1,2,3,4] => 4
[1,3,4,2] => [1,2,3,4] => 4
[1,4,2,3] => [1,2,4,3] => 2
[1,4,3,2] => [1,2,4,3] => 2
[2,1,3,4] => [1,2,3,4] => 4
[2,1,4,3] => [1,2,3,4] => 4
[2,3,1,4] => [1,2,3,4] => 4
[2,3,4,1] => [1,2,3,4] => 4
[2,4,1,3] => [1,2,4,3] => 2
[2,4,3,1] => [1,2,4,3] => 2
[3,1,2,4] => [1,3,2,4] => 2
[3,1,4,2] => [1,3,4,2] => 1
[3,2,1,4] => [1,3,2,4] => 2
[3,2,4,1] => [1,3,4,2] => 1
[3,4,1,2] => [1,3,2,4] => 2
[3,4,2,1] => [1,3,2,4] => 2
[4,1,2,3] => [1,4,3,2] => 0
[4,1,3,2] => [1,4,2,3] => 1
[4,2,1,3] => [1,4,3,2] => 0
[4,2,3,1] => [1,4,2,3] => 1
[4,3,1,2] => [1,4,2,3] => 1
[4,3,2,1] => [1,4,2,3] => 1
[1,2,3,4,5] => [1,2,3,4,5] => 10
[1,2,3,5,4] => [1,2,3,4,5] => 10
[1,2,4,3,5] => [1,2,3,4,5] => 10
[1,2,4,5,3] => [1,2,3,4,5] => 10
[1,2,5,3,4] => [1,2,3,5,4] => 7
[1,2,5,4,3] => [1,2,3,5,4] => 7
[1,3,2,4,5] => [1,2,3,4,5] => 10
[1,3,2,5,4] => [1,2,3,4,5] => 10
[1,3,4,2,5] => [1,2,3,4,5] => 10
[1,3,4,5,2] => [1,2,3,4,5] => 10
[1,3,5,2,4] => [1,2,3,5,4] => 7
[1,3,5,4,2] => [1,2,3,5,4] => 7
[1,4,2,3,5] => [1,2,4,3,5] => 7
[1,4,2,5,3] => [1,2,4,5,3] => 5
[1,4,3,2,5] => [1,2,4,3,5] => 7
[1,4,3,5,2] => [1,2,4,5,3] => 5
[1,4,5,2,3] => [1,2,4,3,5] => 7
Description
The number of occurrences of the pattern 123 in a permutation.
Matching statistic: St000119
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(load all 4 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000119: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00069: Permutations —complement⟶ Permutations
St000119: 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] => 1
[1,3,2] => [1,2,3] => [3,2,1] => 1
[2,1,3] => [1,2,3] => [3,2,1] => 1
[2,3,1] => [1,2,3] => [3,2,1] => 1
[3,1,2] => [1,3,2] => [3,1,2] => 0
[3,2,1] => [1,3,2] => [3,1,2] => 0
[1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 4
[1,2,4,3] => [1,2,3,4] => [4,3,2,1] => 4
[1,3,2,4] => [1,2,3,4] => [4,3,2,1] => 4
[1,3,4,2] => [1,2,3,4] => [4,3,2,1] => 4
[1,4,2,3] => [1,2,4,3] => [4,3,1,2] => 2
[1,4,3,2] => [1,2,4,3] => [4,3,1,2] => 2
[2,1,3,4] => [1,2,3,4] => [4,3,2,1] => 4
[2,1,4,3] => [1,2,3,4] => [4,3,2,1] => 4
[2,3,1,4] => [1,2,3,4] => [4,3,2,1] => 4
[2,3,4,1] => [1,2,3,4] => [4,3,2,1] => 4
[2,4,1,3] => [1,2,4,3] => [4,3,1,2] => 2
[2,4,3,1] => [1,2,4,3] => [4,3,1,2] => 2
[3,1,2,4] => [1,3,2,4] => [4,2,3,1] => 2
[3,1,4,2] => [1,3,4,2] => [4,2,1,3] => 1
[3,2,1,4] => [1,3,2,4] => [4,2,3,1] => 2
[3,2,4,1] => [1,3,4,2] => [4,2,1,3] => 1
[3,4,1,2] => [1,3,2,4] => [4,2,3,1] => 2
[3,4,2,1] => [1,3,2,4] => [4,2,3,1] => 2
[4,1,2,3] => [1,4,3,2] => [4,1,2,3] => 0
[4,1,3,2] => [1,4,2,3] => [4,1,3,2] => 1
[4,2,1,3] => [1,4,3,2] => [4,1,2,3] => 0
[4,2,3,1] => [1,4,2,3] => [4,1,3,2] => 1
[4,3,1,2] => [1,4,2,3] => [4,1,3,2] => 1
[4,3,2,1] => [1,4,2,3] => [4,1,3,2] => 1
[1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 10
[1,2,3,5,4] => [1,2,3,4,5] => [5,4,3,2,1] => 10
[1,2,4,3,5] => [1,2,3,4,5] => [5,4,3,2,1] => 10
[1,2,4,5,3] => [1,2,3,4,5] => [5,4,3,2,1] => 10
[1,2,5,3,4] => [1,2,3,5,4] => [5,4,3,1,2] => 7
[1,2,5,4,3] => [1,2,3,5,4] => [5,4,3,1,2] => 7
[1,3,2,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 10
[1,3,2,5,4] => [1,2,3,4,5] => [5,4,3,2,1] => 10
[1,3,4,2,5] => [1,2,3,4,5] => [5,4,3,2,1] => 10
[1,3,4,5,2] => [1,2,3,4,5] => [5,4,3,2,1] => 10
[1,3,5,2,4] => [1,2,3,5,4] => [5,4,3,1,2] => 7
[1,3,5,4,2] => [1,2,3,5,4] => [5,4,3,1,2] => 7
[1,4,2,3,5] => [1,2,4,3,5] => [5,4,2,3,1] => 7
[1,4,2,5,3] => [1,2,4,5,3] => [5,4,2,1,3] => 5
[1,4,3,2,5] => [1,2,4,3,5] => [5,4,2,3,1] => 7
[1,4,3,5,2] => [1,2,4,5,3] => [5,4,2,1,3] => 5
[1,4,5,2,3] => [1,2,4,3,5] => [5,4,2,3,1] => 7
Description
The number of occurrences of the pattern 321 in a permutation.
Matching statistic: St000095
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000095: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00064: Permutations —reverse⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000095: Graphs ⟶ 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] => ([(0,1)],2)
=> 0
[2,1] => [1,2] => [2,1] => ([(0,1)],2)
=> 0
[1,2,3] => [1,2,3] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[1,3,2] => [1,2,3] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[2,1,3] => [1,2,3] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[2,3,1] => [1,2,3] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[3,1,2] => [1,3,2] => [2,3,1] => ([(0,2),(1,2)],3)
=> 0
[3,2,1] => [1,3,2] => [2,3,1] => ([(0,2),(1,2)],3)
=> 0
[1,2,3,4] => [1,2,3,4] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[1,2,4,3] => [1,2,3,4] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[1,3,2,4] => [1,2,3,4] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[1,3,4,2] => [1,2,3,4] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[1,4,2,3] => [1,2,4,3] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,4,3,2] => [1,2,4,3] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[2,1,3,4] => [1,2,3,4] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[2,1,4,3] => [1,2,3,4] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[2,3,1,4] => [1,2,3,4] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[2,3,4,1] => [1,2,3,4] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[2,4,1,3] => [1,2,4,3] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[2,4,3,1] => [1,2,4,3] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[3,1,2,4] => [1,3,2,4] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[3,1,4,2] => [1,3,4,2] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[3,2,1,4] => [1,3,2,4] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[3,2,4,1] => [1,3,4,2] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[3,4,1,2] => [1,3,2,4] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[3,4,2,1] => [1,3,2,4] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[4,1,2,3] => [1,4,3,2] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 0
[4,1,3,2] => [1,4,2,3] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[4,2,1,3] => [1,4,3,2] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 0
[4,2,3,1] => [1,4,2,3] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[4,3,1,2] => [1,4,2,3] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[4,3,2,1] => [1,4,2,3] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 10
[1,2,3,5,4] => [1,2,3,4,5] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 10
[1,2,4,3,5] => [1,2,3,4,5] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 10
[1,2,4,5,3] => [1,2,3,4,5] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 10
[1,2,5,3,4] => [1,2,3,5,4] => [4,5,3,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 7
[1,2,5,4,3] => [1,2,3,5,4] => [4,5,3,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 7
[1,3,2,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 10
[1,3,2,5,4] => [1,2,3,4,5] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 10
[1,3,4,2,5] => [1,2,3,4,5] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 10
[1,3,4,5,2] => [1,2,3,4,5] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 10
[1,3,5,2,4] => [1,2,3,5,4] => [4,5,3,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 7
[1,3,5,4,2] => [1,2,3,5,4] => [4,5,3,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 7
[1,4,2,3,5] => [1,2,4,3,5] => [5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 7
[1,4,2,5,3] => [1,2,4,5,3] => [3,5,4,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[1,4,3,2,5] => [1,2,4,3,5] => [5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 7
[1,4,3,5,2] => [1,2,4,5,3] => [3,5,4,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[1,4,5,2,3] => [1,2,4,3,5] => [5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 7
Description
The number of triangles of a graph.
A triangle $T$ of a graph $G$ is a collection of three vertices $\{u,v,w\} \in G$ such that they form $K_3$, the complete graph on three vertices.
Matching statistic: St001328
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001328: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00064: Permutations —reverse⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001328: Graphs ⟶ 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] => ([(0,1)],2)
=> 0
[2,1] => [1,2] => [2,1] => ([(0,1)],2)
=> 0
[1,2,3] => [1,2,3] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[1,3,2] => [1,2,3] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[2,1,3] => [1,2,3] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[2,3,1] => [1,2,3] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[3,1,2] => [1,3,2] => [2,3,1] => ([(0,2),(1,2)],3)
=> 0
[3,2,1] => [1,3,2] => [2,3,1] => ([(0,2),(1,2)],3)
=> 0
[1,2,3,4] => [1,2,3,4] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[1,2,4,3] => [1,2,3,4] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[1,3,2,4] => [1,2,3,4] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[1,3,4,2] => [1,2,3,4] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[1,4,2,3] => [1,2,4,3] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,4,3,2] => [1,2,4,3] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[2,1,3,4] => [1,2,3,4] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[2,1,4,3] => [1,2,3,4] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[2,3,1,4] => [1,2,3,4] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[2,3,4,1] => [1,2,3,4] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[2,4,1,3] => [1,2,4,3] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[2,4,3,1] => [1,2,4,3] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[3,1,2,4] => [1,3,2,4] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[3,1,4,2] => [1,3,4,2] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[3,2,1,4] => [1,3,2,4] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[3,2,4,1] => [1,3,4,2] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[3,4,1,2] => [1,3,2,4] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[3,4,2,1] => [1,3,2,4] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[4,1,2,3] => [1,4,3,2] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 0
[4,1,3,2] => [1,4,2,3] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[4,2,1,3] => [1,4,3,2] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 0
[4,2,3,1] => [1,4,2,3] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[4,3,1,2] => [1,4,2,3] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[4,3,2,1] => [1,4,2,3] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 10
[1,2,3,5,4] => [1,2,3,4,5] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 10
[1,2,4,3,5] => [1,2,3,4,5] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 10
[1,2,4,5,3] => [1,2,3,4,5] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 10
[1,2,5,3,4] => [1,2,3,5,4] => [4,5,3,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 7
[1,2,5,4,3] => [1,2,3,5,4] => [4,5,3,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 7
[1,3,2,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 10
[1,3,2,5,4] => [1,2,3,4,5] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 10
[1,3,4,2,5] => [1,2,3,4,5] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 10
[1,3,4,5,2] => [1,2,3,4,5] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 10
[1,3,5,2,4] => [1,2,3,5,4] => [4,5,3,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 7
[1,3,5,4,2] => [1,2,3,5,4] => [4,5,3,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 7
[1,4,2,3,5] => [1,2,4,3,5] => [5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 7
[1,4,2,5,3] => [1,2,4,5,3] => [3,5,4,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[1,4,3,2,5] => [1,2,4,3,5] => [5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 7
[1,4,3,5,2] => [1,2,4,5,3] => [3,5,4,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[1,4,5,2,3] => [1,2,4,3,5] => [5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 7
Description
The minimal number of occurrences of the bipartite-pattern in a linear ordering of the vertices of the graph.
A graph is bipartite if and only if in any linear ordering of its vertices, there are no three vertices $a < b < c$ such that $(a,b)$ and $(b,c)$ are edges. This statistic is the minimal number of occurrences of this pattern, in the set of all linear orderings of the vertices.
Matching statistic: St001396
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001396: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00069: Permutations —complement⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001396: Posets ⟶ 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] => ([],2)
=> 0
[2,1] => [1,2] => [2,1] => ([],2)
=> 0
[1,2,3] => [1,2,3] => [3,2,1] => ([],3)
=> 1
[1,3,2] => [1,2,3] => [3,2,1] => ([],3)
=> 1
[2,1,3] => [1,2,3] => [3,2,1] => ([],3)
=> 1
[2,3,1] => [1,2,3] => [3,2,1] => ([],3)
=> 1
[3,1,2] => [1,3,2] => [3,1,2] => ([(1,2)],3)
=> 0
[3,2,1] => [1,3,2] => [3,1,2] => ([(1,2)],3)
=> 0
[1,2,3,4] => [1,2,3,4] => [4,3,2,1] => ([],4)
=> 4
[1,2,4,3] => [1,2,3,4] => [4,3,2,1] => ([],4)
=> 4
[1,3,2,4] => [1,2,3,4] => [4,3,2,1] => ([],4)
=> 4
[1,3,4,2] => [1,2,3,4] => [4,3,2,1] => ([],4)
=> 4
[1,4,2,3] => [1,2,4,3] => [4,3,1,2] => ([(2,3)],4)
=> 2
[1,4,3,2] => [1,2,4,3] => [4,3,1,2] => ([(2,3)],4)
=> 2
[2,1,3,4] => [1,2,3,4] => [4,3,2,1] => ([],4)
=> 4
[2,1,4,3] => [1,2,3,4] => [4,3,2,1] => ([],4)
=> 4
[2,3,1,4] => [1,2,3,4] => [4,3,2,1] => ([],4)
=> 4
[2,3,4,1] => [1,2,3,4] => [4,3,2,1] => ([],4)
=> 4
[2,4,1,3] => [1,2,4,3] => [4,3,1,2] => ([(2,3)],4)
=> 2
[2,4,3,1] => [1,2,4,3] => [4,3,1,2] => ([(2,3)],4)
=> 2
[3,1,2,4] => [1,3,2,4] => [4,2,3,1] => ([(2,3)],4)
=> 2
[3,1,4,2] => [1,3,4,2] => [4,2,1,3] => ([(1,3),(2,3)],4)
=> 1
[3,2,1,4] => [1,3,2,4] => [4,2,3,1] => ([(2,3)],4)
=> 2
[3,2,4,1] => [1,3,4,2] => [4,2,1,3] => ([(1,3),(2,3)],4)
=> 1
[3,4,1,2] => [1,3,2,4] => [4,2,3,1] => ([(2,3)],4)
=> 2
[3,4,2,1] => [1,3,2,4] => [4,2,3,1] => ([(2,3)],4)
=> 2
[4,1,2,3] => [1,4,3,2] => [4,1,2,3] => ([(1,2),(2,3)],4)
=> 0
[4,1,3,2] => [1,4,2,3] => [4,1,3,2] => ([(1,2),(1,3)],4)
=> 1
[4,2,1,3] => [1,4,3,2] => [4,1,2,3] => ([(1,2),(2,3)],4)
=> 0
[4,2,3,1] => [1,4,2,3] => [4,1,3,2] => ([(1,2),(1,3)],4)
=> 1
[4,3,1,2] => [1,4,2,3] => [4,1,3,2] => ([(1,2),(1,3)],4)
=> 1
[4,3,2,1] => [1,4,2,3] => [4,1,3,2] => ([(1,2),(1,3)],4)
=> 1
[1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => ([],5)
=> 10
[1,2,3,5,4] => [1,2,3,4,5] => [5,4,3,2,1] => ([],5)
=> 10
[1,2,4,3,5] => [1,2,3,4,5] => [5,4,3,2,1] => ([],5)
=> 10
[1,2,4,5,3] => [1,2,3,4,5] => [5,4,3,2,1] => ([],5)
=> 10
[1,2,5,3,4] => [1,2,3,5,4] => [5,4,3,1,2] => ([(3,4)],5)
=> 7
[1,2,5,4,3] => [1,2,3,5,4] => [5,4,3,1,2] => ([(3,4)],5)
=> 7
[1,3,2,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => ([],5)
=> 10
[1,3,2,5,4] => [1,2,3,4,5] => [5,4,3,2,1] => ([],5)
=> 10
[1,3,4,2,5] => [1,2,3,4,5] => [5,4,3,2,1] => ([],5)
=> 10
[1,3,4,5,2] => [1,2,3,4,5] => [5,4,3,2,1] => ([],5)
=> 10
[1,3,5,2,4] => [1,2,3,5,4] => [5,4,3,1,2] => ([(3,4)],5)
=> 7
[1,3,5,4,2] => [1,2,3,5,4] => [5,4,3,1,2] => ([(3,4)],5)
=> 7
[1,4,2,3,5] => [1,2,4,3,5] => [5,4,2,3,1] => ([(3,4)],5)
=> 7
[1,4,2,5,3] => [1,2,4,5,3] => [5,4,2,1,3] => ([(2,4),(3,4)],5)
=> 5
[1,4,3,2,5] => [1,2,4,3,5] => [5,4,2,3,1] => ([(3,4)],5)
=> 7
[1,4,3,5,2] => [1,2,4,5,3] => [5,4,2,1,3] => ([(2,4),(3,4)],5)
=> 5
[1,4,5,2,3] => [1,2,4,3,5] => [5,4,2,3,1] => ([(3,4)],5)
=> 7
Description
Number of triples of incomparable elements in a finite poset.
For a finite poset this is the number of 3-element sets $S \in \binom{P}{3}$ that are pairwise incomparable.
Matching statistic: St000422
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00010: Binary trees —to ordered tree: left child = left brother⟶ Ordered trees
Mp00046: Ordered trees —to graph⟶ Graphs
St000422: Graphs ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 25%
Mp00010: Binary trees —to ordered tree: left child = left brother⟶ Ordered trees
Mp00046: Ordered trees —to graph⟶ Graphs
St000422: Graphs ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 25%
Values
[1] => [.,.]
=> [[]]
=> ([(0,1)],2)
=> 2 = 0 + 2
[1,2] => [.,[.,.]]
=> [[[]]]
=> ([(0,2),(1,2)],3)
=> ? ∊ {0,0} + 2
[2,1] => [[.,.],.]
=> [[],[]]
=> ([(0,2),(1,2)],3)
=> ? ∊ {0,0} + 2
[1,2,3] => [.,[.,[.,.]]]
=> [[[[]]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? ∊ {0,0,1,1,1,1} + 2
[1,3,2] => [.,[[.,.],.]]
=> [[[],[]]]
=> ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {0,0,1,1,1,1} + 2
[2,1,3] => [[.,.],[.,.]]
=> [[],[[]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? ∊ {0,0,1,1,1,1} + 2
[2,3,1] => [[.,.],[.,.]]
=> [[],[[]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? ∊ {0,0,1,1,1,1} + 2
[3,1,2] => [[.,[.,.]],.]
=> [[[]],[]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? ∊ {0,0,1,1,1,1} + 2
[3,2,1] => [[[.,.],.],.]
=> [[],[],[]]
=> ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {0,0,1,1,1,1} + 2
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [[[[[]]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,2,2,4,4,4,4,4,4,4,4} + 2
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [[[[],[]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,2,2,4,4,4,4,4,4,4,4} + 2
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [[[],[[]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,2,2,4,4,4,4,4,4,4,4} + 2
[1,3,4,2] => [.,[[.,.],[.,.]]]
=> [[[],[[]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,2,2,4,4,4,4,4,4,4,4} + 2
[1,4,2,3] => [.,[[.,[.,.]],.]]
=> [[[[]],[]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,2,2,4,4,4,4,4,4,4,4} + 2
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [[[],[],[]]]
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4 = 2 + 2
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [[],[[[]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,2,2,4,4,4,4,4,4,4,4} + 2
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [[],[[],[]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,2,2,4,4,4,4,4,4,4,4} + 2
[2,3,1,4] => [[.,.],[.,[.,.]]]
=> [[],[[[]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,2,2,4,4,4,4,4,4,4,4} + 2
[2,3,4,1] => [[.,.],[.,[.,.]]]
=> [[],[[[]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,2,2,4,4,4,4,4,4,4,4} + 2
[2,4,1,3] => [[.,.],[[.,.],.]]
=> [[],[[],[]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,2,2,4,4,4,4,4,4,4,4} + 2
[2,4,3,1] => [[.,.],[[.,.],.]]
=> [[],[[],[]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,2,2,4,4,4,4,4,4,4,4} + 2
[3,1,2,4] => [[.,[.,.]],[.,.]]
=> [[[]],[[]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,2,2,4,4,4,4,4,4,4,4} + 2
[3,1,4,2] => [[.,[.,.]],[.,.]]
=> [[[]],[[]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,2,2,4,4,4,4,4,4,4,4} + 2
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [[],[],[[]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,2,2,4,4,4,4,4,4,4,4} + 2
[3,2,4,1] => [[[.,.],.],[.,.]]
=> [[],[],[[]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,2,2,4,4,4,4,4,4,4,4} + 2
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [[[]],[[]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,2,2,4,4,4,4,4,4,4,4} + 2
[3,4,2,1] => [[[.,.],.],[.,.]]
=> [[],[],[[]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,2,2,4,4,4,4,4,4,4,4} + 2
[4,1,2,3] => [[.,[.,[.,.]]],.]
=> [[[[]]],[]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,2,2,4,4,4,4,4,4,4,4} + 2
[4,1,3,2] => [[.,[[.,.],.]],.]
=> [[[],[]],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,2,2,4,4,4,4,4,4,4,4} + 2
[4,2,1,3] => [[[.,.],[.,.]],.]
=> [[],[[]],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,2,2,4,4,4,4,4,4,4,4} + 2
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [[],[[]],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,2,2,4,4,4,4,4,4,4,4} + 2
[4,3,1,2] => [[[.,[.,.]],.],.]
=> [[[]],[],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,2,2,4,4,4,4,4,4,4,4} + 2
[4,3,2,1] => [[[[.,.],.],.],.]
=> [[],[],[],[]]
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4 = 2 + 2
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [[[[[[]]]]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10} + 2
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [[[[[],[]]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10} + 2
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [[[[],[[]]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10} + 2
[1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
=> [[[[],[[]]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10} + 2
[1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
=> [[[[[]],[]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10} + 2
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [[[[],[],[]]]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10} + 2
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [[[],[[[]]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10} + 2
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [[[],[[],[]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 4 + 2
[1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
=> [[[],[[[]]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10} + 2
[1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
=> [[[],[[[]]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10} + 2
[1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
=> [[[],[[],[]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 4 + 2
[1,3,5,4,2] => [.,[[.,.],[[.,.],.]]]
=> [[[],[[],[]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 4 + 2
[1,4,2,3,5] => [.,[[.,[.,.]],[.,.]]]
=> [[[[]],[[]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10} + 2
[1,4,2,5,3] => [.,[[.,[.,.]],[.,.]]]
=> [[[[]],[[]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10} + 2
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [[[],[],[[]]]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10} + 2
[1,4,3,5,2] => [.,[[[.,.],.],[.,.]]]
=> [[[],[],[[]]]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10} + 2
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [[[[]],[[]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10} + 2
[1,4,5,3,2] => [.,[[[.,.],.],[.,.]]]
=> [[[],[],[[]]]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10} + 2
[1,5,2,3,4] => [.,[[.,[.,[.,.]]],.]]
=> [[[[[]]],[]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10} + 2
[1,5,2,4,3] => [.,[[.,[[.,.],.]],.]]
=> [[[[],[]],[]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 4 + 2
[1,5,3,2,4] => [.,[[[.,.],[.,.]],.]]
=> [[[],[[]],[]]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10} + 2
[1,5,3,4,2] => [.,[[[.,.],[.,.]],.]]
=> [[[],[[]],[]]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10} + 2
[1,5,4,2,3] => [.,[[[.,[.,.]],.],.]]
=> [[[[]],[],[]]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10} + 2
[1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [[[],[],[],[]]]
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10} + 2
[3,2,1,5,4] => [[[.,.],.],[[.,.],.]]
=> [[],[],[[],[]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 4 + 2
[3,2,5,1,4] => [[[.,.],.],[[.,.],.]]
=> [[],[],[[],[]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 4 + 2
[3,2,5,4,1] => [[[.,.],.],[[.,.],.]]
=> [[],[],[[],[]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 4 + 2
[3,5,2,1,4] => [[[.,.],.],[[.,.],.]]
=> [[],[],[[],[]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 4 + 2
[3,5,2,4,1] => [[[.,.],.],[[.,.],.]]
=> [[],[],[[],[]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 4 + 2
[3,5,4,2,1] => [[[.,.],.],[[.,.],.]]
=> [[],[],[[],[]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 4 + 2
[5,2,1,4,3] => [[[.,.],[[.,.],.]],.]
=> [[],[[],[]],[]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 4 + 2
[5,2,4,1,3] => [[[.,.],[[.,.],.]],.]
=> [[],[[],[]],[]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 4 + 2
[5,2,4,3,1] => [[[.,.],[[.,.],.]],.]
=> [[],[[],[]],[]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 4 + 2
[5,4,1,3,2] => [[[.,[[.,.],.]],.],.]
=> [[[],[]],[],[]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 4 + 2
[1,2,5,3,4,6] => [.,[.,[[.,[.,.]],[.,.]]]]
=> [[[[[]],[[]]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 6 + 2
[1,2,5,3,6,4] => [.,[.,[[.,[.,.]],[.,.]]]]
=> [[[[[]],[[]]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 6 + 2
[1,2,5,6,3,4] => [.,[.,[[.,[.,.]],[.,.]]]]
=> [[[[[]],[[]]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 6 + 2
[2,1,5,3,4,6] => [[.,.],[[.,[.,.]],[.,.]]]
=> [[],[[[]],[[]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 6 + 2
[2,1,5,3,6,4] => [[.,.],[[.,[.,.]],[.,.]]]
=> [[],[[[]],[[]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 6 + 2
[2,1,5,6,3,4] => [[.,.],[[.,[.,.]],[.,.]]]
=> [[],[[[]],[[]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 6 + 2
[2,5,1,3,4,6] => [[.,.],[[.,[.,.]],[.,.]]]
=> [[],[[[]],[[]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 6 + 2
[2,5,1,3,6,4] => [[.,.],[[.,[.,.]],[.,.]]]
=> [[],[[[]],[[]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 6 + 2
[2,5,1,6,3,4] => [[.,.],[[.,[.,.]],[.,.]]]
=> [[],[[[]],[[]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 6 + 2
[2,5,3,1,4,6] => [[.,.],[[.,[.,.]],[.,.]]]
=> [[],[[[]],[[]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 6 + 2
[2,5,3,1,6,4] => [[.,.],[[.,[.,.]],[.,.]]]
=> [[],[[[]],[[]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 6 + 2
[2,5,3,4,1,6] => [[.,.],[[.,[.,.]],[.,.]]]
=> [[],[[[]],[[]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 6 + 2
[2,5,3,4,6,1] => [[.,.],[[.,[.,.]],[.,.]]]
=> [[],[[[]],[[]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 6 + 2
[2,5,3,6,1,4] => [[.,.],[[.,[.,.]],[.,.]]]
=> [[],[[[]],[[]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 6 + 2
[2,5,3,6,4,1] => [[.,.],[[.,[.,.]],[.,.]]]
=> [[],[[[]],[[]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 6 + 2
[2,5,6,1,3,4] => [[.,.],[[.,[.,.]],[.,.]]]
=> [[],[[[]],[[]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 6 + 2
[2,5,6,3,1,4] => [[.,.],[[.,[.,.]],[.,.]]]
=> [[],[[[]],[[]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 6 + 2
[2,5,6,3,4,1] => [[.,.],[[.,[.,.]],[.,.]]]
=> [[],[[[]],[[]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 6 + 2
[5,3,1,2,4,6] => [[[.,[.,.]],[.,.]],[.,.]]
=> [[[]],[[]],[[]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 6 + 2
[5,3,1,2,6,4] => [[[.,[.,.]],[.,.]],[.,.]]
=> [[[]],[[]],[[]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 6 + 2
[5,3,1,4,2,6] => [[[.,[.,.]],[.,.]],[.,.]]
=> [[[]],[[]],[[]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 6 + 2
[5,3,1,4,6,2] => [[[.,[.,.]],[.,.]],[.,.]]
=> [[[]],[[]],[[]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 6 + 2
[5,3,1,6,2,4] => [[[.,[.,.]],[.,.]],[.,.]]
=> [[[]],[[]],[[]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 6 + 2
[5,3,1,6,4,2] => [[[.,[.,.]],[.,.]],[.,.]]
=> [[[]],[[]],[[]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 6 + 2
[5,3,4,1,2,6] => [[[.,[.,.]],[.,.]],[.,.]]
=> [[[]],[[]],[[]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 6 + 2
[5,3,4,1,6,2] => [[[.,[.,.]],[.,.]],[.,.]]
=> [[[]],[[]],[[]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 6 + 2
[5,3,4,6,1,2] => [[[.,[.,.]],[.,.]],[.,.]]
=> [[[]],[[]],[[]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 6 + 2
[5,3,6,1,2,4] => [[[.,[.,.]],[.,.]],[.,.]]
=> [[[]],[[]],[[]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 6 + 2
[5,3,6,1,4,2] => [[[.,[.,.]],[.,.]],[.,.]]
=> [[[]],[[]],[[]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 6 + 2
[5,3,6,4,1,2] => [[[.,[.,.]],[.,.]],[.,.]]
=> [[[]],[[]],[[]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 6 + 2
[5,6,3,1,2,4] => [[[.,[.,.]],[.,.]],[.,.]]
=> [[[]],[[]],[[]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 6 + 2
[5,6,3,1,4,2] => [[[.,[.,.]],[.,.]],[.,.]]
=> [[[]],[[]],[[]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 6 + 2
[5,6,3,4,1,2] => [[[.,[.,.]],[.,.]],[.,.]]
=> [[[]],[[]],[[]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 6 + 2
Description
The energy of a graph, if it is integral.
The energy of a graph is the sum of the absolute values of its eigenvalues. This statistic is only defined for graphs with integral energy. It is known, that the energy is never an odd integer [2]. In fact, it is never the square root of an odd integer [3].
The energy of a graph is the sum of the energies of the connected components of a graph. The energy of the complete graph $K_n$ equals $2n-2$. For this reason, we do not define the energy of the empty graph.
Matching statistic: St001879
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00252: Permutations —restriction⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001879: Posets ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 38%
Mp00065: Permutations —permutation poset⟶ Posets
St001879: Posets ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 38%
Values
[1] => [] => ([],0)
=> ? = 0
[1,2] => [1] => ([],1)
=> ? ∊ {0,0}
[2,1] => [1] => ([],1)
=> ? ∊ {0,0}
[1,2,3] => [1,2] => ([(0,1)],2)
=> ? ∊ {0,0,1,1,1,1}
[1,3,2] => [1,2] => ([(0,1)],2)
=> ? ∊ {0,0,1,1,1,1}
[2,1,3] => [2,1] => ([],2)
=> ? ∊ {0,0,1,1,1,1}
[2,3,1] => [2,1] => ([],2)
=> ? ∊ {0,0,1,1,1,1}
[3,1,2] => [1,2] => ([(0,1)],2)
=> ? ∊ {0,0,1,1,1,1}
[3,2,1] => [2,1] => ([],2)
=> ? ∊ {0,0,1,1,1,1}
[1,2,3,4] => [1,2,3] => ([(0,2),(2,1)],3)
=> 2
[1,2,4,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 2
[1,3,2,4] => [1,3,2] => ([(0,1),(0,2)],3)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,4,4,4,4,4,4,4,4}
[1,3,4,2] => [1,3,2] => ([(0,1),(0,2)],3)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,4,4,4,4,4,4,4,4}
[1,4,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 2
[1,4,3,2] => [1,3,2] => ([(0,1),(0,2)],3)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,4,4,4,4,4,4,4,4}
[2,1,3,4] => [2,1,3] => ([(0,2),(1,2)],3)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,4,4,4,4,4,4,4,4}
[2,1,4,3] => [2,1,3] => ([(0,2),(1,2)],3)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,4,4,4,4,4,4,4,4}
[2,3,1,4] => [2,3,1] => ([(1,2)],3)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,4,4,4,4,4,4,4,4}
[2,3,4,1] => [2,3,1] => ([(1,2)],3)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,4,4,4,4,4,4,4,4}
[2,4,1,3] => [2,1,3] => ([(0,2),(1,2)],3)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,4,4,4,4,4,4,4,4}
[2,4,3,1] => [2,3,1] => ([(1,2)],3)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,4,4,4,4,4,4,4,4}
[3,1,2,4] => [3,1,2] => ([(1,2)],3)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,4,4,4,4,4,4,4,4}
[3,1,4,2] => [3,1,2] => ([(1,2)],3)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,4,4,4,4,4,4,4,4}
[3,2,1,4] => [3,2,1] => ([],3)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,4,4,4,4,4,4,4,4}
[3,2,4,1] => [3,2,1] => ([],3)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,4,4,4,4,4,4,4,4}
[3,4,1,2] => [3,1,2] => ([(1,2)],3)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,4,4,4,4,4,4,4,4}
[3,4,2,1] => [3,2,1] => ([],3)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,4,4,4,4,4,4,4,4}
[4,1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 2
[4,1,3,2] => [1,3,2] => ([(0,1),(0,2)],3)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,4,4,4,4,4,4,4,4}
[4,2,1,3] => [2,1,3] => ([(0,2),(1,2)],3)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,4,4,4,4,4,4,4,4}
[4,2,3,1] => [2,3,1] => ([(1,2)],3)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,4,4,4,4,4,4,4,4}
[4,3,1,2] => [3,1,2] => ([(1,2)],3)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,4,4,4,4,4,4,4,4}
[4,3,2,1] => [3,2,1] => ([],3)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,4,4,4,4,4,4,4,4}
[1,2,3,4,5] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 3
[1,2,3,5,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 3
[1,2,4,3,5] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10}
[1,2,4,5,3] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10}
[1,2,5,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 3
[1,2,5,4,3] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10}
[1,3,2,4,5] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
[1,3,2,5,4] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
[1,3,4,2,5] => [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10}
[1,3,4,5,2] => [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10}
[1,3,5,2,4] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
[1,3,5,4,2] => [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10}
[1,4,2,3,5] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10}
[1,4,2,5,3] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10}
[1,4,3,2,5] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10}
[1,4,3,5,2] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10}
[1,4,5,2,3] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10}
[1,4,5,3,2] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10}
[1,5,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 3
[1,5,2,4,3] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10}
[1,5,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
[1,5,3,4,2] => [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10}
[1,5,4,2,3] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10}
[1,5,4,3,2] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10}
[2,1,3,4,5] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10}
[2,1,3,5,4] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10}
[2,1,4,3,5] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10}
[2,1,4,5,3] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10}
[2,1,5,3,4] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10}
[5,1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 3
[5,1,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
[1,2,3,4,5,6] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[1,2,3,4,6,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[1,2,3,6,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[1,2,4,3,5,6] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 5
[1,2,4,3,6,5] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 5
[1,2,4,6,3,5] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 5
[1,2,6,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[1,2,6,4,3,5] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 5
[1,3,2,4,5,6] => [1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 5
[1,3,2,4,6,5] => [1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 5
[1,3,2,6,4,5] => [1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 5
[1,3,4,2,5,6] => [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 6
[1,3,4,2,6,5] => [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 6
[1,3,4,6,2,5] => [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 6
[1,3,6,2,4,5] => [1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 5
[1,3,6,4,2,5] => [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 6
[1,4,2,3,5,6] => [1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 6
[1,4,2,3,6,5] => [1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 6
[1,4,2,6,3,5] => [1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 6
[1,4,3,2,5,6] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 9
[1,4,3,2,6,5] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 9
[1,4,3,6,2,5] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 9
[1,4,6,2,3,5] => [1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 6
[1,4,6,3,2,5] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 9
[1,6,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[1,6,2,4,3,5] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 5
[1,6,3,2,4,5] => [1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 5
[1,6,3,4,2,5] => [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 6
[1,6,4,2,3,5] => [1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 6
[1,6,4,3,2,5] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 9
[6,1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[6,1,2,4,3,5] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 5
[6,1,3,2,4,5] => [1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 5
[6,1,3,4,2,5] => [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 6
[6,1,4,2,3,5] => [1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 6
[6,1,4,3,2,5] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 9
Description
The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice.
Matching statistic: St000264
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000264: Graphs ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 6%
Mp00223: Permutations —runsort⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000264: Graphs ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 6%
Values
[1] => [1] => [1] => ([],1)
=> ? = 0
[1,2] => [1,2] => [1,2] => ([],2)
=> ? ∊ {0,0}
[2,1] => [2,1] => [1,2] => ([],2)
=> ? ∊ {0,0}
[1,2,3] => [1,3,2] => [1,3,2] => ([(1,2)],3)
=> ? ∊ {0,0,1,1,1,1}
[1,3,2] => [1,3,2] => [1,3,2] => ([(1,2)],3)
=> ? ∊ {0,0,1,1,1,1}
[2,1,3] => [2,1,3] => [1,3,2] => ([(1,2)],3)
=> ? ∊ {0,0,1,1,1,1}
[2,3,1] => [2,3,1] => [1,2,3] => ([],3)
=> ? ∊ {0,0,1,1,1,1}
[3,1,2] => [3,1,2] => [1,2,3] => ([],3)
=> ? ∊ {0,0,1,1,1,1}
[3,2,1] => [3,2,1] => [1,2,3] => ([],3)
=> ? ∊ {0,0,1,1,1,1}
[1,2,3,4] => [1,4,3,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4}
[1,2,4,3] => [1,4,3,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4}
[1,3,2,4] => [1,4,3,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4}
[1,3,4,2] => [1,4,3,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4}
[1,4,2,3] => [1,4,3,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4}
[1,4,3,2] => [1,4,3,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4}
[2,1,3,4] => [2,1,4,3] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4}
[2,1,4,3] => [2,1,4,3] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4}
[2,3,1,4] => [2,4,1,3] => [1,3,2,4] => ([(2,3)],4)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4}
[2,3,4,1] => [2,4,3,1] => [1,2,4,3] => ([(2,3)],4)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4}
[2,4,1,3] => [2,4,1,3] => [1,3,2,4] => ([(2,3)],4)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4}
[2,4,3,1] => [2,4,3,1] => [1,2,4,3] => ([(2,3)],4)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4}
[3,1,2,4] => [3,1,4,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4}
[3,1,4,2] => [3,1,4,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4}
[3,2,1,4] => [3,2,1,4] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4}
[3,2,4,1] => [3,2,4,1] => [1,2,4,3] => ([(2,3)],4)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4}
[3,4,1,2] => [3,4,1,2] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4}
[3,4,2,1] => [3,4,2,1] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4}
[4,1,2,3] => [4,1,3,2] => [1,3,2,4] => ([(2,3)],4)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4}
[4,1,3,2] => [4,1,3,2] => [1,3,2,4] => ([(2,3)],4)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4}
[4,2,1,3] => [4,2,1,3] => [1,3,2,4] => ([(2,3)],4)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4}
[4,2,3,1] => [4,2,3,1] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4}
[4,3,1,2] => [4,3,1,2] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4}
[4,3,2,1] => [4,3,2,1] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4}
[1,2,3,4,5] => [1,5,4,3,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10}
[1,2,3,5,4] => [1,5,4,3,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10}
[1,2,4,3,5] => [1,5,4,3,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10}
[1,2,4,5,3] => [1,5,4,3,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10}
[1,2,5,3,4] => [1,5,4,3,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10}
[1,2,5,4,3] => [1,5,4,3,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10}
[1,3,2,4,5] => [1,5,4,3,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10}
[1,3,2,5,4] => [1,5,4,3,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10}
[1,3,4,2,5] => [1,5,4,3,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10}
[1,3,4,5,2] => [1,5,4,3,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10}
[1,3,5,2,4] => [1,5,4,3,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10}
[1,3,5,4,2] => [1,5,4,3,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10}
[1,4,2,3,5] => [1,5,4,3,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10}
[1,4,2,5,3] => [1,5,4,3,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10}
[1,4,3,2,5] => [1,5,4,3,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10}
[1,4,3,5,2] => [1,5,4,3,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10}
[1,4,5,2,3] => [1,5,4,3,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10}
[2,3,1,4,5,6] => [2,6,1,5,4,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
[2,3,1,4,6,5] => [2,6,1,5,4,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
[2,3,1,5,4,6] => [2,6,1,5,4,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
[2,3,1,5,6,4] => [2,6,1,5,4,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
[2,3,1,6,4,5] => [2,6,1,5,4,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
[2,3,1,6,5,4] => [2,6,1,5,4,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
[2,4,1,3,5,6] => [2,6,1,5,4,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
[2,4,1,3,6,5] => [2,6,1,5,4,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
[2,4,1,5,3,6] => [2,6,1,5,4,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
[2,4,1,5,6,3] => [2,6,1,5,4,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
[2,4,1,6,3,5] => [2,6,1,5,4,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
[2,4,1,6,5,3] => [2,6,1,5,4,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
[2,5,1,3,4,6] => [2,6,1,5,4,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
[2,5,1,3,6,4] => [2,6,1,5,4,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
[2,5,1,4,3,6] => [2,6,1,5,4,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
[2,5,1,4,6,3] => [2,6,1,5,4,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
[2,5,1,6,3,4] => [2,6,1,5,4,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
[2,5,1,6,4,3] => [2,6,1,5,4,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
[2,6,1,3,4,5] => [2,6,1,5,4,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
[2,6,1,3,5,4] => [2,6,1,5,4,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
[2,6,1,4,3,5] => [2,6,1,5,4,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
[2,6,1,4,5,3] => [2,6,1,5,4,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
[2,6,1,5,3,4] => [2,6,1,5,4,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
[2,6,1,5,4,3] => [2,6,1,5,4,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
[3,2,4,1,5,6] => [3,2,6,1,5,4] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
[3,2,4,1,6,5] => [3,2,6,1,5,4] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
[3,2,5,1,4,6] => [3,2,6,1,5,4] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
[3,2,5,1,6,4] => [3,2,6,1,5,4] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
[3,2,6,1,4,5] => [3,2,6,1,5,4] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
[3,2,6,1,5,4] => [3,2,6,1,5,4] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
[4,2,3,1,5,6] => [4,2,6,1,5,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
[4,2,3,1,6,5] => [4,2,6,1,5,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
[4,2,5,1,3,6] => [4,2,6,1,5,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
[4,2,5,1,6,3] => [4,2,6,1,5,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
[4,2,6,1,3,5] => [4,2,6,1,5,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
[4,2,6,1,5,3] => [4,2,6,1,5,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
[4,3,2,5,1,6] => [4,3,2,6,1,5] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
[4,3,2,6,1,5] => [4,3,2,6,1,5] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
Description
The girth of a graph, which is not a tree.
This is the length of the shortest cycle in the graph.
Matching statistic: St001605
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001605: Integer partitions ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 12%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001605: Integer partitions ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 12%
Values
[1] => [1]
=> []
=> ?
=> ? = 0
[1,2] => [2]
=> []
=> ?
=> ? ∊ {0,0}
[2,1] => [1,1]
=> [1]
=> []
=> ? ∊ {0,0}
[1,2,3] => [3]
=> []
=> ?
=> ? ∊ {0,0,1,1,1,1}
[1,3,2] => [2,1]
=> [1]
=> []
=> ? ∊ {0,0,1,1,1,1}
[2,1,3] => [2,1]
=> [1]
=> []
=> ? ∊ {0,0,1,1,1,1}
[2,3,1] => [2,1]
=> [1]
=> []
=> ? ∊ {0,0,1,1,1,1}
[3,1,2] => [2,1]
=> [1]
=> []
=> ? ∊ {0,0,1,1,1,1}
[3,2,1] => [1,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1}
[1,2,3,4] => [4]
=> []
=> ?
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4}
[1,2,4,3] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4}
[1,3,2,4] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4}
[1,3,4,2] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4}
[1,4,2,3] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4}
[1,4,3,2] => [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4}
[2,1,3,4] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4}
[2,1,4,3] => [2,2]
=> [2]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4}
[2,3,1,4] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4}
[2,3,4,1] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4}
[2,4,1,3] => [2,2]
=> [2]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4}
[2,4,3,1] => [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4}
[3,1,2,4] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4}
[3,1,4,2] => [2,2]
=> [2]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4}
[3,2,1,4] => [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4}
[3,2,4,1] => [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4}
[3,4,1,2] => [2,2]
=> [2]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4}
[3,4,2,1] => [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4}
[4,1,2,3] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4}
[4,1,3,2] => [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4}
[4,2,1,3] => [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4}
[4,2,3,1] => [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4}
[4,3,1,2] => [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4}
[4,3,2,1] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4}
[1,2,3,4,5] => [5]
=> []
=> ?
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10}
[1,2,3,5,4] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10}
[1,2,4,3,5] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10}
[1,2,4,5,3] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10}
[1,2,5,3,4] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10}
[1,2,5,4,3] => [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10}
[1,3,2,4,5] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10}
[1,3,2,5,4] => [3,2]
=> [2]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10}
[1,3,4,2,5] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10}
[1,3,4,5,2] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10}
[1,3,5,2,4] => [3,2]
=> [2]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10}
[1,3,5,4,2] => [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10}
[1,4,2,3,5] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10}
[1,4,2,5,3] => [3,2]
=> [2]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10}
[1,4,3,2,5] => [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10}
[1,4,3,5,2] => [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10}
[1,4,5,2,3] => [3,2]
=> [2]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10}
[5,4,3,2,1] => [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 2
[1,6,5,4,3,2] => [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 2
[2,6,5,4,3,1] => [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 2
[3,6,5,4,2,1] => [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 2
[4,6,5,3,2,1] => [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 2
[5,4,3,2,1,6] => [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 2
[5,4,3,2,6,1] => [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 2
[5,4,3,6,2,1] => [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 2
[5,4,6,3,2,1] => [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 2
[5,6,4,3,2,1] => [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 2
[6,1,5,4,3,2] => [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 2
[6,2,5,4,3,1] => [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 2
[6,3,5,4,2,1] => [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 2
[6,4,3,2,1,5] => [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 2
[6,4,3,2,5,1] => [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 2
[6,4,3,5,2,1] => [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 2
[6,4,5,3,2,1] => [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 2
[6,5,1,4,3,2] => [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 2
[6,5,2,4,3,1] => [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 2
[6,5,3,2,1,4] => [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 2
[6,5,3,2,4,1] => [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 2
[6,5,3,4,2,1] => [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 2
[6,5,4,1,3,2] => [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 2
[6,5,4,2,1,3] => [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 2
[6,5,4,2,3,1] => [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 2
[6,5,4,3,1,2] => [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 2
[6,5,4,3,2,1] => [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 6
Description
The number of colourings of a cycle such that the multiplicities of colours are given by a partition.
Two colourings are considered equal, if they are obtained by an action of the cyclic group.
This statistic is only defined for partitions of size at least 3, to avoid ambiguity.
Matching statistic: St001117
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Values
[1] => ([],1)
=> ([],1)
=> 0
[1,2] => ([(0,1)],2)
=> ([],2)
=> 0
[2,1] => ([(0,1)],2)
=> ([],2)
=> 0
[1,2,3] => ([(0,2),(2,1)],3)
=> ([],3)
=> 0
[1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 1
[2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 1
[2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 1
[3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 1
[3,2,1] => ([(0,2),(2,1)],3)
=> ([],3)
=> 0
[1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0
[1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 2
[1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> ? ∊ {1,1,1,1,4,4,4,4,4,4,4,4}
[1,3,4,2] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> ? ∊ {1,1,1,1,4,4,4,4,4,4,4,4}
[1,4,2,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> ? ∊ {1,1,1,1,4,4,4,4,4,4,4,4}
[1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 2
[2,1,3,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 2
[2,1,4,3] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(2,5),(3,4)],6)
=> 1
[2,3,1,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> ? ∊ {1,1,1,1,4,4,4,4,4,4,4,4}
[2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 2
[2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ([(2,3),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {1,1,1,1,4,4,4,4,4,4,4,4}
[2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> ? ∊ {1,1,1,1,4,4,4,4,4,4,4,4}
[3,1,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> ? ∊ {1,1,1,1,4,4,4,4,4,4,4,4}
[3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ([(2,3),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {1,1,1,1,4,4,4,4,4,4,4,4}
[3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 2
[3,2,4,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> ? ∊ {1,1,1,1,4,4,4,4,4,4,4,4}
[3,4,1,2] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(2,5),(3,4)],6)
=> 1
[3,4,2,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 2
[4,1,2,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 2
[4,1,3,2] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> ? ∊ {1,1,1,1,4,4,4,4,4,4,4,4}
[4,2,1,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> ? ∊ {1,1,1,1,4,4,4,4,4,4,4,4}
[4,2,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> ? ∊ {1,1,1,1,4,4,4,4,4,4,4,4}
[4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 2
[4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0
[1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(2,7),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10}
[1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(2,7),(3,6),(3,9),(4,5),(4,7),(4,8),(5,8),(5,9),(6,8),(6,9),(7,9),(8,9)],10)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10}
[1,2,4,5,3] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(2,7),(3,6),(3,9),(4,5),(4,7),(4,8),(5,8),(5,9),(6,8),(6,9),(7,9),(8,9)],10)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10}
[1,2,5,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(2,7),(3,6),(3,9),(4,5),(4,7),(4,8),(5,8),(5,9),(6,8),(6,9),(7,9),(8,9)],10)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10}
[1,2,5,4,3] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ([(2,5),(2,8),(3,4),(3,8),(4,7),(5,7),(6,7),(6,8),(7,8)],9)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10}
[1,3,2,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(2,7),(3,6),(3,9),(4,5),(4,7),(4,8),(5,8),(5,9),(6,8),(6,9),(7,9),(8,9)],10)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10}
[1,3,2,5,4] => ([(0,1),(0,2),(0,3),(1,7),(1,8),(2,5),(2,8),(3,5),(3,7),(3,8),(5,9),(6,4),(7,6),(7,9),(8,6),(8,9),(9,4)],10)
=> ([(2,5),(3,6),(3,9),(4,7),(4,8),(5,9),(6,8),(6,9),(7,8),(7,9)],10)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10}
[1,3,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ([(2,3),(3,7),(3,11),(4,5),(4,6),(4,10),(4,11),(5,6),(5,9),(5,11),(6,8),(6,11),(7,8),(7,9),(7,10),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10}
[1,3,4,5,2] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(2,7),(3,6),(3,9),(4,5),(4,7),(4,8),(5,8),(5,9),(6,8),(6,9),(7,9),(8,9)],10)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10}
[1,3,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ([(2,3),(3,9),(4,9),(4,11),(4,12),(4,13),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,10),(6,12),(6,13),(7,8),(7,10),(7,11),(7,13),(8,10),(8,11),(8,12),(9,11),(9,12),(9,13),(10,11),(10,12),(10,13),(11,12),(11,13),(12,13)],14)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10}
[1,3,5,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> ([(2,3),(2,11),(3,10),(4,9),(4,10),(4,11),(5,6),(5,7),(5,8),(5,10),(6,7),(6,8),(6,11),(7,8),(7,9),(7,10),(8,9),(8,11),(9,10),(9,11),(10,11)],12)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10}
[1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ([(2,3),(3,7),(3,11),(4,5),(4,6),(4,10),(4,11),(5,6),(5,9),(5,11),(6,8),(6,11),(7,8),(7,9),(7,10),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10}
[1,4,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ([(2,3),(3,9),(4,9),(4,11),(4,12),(4,13),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,10),(6,12),(6,13),(7,8),(7,10),(7,11),(7,13),(8,10),(8,11),(8,12),(9,11),(9,12),(9,13),(10,11),(10,12),(10,13),(11,12),(11,13),(12,13)],14)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10}
[1,4,3,2,5] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ([(2,3),(2,10),(3,9),(4,7),(4,8),(4,9),(5,6),(5,8),(5,9),(5,10),(6,7),(6,9),(6,10),(7,8),(7,10),(8,10),(9,10)],11)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10}
[1,4,3,5,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ([(2,3),(2,12),(3,11),(4,5),(4,6),(4,7),(4,8),(4,12),(5,6),(5,8),(5,10),(5,11),(6,8),(6,9),(6,11),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,12),(9,10),(9,11),(9,12),(10,11),(10,12),(11,12)],13)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10}
[1,4,5,2,3] => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(5,9),(5,10),(6,9),(6,10),(7,10),(8,9),(8,10),(9,4),(10,4)],11)
=> ([(2,3),(3,10),(4,5),(4,8),(4,9),(5,7),(5,9),(6,7),(6,8),(6,9),(6,10),(7,8),(7,10),(8,10),(9,10)],11)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10}
[1,4,5,3,2] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ([(2,3),(2,10),(3,9),(4,7),(4,8),(4,9),(5,6),(5,8),(5,9),(5,10),(6,7),(6,9),(6,10),(7,8),(7,10),(8,10),(9,10)],11)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10}
[1,5,2,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(2,7),(3,6),(3,9),(4,5),(4,7),(4,8),(5,8),(5,9),(6,8),(6,9),(7,9),(8,9)],10)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10}
[1,5,2,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> ([(2,3),(2,11),(3,10),(4,9),(4,10),(4,11),(5,6),(5,7),(5,8),(5,10),(6,7),(6,8),(6,11),(7,8),(7,9),(7,10),(8,9),(8,11),(9,10),(9,11),(10,11)],12)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10}
[1,5,3,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ([(2,3),(2,12),(3,11),(4,5),(4,6),(4,7),(4,8),(4,12),(5,6),(5,8),(5,10),(5,11),(6,8),(6,9),(6,11),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,12),(9,10),(9,11),(9,12),(10,11),(10,12),(11,12)],13)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10}
[1,5,3,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ([(2,3),(2,12),(3,11),(4,5),(4,6),(4,7),(4,8),(4,12),(5,6),(5,8),(5,10),(5,11),(6,8),(6,9),(6,11),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,12),(9,10),(9,11),(9,12),(10,11),(10,12),(11,12)],13)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10}
[1,5,4,2,3] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ([(2,3),(2,10),(3,9),(4,7),(4,8),(4,9),(5,6),(5,8),(5,9),(5,10),(6,7),(6,9),(6,10),(7,8),(7,10),(8,10),(9,10)],11)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10}
[1,5,4,3,2] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(2,7),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10}
[2,1,3,4,5] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(2,7),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10}
[2,1,3,5,4] => ([(0,1),(0,2),(0,3),(1,7),(1,8),(2,5),(2,8),(3,5),(3,7),(5,9),(6,4),(7,6),(7,9),(8,6),(8,9),(9,4)],10)
=> ([(2,3),(3,9),(4,5),(4,7),(4,8),(5,6),(5,8),(6,7),(6,9),(7,9),(8,9)],10)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10}
[2,1,4,3,5] => ([(0,1),(0,2),(0,3),(1,7),(1,8),(2,5),(2,8),(3,5),(3,7),(3,8),(5,9),(6,4),(7,6),(7,9),(8,6),(8,9),(9,4)],10)
=> ([(2,5),(3,6),(3,9),(4,7),(4,8),(5,9),(6,8),(6,9),(7,8),(7,9)],10)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10}
[2,1,4,5,3] => ([(0,2),(0,3),(0,4),(1,9),(1,10),(2,6),(2,7),(3,5),(3,6),(4,1),(4,5),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ([(2,3),(3,10),(4,6),(4,7),(4,8),(5,6),(5,8),(5,9),(5,10),(6,7),(6,9),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10}
[2,1,5,3,4] => ([(0,2),(0,3),(0,4),(1,9),(1,10),(2,6),(2,7),(3,5),(3,6),(4,1),(4,5),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ([(2,3),(3,10),(4,6),(4,7),(4,8),(5,6),(5,8),(5,9),(5,10),(6,7),(6,9),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10}
[2,1,5,4,3] => ([(0,3),(0,4),(1,8),(2,7),(2,8),(3,1),(3,5),(4,2),(4,5),(5,7),(5,8),(7,6),(8,6)],9)
=> ([(2,6),(3,4),(3,8),(4,7),(5,7),(5,8),(6,8),(7,8)],9)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10}
[2,3,1,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(2,7),(3,6),(3,9),(4,5),(4,7),(4,8),(5,8),(5,9),(6,8),(6,9),(7,9),(8,9)],10)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10}
[2,3,1,5,4] => ([(0,2),(0,3),(0,4),(1,9),(1,10),(2,6),(2,7),(3,5),(3,6),(4,1),(4,5),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ([(2,3),(3,10),(4,6),(4,7),(4,8),(5,6),(5,8),(5,9),(5,10),(6,7),(6,9),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10}
[2,3,4,1,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(2,7),(3,6),(3,9),(4,5),(4,7),(4,8),(5,8),(5,9),(6,8),(6,9),(7,9),(8,9)],10)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10}
[2,3,4,5,1] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(2,7),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10}
[2,3,5,1,4] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,9),(2,10),(3,6),(3,9),(4,7),(4,9),(4,10),(5,1),(5,6),(5,7),(5,10),(6,11),(6,12),(7,11),(7,12),(9,12),(10,11),(10,12),(11,8),(12,8)],13)
=> ([(2,3),(3,11),(4,6),(4,9),(4,10),(4,11),(5,7),(5,8),(5,10),(5,11),(5,12),(6,8),(6,9),(6,10),(6,12),(7,8),(7,9),(7,10),(7,11),(7,12),(8,9),(8,11),(8,12),(9,10),(9,12),(10,12),(11,12)],13)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10}
[2,3,5,4,1] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ([(2,3),(2,10),(3,9),(4,7),(4,8),(4,9),(5,6),(5,8),(5,9),(5,10),(6,7),(6,9),(6,10),(7,8),(7,10),(8,10),(9,10)],11)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10}
[2,4,1,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ([(2,3),(3,9),(4,9),(4,11),(4,12),(4,13),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,10),(6,12),(6,13),(7,8),(7,10),(7,11),(7,13),(8,10),(8,11),(8,12),(9,11),(9,12),(9,13),(10,11),(10,12),(10,13),(11,12),(11,13),(12,13)],14)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10}
[2,4,1,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,6),(2,9),(2,11),(3,6),(3,9),(3,10),(4,7),(4,9),(4,10),(4,11),(5,7),(5,9),(5,10),(5,11),(6,13),(7,12),(7,13),(9,12),(9,13),(10,12),(10,13),(11,12),(11,13),(12,8),(13,8)],14)
=> ([(2,3),(3,13),(4,5),(4,6),(4,11),(4,12),(4,13),(5,6),(5,10),(5,11),(5,13),(6,9),(6,11),(6,13),(7,8),(7,9),(7,10),(7,12),(7,13),(8,9),(8,10),(8,12),(8,13),(9,10),(9,11),(9,12),(10,11),(10,12),(11,12),(11,13),(12,13)],14)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10}
[2,4,3,1,5] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ([(2,3),(2,12),(3,11),(4,5),(4,6),(4,7),(4,8),(4,12),(5,6),(5,8),(5,10),(5,11),(6,8),(6,9),(6,11),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,12),(9,10),(9,11),(9,12),(10,11),(10,12),(11,12)],13)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10}
[5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 0
[6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 0
Description
The game chromatic index of a graph.
Two players, Alice and Bob, take turns colouring properly any uncolored edge of the graph. Alice begins. If it is not possible for either player to colour a edge, then Bob wins. If the graph is completely colored, Alice wins.
The game chromatic index is the smallest number of colours such that Alice has a winning strategy.
The following 6 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001574The minimal number of edges to add or remove to make a graph regular. St001576The minimal number of edges to add or remove to make a graph vertex transitive. St001742The difference of the maximal and the minimal degree in a graph. St000086The number of subgraphs. St000274The number of perfect matchings of a graph. St001575The minimal number of edges to add or remove to make a graph edge transitive.
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