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Your data matches 48 different statistics following compositions of up to 3 maps.
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Matching statistic: St000264
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(load all 2 compositions to match this statistic)
Mp00223: Permutations —runsort⟶ Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000264: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000264: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,3,2,5,4] => [1,3,2,5,4] => [3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,4,2,5,3] => [1,4,2,5,3] => [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 4
[1,4,3,2,5] => [1,4,2,5,3] => [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 4
[1,5,2,4,3] => [1,5,2,4,3] => [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 3
[1,5,3,2,4] => [1,5,2,4,3] => [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 3
[2,4,1,5,3] => [1,5,2,4,3] => [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 3
[2,4,3,1,5] => [1,5,2,4,3] => [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 3
[2,5,1,4,3] => [1,4,2,5,3] => [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 4
[2,5,3,1,4] => [1,4,2,5,3] => [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 4
[2,5,4,1,3] => [1,3,2,5,4] => [3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[3,1,4,2,5] => [1,4,2,5,3] => [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 4
[3,1,5,2,4] => [1,5,2,4,3] => [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 3
[3,2,4,1,5] => [1,5,2,4,3] => [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 3
[3,2,5,1,4] => [1,4,2,5,3] => [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 4
[4,1,3,2,5] => [1,3,2,5,4] => [3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[4,2,5,1,3] => [1,3,2,5,4] => [3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,2,4,3,6,5] => [1,2,4,3,6,5] => [3,4,5,2,6,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,2,5,3,6,4] => [1,2,5,3,6,4] => [3,4,5,6,1,2] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 4
[1,2,5,4,3,6] => [1,2,5,3,6,4] => [3,4,5,6,1,2] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 4
[1,2,6,3,5,4] => [1,2,6,3,5,4] => [3,4,2,6,5,1] => ([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,2,6,4,3,5] => [1,2,6,3,5,4] => [3,4,2,6,5,1] => ([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,3,2,4,6,5] => [1,3,2,4,6,5] => [3,4,2,5,6,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,3,2,5,4,6] => [1,3,2,5,4,6] => [3,4,2,5,1,6] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,3,2,5,6,4] => [1,3,2,5,6,4] => [3,4,2,6,1,5] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6)
=> 3
[1,3,2,6,4,5] => [1,3,2,6,4,5] => [3,4,2,1,6,5] => ([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,3,2,6,5,4] => [1,3,2,6,4,5] => [3,4,2,1,6,5] => ([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,3,4,2,6,5] => [1,3,4,2,6,5] => [3,5,2,4,6,1] => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,3,5,2,6,4] => [1,3,5,2,6,4] => [3,5,2,6,1,4] => ([(0,1),(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,3,5,4,2,6] => [1,3,5,2,6,4] => [3,5,2,6,1,4] => ([(0,1),(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,3,6,2,5,4] => [1,3,6,2,5,4] => [3,2,5,6,4,1] => ([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,3,6,4,2,5] => [1,3,6,2,5,4] => [3,2,5,6,4,1] => ([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,4,2,3,6,5] => [1,4,2,3,6,5] => [3,2,5,4,6,1] => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> 3
[1,4,2,5,3,6] => [1,4,2,5,3,6] => [3,4,5,1,2,6] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 4
[1,4,2,5,6,3] => [1,4,2,5,6,3] => [3,4,6,1,2,5] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
[1,4,2,6,3,5] => [1,4,2,6,3,5] => [3,4,1,5,6,2] => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
[1,4,2,6,5,3] => [1,4,2,6,3,5] => [3,4,1,5,6,2] => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
[1,4,3,2,5,6] => [1,4,2,5,6,3] => [3,4,6,1,2,5] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
[1,4,3,2,6,5] => [1,4,2,6,3,5] => [3,4,1,5,6,2] => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
[1,4,3,5,2,6] => [1,4,2,6,3,5] => [3,4,1,5,6,2] => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
[1,4,3,6,2,5] => [1,4,2,5,3,6] => [3,4,5,1,2,6] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 4
[1,4,3,6,5,2] => [1,4,2,3,6,5] => [3,2,5,4,6,1] => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> 3
[1,4,5,2,6,3] => [1,4,5,2,6,3] => [3,5,6,1,2,4] => ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> 4
[1,4,5,3,2,6] => [1,4,5,2,6,3] => [3,5,6,1,2,4] => ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> 4
[1,4,6,2,5,3] => [1,4,6,2,5,3] => [3,2,6,4,5,1] => ([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,4,6,3,2,5] => [1,4,6,2,5,3] => [3,2,6,4,5,1] => ([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,5,2,3,6,4] => [1,5,2,3,6,4] => [3,2,5,6,1,4] => ([(0,3),(0,4),(1,2),(1,5),(2,5),(3,5),(4,5)],6)
=> 3
[1,5,2,4,3,6] => [1,5,2,4,3,6] => [3,2,5,4,1,6] => ([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> 3
[1,5,2,4,6,3] => [1,5,2,4,6,3] => [3,2,6,4,1,5] => ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,5,2,6,3,4] => [1,5,2,6,3,4] => [3,4,1,6,2,5] => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> 4
[1,5,2,6,4,3] => [1,5,2,6,3,4] => [3,4,1,6,2,5] => ([(0,4),(1,2),(1,3),(2,5),(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: St001934
Mp00223: Permutations —runsort⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St001934: Integer partitions ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 100%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St001934: Integer partitions ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 100%
Values
[1,3,2,5,4] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> ? = 3 - 2
[1,4,2,5,3] => [1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> 2 = 4 - 2
[1,4,3,2,5] => [1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> 2 = 4 - 2
[1,5,2,4,3] => [1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 1 = 3 - 2
[1,5,3,2,4] => [1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 1 = 3 - 2
[2,4,1,5,3] => [1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 1 = 3 - 2
[2,4,3,1,5] => [1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 1 = 3 - 2
[2,5,1,4,3] => [1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> 2 = 4 - 2
[2,5,3,1,4] => [1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> 2 = 4 - 2
[2,5,4,1,3] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> ? = 3 - 2
[3,1,4,2,5] => [1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> 2 = 4 - 2
[3,1,5,2,4] => [1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 1 = 3 - 2
[3,2,4,1,5] => [1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 1 = 3 - 2
[3,2,5,1,4] => [1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> 2 = 4 - 2
[4,1,3,2,5] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> ? = 3 - 2
[4,2,5,1,3] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> ? = 3 - 2
[1,2,4,3,6,5] => [1,2,4,3,6,5] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2,1]
=> ? = 3 - 2
[1,2,5,3,6,4] => [1,2,5,3,6,4] => [1,0,1,0,1,1,1,0,0,1,0,0]
=> [4,2,2,2,1]
=> ? = 4 - 2
[1,2,5,4,3,6] => [1,2,5,3,6,4] => [1,0,1,0,1,1,1,0,0,1,0,0]
=> [4,2,2,2,1]
=> ? = 4 - 2
[1,2,6,3,5,4] => [1,2,6,3,5,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2,1]
=> ? = 3 - 2
[1,2,6,4,3,5] => [1,2,6,3,5,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2,1]
=> ? = 3 - 2
[1,3,2,4,6,5] => [1,3,2,4,6,5] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [4,4,3,1,1]
=> ? = 3 - 2
[1,3,2,5,4,6] => [1,3,2,5,4,6] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1,1]
=> ? = 3 - 2
[1,3,2,5,6,4] => [1,3,2,5,6,4] => [1,0,1,1,0,0,1,1,0,1,0,0]
=> [4,3,3,1,1]
=> ? = 3 - 2
[1,3,2,6,4,5] => [1,3,2,6,4,5] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,3,3,1,1]
=> ? = 3 - 2
[1,3,2,6,5,4] => [1,3,2,6,4,5] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,3,3,1,1]
=> ? = 3 - 2
[1,3,4,2,6,5] => [1,3,4,2,6,5] => [1,0,1,1,0,1,0,0,1,1,0,0]
=> [4,4,2,1,1]
=> ? = 3 - 2
[1,3,5,2,6,4] => [1,3,5,2,6,4] => [1,0,1,1,0,1,1,0,0,1,0,0]
=> [4,2,2,1,1]
=> ? = 3 - 2
[1,3,5,4,2,6] => [1,3,5,2,6,4] => [1,0,1,1,0,1,1,0,0,1,0,0]
=> [4,2,2,1,1]
=> ? = 3 - 2
[1,3,6,2,5,4] => [1,3,6,2,5,4] => [1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,2,2,1,1]
=> ? = 3 - 2
[1,3,6,4,2,5] => [1,3,6,2,5,4] => [1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,2,2,1,1]
=> ? = 3 - 2
[1,4,2,3,6,5] => [1,4,2,3,6,5] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [4,4,1,1,1]
=> ? = 3 - 2
[1,4,2,5,3,6] => [1,4,2,5,3,6] => [1,0,1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,1,1]
=> ? = 4 - 2
[1,4,2,5,6,3] => [1,4,2,5,6,3] => [1,0,1,1,1,0,0,1,0,1,0,0]
=> [4,3,1,1,1]
=> ? = 4 - 2
[1,4,2,6,3,5] => [1,4,2,6,3,5] => [1,0,1,1,1,0,0,1,1,0,0,0]
=> [3,3,1,1,1]
=> ? = 4 - 2
[1,4,2,6,5,3] => [1,4,2,6,3,5] => [1,0,1,1,1,0,0,1,1,0,0,0]
=> [3,3,1,1,1]
=> ? = 4 - 2
[1,4,3,2,5,6] => [1,4,2,5,6,3] => [1,0,1,1,1,0,0,1,0,1,0,0]
=> [4,3,1,1,1]
=> ? = 4 - 2
[1,4,3,2,6,5] => [1,4,2,6,3,5] => [1,0,1,1,1,0,0,1,1,0,0,0]
=> [3,3,1,1,1]
=> ? = 4 - 2
[1,4,3,5,2,6] => [1,4,2,6,3,5] => [1,0,1,1,1,0,0,1,1,0,0,0]
=> [3,3,1,1,1]
=> ? = 4 - 2
[1,4,3,6,2,5] => [1,4,2,5,3,6] => [1,0,1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,1,1]
=> ? = 4 - 2
[1,4,3,6,5,2] => [1,4,2,3,6,5] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [4,4,1,1,1]
=> ? = 3 - 2
[1,4,5,2,6,3] => [1,4,5,2,6,3] => [1,0,1,1,1,0,1,0,0,1,0,0]
=> [4,2,1,1,1]
=> ? = 4 - 2
[1,4,5,3,2,6] => [1,4,5,2,6,3] => [1,0,1,1,1,0,1,0,0,1,0,0]
=> [4,2,1,1,1]
=> ? = 4 - 2
[1,4,6,2,5,3] => [1,4,6,2,5,3] => [1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,2,1,1,1]
=> 1 = 3 - 2
[1,4,6,3,2,5] => [1,4,6,2,5,3] => [1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,2,1,1,1]
=> 1 = 3 - 2
[1,5,2,3,6,4] => [1,5,2,3,6,4] => [1,0,1,1,1,1,0,0,0,1,0,0]
=> [4,1,1,1,1]
=> ? = 3 - 2
[1,5,2,4,3,6] => [1,5,2,4,3,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [5,1,1,1,1]
=> ? = 3 - 2
[1,5,2,4,6,3] => [1,5,2,4,6,3] => [1,0,1,1,1,1,0,0,0,1,0,0]
=> [4,1,1,1,1]
=> ? = 3 - 2
[1,5,2,6,3,4] => [1,5,2,6,3,4] => [1,0,1,1,1,1,0,0,1,0,0,0]
=> [3,1,1,1,1]
=> 2 = 4 - 2
[1,5,2,6,4,3] => [1,5,2,6,3,4] => [1,0,1,1,1,1,0,0,1,0,0,0]
=> [3,1,1,1,1]
=> 2 = 4 - 2
[1,5,3,2,4,6] => [1,5,2,4,6,3] => [1,0,1,1,1,1,0,0,0,1,0,0]
=> [4,1,1,1,1]
=> ? = 3 - 2
[1,5,3,2,6,4] => [1,5,2,6,3,4] => [1,0,1,1,1,1,0,0,1,0,0,0]
=> [3,1,1,1,1]
=> 2 = 4 - 2
[1,5,3,4,2,6] => [1,5,2,6,3,4] => [1,0,1,1,1,1,0,0,1,0,0,0]
=> [3,1,1,1,1]
=> 2 = 4 - 2
[1,5,3,6,2,4] => [1,5,2,4,3,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [5,1,1,1,1]
=> ? = 3 - 2
[1,5,3,6,4,2] => [1,5,2,3,6,4] => [1,0,1,1,1,1,0,0,0,1,0,0]
=> [4,1,1,1,1]
=> ? = 3 - 2
[1,5,4,2,3,6] => [1,5,2,3,6,4] => [1,0,1,1,1,1,0,0,0,1,0,0]
=> [4,1,1,1,1]
=> ? = 3 - 2
[1,5,4,2,6,3] => [1,5,2,6,3,4] => [1,0,1,1,1,1,0,0,1,0,0,0]
=> [3,1,1,1,1]
=> 2 = 4 - 2
[1,5,4,3,2,6] => [1,5,2,6,3,4] => [1,0,1,1,1,1,0,0,1,0,0,0]
=> [3,1,1,1,1]
=> 2 = 4 - 2
[1,5,4,3,6,2] => [1,5,2,3,6,4] => [1,0,1,1,1,1,0,0,0,1,0,0]
=> [4,1,1,1,1]
=> ? = 3 - 2
[1,5,6,2,4,3] => [1,5,6,2,4,3] => [1,0,1,1,1,1,0,1,0,0,0,0]
=> [2,1,1,1,1]
=> 1 = 3 - 2
[1,5,6,3,2,4] => [1,5,6,2,4,3] => [1,0,1,1,1,1,0,1,0,0,0,0]
=> [2,1,1,1,1]
=> 1 = 3 - 2
[1,6,2,3,5,4] => [1,6,2,3,5,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1]
=> 1 = 3 - 2
[1,6,2,4,3,5] => [1,6,2,4,3,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1]
=> 1 = 3 - 2
[1,6,2,4,5,3] => [1,6,2,4,5,3] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1]
=> 1 = 3 - 2
[1,6,2,5,3,4] => [1,6,2,5,3,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1]
=> 1 = 3 - 2
[1,6,2,5,4,3] => [1,6,2,5,3,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1]
=> 1 = 3 - 2
[1,6,3,2,4,5] => [1,6,2,4,5,3] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1]
=> 1 = 3 - 2
[1,6,3,2,5,4] => [1,6,2,5,3,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1]
=> 1 = 3 - 2
[1,6,3,4,2,5] => [1,6,2,5,3,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1]
=> 1 = 3 - 2
[1,6,3,5,2,4] => [1,6,2,4,3,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1]
=> 1 = 3 - 2
[1,6,3,5,4,2] => [1,6,2,3,5,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1]
=> 1 = 3 - 2
[1,6,4,2,3,5] => [1,6,2,3,5,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1]
=> 1 = 3 - 2
[1,6,4,2,5,3] => [1,6,2,5,3,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1]
=> 1 = 3 - 2
[1,6,4,3,2,5] => [1,6,2,5,3,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1]
=> 1 = 3 - 2
[1,6,4,3,5,2] => [1,6,2,3,5,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1]
=> 1 = 3 - 2
[1,6,5,2,4,3] => [1,6,2,4,3,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1]
=> 1 = 3 - 2
[1,6,5,3,2,4] => [1,6,2,4,3,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1]
=> 1 = 3 - 2
[2,1,4,3,6,5] => [1,4,2,3,6,5] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [4,4,1,1,1]
=> ? = 3 - 2
[2,1,5,3,6,4] => [1,5,2,3,6,4] => [1,0,1,1,1,1,0,0,0,1,0,0]
=> [4,1,1,1,1]
=> ? = 3 - 2
[2,1,5,4,3,6] => [1,5,2,3,6,4] => [1,0,1,1,1,1,0,0,0,1,0,0]
=> [4,1,1,1,1]
=> ? = 3 - 2
[2,1,6,3,5,4] => [1,6,2,3,5,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1]
=> 1 = 3 - 2
[2,1,6,4,3,5] => [1,6,2,3,5,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1]
=> 1 = 3 - 2
[2,3,5,1,6,4] => [1,6,2,3,5,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1]
=> 1 = 3 - 2
[2,3,5,4,1,6] => [1,6,2,3,5,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1]
=> 1 = 3 - 2
[2,3,6,1,5,4] => [1,5,2,3,6,4] => [1,0,1,1,1,1,0,0,0,1,0,0]
=> [4,1,1,1,1]
=> ? = 3 - 2
[2,3,6,4,1,5] => [1,5,2,3,6,4] => [1,0,1,1,1,1,0,0,0,1,0,0]
=> [4,1,1,1,1]
=> ? = 3 - 2
[2,3,6,5,1,4] => [1,4,2,3,6,5] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [4,4,1,1,1]
=> ? = 3 - 2
[2,4,1,5,3,6] => [1,5,2,4,3,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [5,1,1,1,1]
=> ? = 3 - 2
[2,4,1,5,6,3] => [1,5,6,2,4,3] => [1,0,1,1,1,1,0,1,0,0,0,0]
=> [2,1,1,1,1]
=> 1 = 3 - 2
[2,4,1,6,3,5] => [1,6,2,4,3,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1]
=> 1 = 3 - 2
[2,4,1,6,5,3] => [1,6,2,4,3,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1]
=> 1 = 3 - 2
[2,4,3,1,5,6] => [1,5,6,2,4,3] => [1,0,1,1,1,1,0,1,0,0,0,0]
=> [2,1,1,1,1]
=> 1 = 3 - 2
[2,4,3,1,6,5] => [1,6,2,4,3,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1]
=> 1 = 3 - 2
[2,4,3,5,1,6] => [1,6,2,4,3,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1]
=> 1 = 3 - 2
[2,4,3,6,1,5] => [1,5,2,4,3,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [5,1,1,1,1]
=> ? = 3 - 2
[2,4,3,6,5,1] => [1,2,4,3,6,5] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2,1]
=> ? = 3 - 2
[2,4,5,1,6,3] => [1,6,2,4,5,3] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1]
=> 1 = 3 - 2
[2,4,5,3,1,6] => [1,6,2,4,5,3] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1]
=> 1 = 3 - 2
[2,4,6,1,5,3] => [1,5,2,4,6,3] => [1,0,1,1,1,1,0,0,0,1,0,0]
=> [4,1,1,1,1]
=> ? = 3 - 2
[2,4,6,3,1,5] => [1,5,2,4,6,3] => [1,0,1,1,1,1,0,0,0,1,0,0]
=> [4,1,1,1,1]
=> ? = 3 - 2
Description
The number of monotone factorisations of genus zero of a permutation of given cycle type.
A monotone factorisation of genus zero of a permutation π∈Sn with ℓ cycles, including fixed points, is a tuple of r=n−ℓ transpositions
(a1,b1),…,(ar,br)
with b1≤⋯≤br and ai<bi for all i, whose product, in this order, is π.
For example, the cycle (2,3,1) has the two factorizations (2,3)(1,3) and (1,2)(2,3).
Matching statistic: St001330
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Mp00223: Permutations —runsort⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00074: Posets —to graph⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 50%
Mp00065: Permutations —permutation poset⟶ Posets
Mp00074: Posets —to graph⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 50%
Values
[1,3,2,5,4] => [1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ? = 3 - 1
[1,4,2,5,3] => [1,4,2,5,3] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ? = 4 - 1
[1,4,3,2,5] => [1,4,2,5,3] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ? = 4 - 1
[1,5,2,4,3] => [1,5,2,4,3] => ([(0,3),(0,4),(4,1),(4,2)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2 = 3 - 1
[1,5,3,2,4] => [1,5,2,4,3] => ([(0,3),(0,4),(4,1),(4,2)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2 = 3 - 1
[2,4,1,5,3] => [1,5,2,4,3] => ([(0,3),(0,4),(4,1),(4,2)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2 = 3 - 1
[2,4,3,1,5] => [1,5,2,4,3] => ([(0,3),(0,4),(4,1),(4,2)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2 = 3 - 1
[2,5,1,4,3] => [1,4,2,5,3] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ? = 4 - 1
[2,5,3,1,4] => [1,4,2,5,3] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ? = 4 - 1
[2,5,4,1,3] => [1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ? = 3 - 1
[3,1,4,2,5] => [1,4,2,5,3] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ? = 4 - 1
[3,1,5,2,4] => [1,5,2,4,3] => ([(0,3),(0,4),(4,1),(4,2)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2 = 3 - 1
[3,2,4,1,5] => [1,5,2,4,3] => ([(0,3),(0,4),(4,1),(4,2)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2 = 3 - 1
[3,2,5,1,4] => [1,4,2,5,3] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ? = 4 - 1
[4,1,3,2,5] => [1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ? = 3 - 1
[4,2,5,1,3] => [1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ? = 3 - 1
[1,2,4,3,6,5] => [1,2,4,3,6,5] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,1),(3,2)],6)
=> ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ? = 3 - 1
[1,2,5,3,6,4] => [1,2,5,3,6,4] => ([(0,4),(2,5),(3,1),(3,5),(4,2),(4,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ? = 4 - 1
[1,2,5,4,3,6] => [1,2,5,3,6,4] => ([(0,4),(2,5),(3,1),(3,5),(4,2),(4,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ? = 4 - 1
[1,2,6,3,5,4] => [1,2,6,3,5,4] => ([(0,5),(4,2),(4,3),(5,1),(5,4)],6)
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2 = 3 - 1
[1,2,6,4,3,5] => [1,2,6,3,5,4] => ([(0,5),(4,2),(4,3),(5,1),(5,4)],6)
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2 = 3 - 1
[1,3,2,4,6,5] => [1,3,2,4,6,5] => ([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
=> ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ? = 3 - 1
[1,3,2,5,4,6] => [1,3,2,5,4,6] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 3 - 1
[1,3,2,5,6,4] => [1,3,2,5,6,4] => ([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(5,1)],6)
=> ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ? = 3 - 1
[1,3,2,6,4,5] => [1,3,2,6,4,5] => ([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(5,1)],6)
=> ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ? = 3 - 1
[1,3,2,6,5,4] => [1,3,2,6,4,5] => ([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(5,1)],6)
=> ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ? = 3 - 1
[1,3,4,2,6,5] => [1,3,4,2,6,5] => ([(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,1)],6)
=> ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 3 - 1
[1,3,5,2,6,4] => [1,3,5,2,6,4] => ([(0,2),(0,3),(1,4),(2,4),(2,5),(3,1),(3,5)],6)
=> ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 3 - 1
[1,3,5,4,2,6] => [1,3,5,2,6,4] => ([(0,2),(0,3),(1,4),(2,4),(2,5),(3,1),(3,5)],6)
=> ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 3 - 1
[1,3,6,2,5,4] => [1,3,6,2,5,4] => ([(0,2),(0,3),(2,4),(2,5),(3,1),(3,4),(3,5)],6)
=> ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 3 - 1
[1,3,6,4,2,5] => [1,3,6,2,5,4] => ([(0,2),(0,3),(2,4),(2,5),(3,1),(3,4),(3,5)],6)
=> ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 3 - 1
[1,4,2,3,6,5] => [1,4,2,3,6,5] => ([(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,1)],6)
=> ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 3 - 1
[1,4,2,5,3,6] => [1,4,2,5,3,6] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
=> ? = 4 - 1
[1,4,2,5,6,3] => [1,4,2,5,6,3] => ([(0,3),(0,4),(3,5),(4,1),(4,5),(5,2)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ? = 4 - 1
[1,4,2,6,3,5] => [1,4,2,6,3,5] => ([(0,2),(0,3),(1,4),(2,4),(2,5),(3,1),(3,5)],6)
=> ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 4 - 1
[1,4,2,6,5,3] => [1,4,2,6,3,5] => ([(0,2),(0,3),(1,4),(2,4),(2,5),(3,1),(3,5)],6)
=> ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 4 - 1
[1,4,3,2,5,6] => [1,4,2,5,6,3] => ([(0,3),(0,4),(3,5),(4,1),(4,5),(5,2)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ? = 4 - 1
[1,4,3,2,6,5] => [1,4,2,6,3,5] => ([(0,2),(0,3),(1,4),(2,4),(2,5),(3,1),(3,5)],6)
=> ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 4 - 1
[1,4,3,5,2,6] => [1,4,2,6,3,5] => ([(0,2),(0,3),(1,4),(2,4),(2,5),(3,1),(3,5)],6)
=> ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 4 - 1
[1,4,3,6,2,5] => [1,4,2,5,3,6] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
=> ? = 4 - 1
[1,4,3,6,5,2] => [1,4,2,3,6,5] => ([(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,1)],6)
=> ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 3 - 1
[1,4,5,2,6,3] => [1,4,5,2,6,3] => ([(0,3),(0,4),(2,5),(3,2),(4,1),(4,5)],6)
=> ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> ? = 4 - 1
[1,4,5,3,2,6] => [1,4,5,2,6,3] => ([(0,3),(0,4),(2,5),(3,2),(4,1),(4,5)],6)
=> ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> ? = 4 - 1
[1,4,6,2,5,3] => [1,4,6,2,5,3] => ([(0,3),(0,4),(3,2),(3,5),(4,1),(4,5)],6)
=> ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 3 - 1
[1,4,6,3,2,5] => [1,4,6,2,5,3] => ([(0,3),(0,4),(3,2),(3,5),(4,1),(4,5)],6)
=> ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 3 - 1
[1,5,2,3,6,4] => [1,5,2,3,6,4] => ([(0,2),(0,4),(2,5),(3,1),(3,5),(4,3)],6)
=> ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> ? = 3 - 1
[1,5,2,4,3,6] => [1,5,2,4,3,6] => ([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6)
=> ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 3 - 1
[1,5,2,4,6,3] => [1,5,2,4,6,3] => ([(0,3),(0,4),(2,5),(3,5),(4,1),(4,2)],6)
=> ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> ? = 3 - 1
[1,5,2,6,3,4] => [1,5,2,6,3,4] => ([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
=> ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> ? = 4 - 1
[1,5,2,6,4,3] => [1,5,2,6,3,4] => ([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
=> ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> ? = 4 - 1
[1,5,3,2,4,6] => [1,5,2,4,6,3] => ([(0,3),(0,4),(2,5),(3,5),(4,1),(4,2)],6)
=> ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> ? = 3 - 1
[1,5,3,2,6,4] => [1,5,2,6,3,4] => ([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
=> ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> ? = 4 - 1
[1,5,3,4,2,6] => [1,5,2,6,3,4] => ([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
=> ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> ? = 4 - 1
[1,5,3,6,2,4] => [1,5,2,4,3,6] => ([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6)
=> ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 3 - 1
[1,5,3,6,4,2] => [1,5,2,3,6,4] => ([(0,2),(0,4),(2,5),(3,1),(3,5),(4,3)],6)
=> ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> ? = 3 - 1
[1,5,4,2,3,6] => [1,5,2,3,6,4] => ([(0,2),(0,4),(2,5),(3,1),(3,5),(4,3)],6)
=> ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> ? = 3 - 1
[1,5,4,2,6,3] => [1,5,2,6,3,4] => ([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
=> ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> ? = 4 - 1
[1,5,4,3,2,6] => [1,5,2,6,3,4] => ([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
=> ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> ? = 4 - 1
[1,5,6,2,4,3] => [1,5,6,2,4,3] => ([(0,4),(0,5),(4,3),(5,1),(5,2)],6)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 2 = 3 - 1
[1,5,6,3,2,4] => [1,5,6,2,4,3] => ([(0,4),(0,5),(4,3),(5,1),(5,2)],6)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 2 = 3 - 1
[1,6,2,3,5,4] => [1,6,2,3,5,4] => ([(0,3),(0,4),(4,5),(5,1),(5,2)],6)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 2 = 3 - 1
[1,6,2,4,5,3] => [1,6,2,4,5,3] => ([(0,3),(0,5),(4,2),(5,1),(5,4)],6)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 2 = 3 - 1
[1,6,2,5,3,4] => [1,6,2,5,3,4] => ([(0,3),(0,5),(4,2),(5,1),(5,4)],6)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 2 = 3 - 1
[1,6,2,5,4,3] => [1,6,2,5,3,4] => ([(0,3),(0,5),(4,2),(5,1),(5,4)],6)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 2 = 3 - 1
[1,6,3,2,4,5] => [1,6,2,4,5,3] => ([(0,3),(0,5),(4,2),(5,1),(5,4)],6)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 2 = 3 - 1
[1,6,3,2,5,4] => [1,6,2,5,3,4] => ([(0,3),(0,5),(4,2),(5,1),(5,4)],6)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 2 = 3 - 1
[1,6,3,4,2,5] => [1,6,2,5,3,4] => ([(0,3),(0,5),(4,2),(5,1),(5,4)],6)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 2 = 3 - 1
[1,6,3,5,4,2] => [1,6,2,3,5,4] => ([(0,3),(0,4),(4,5),(5,1),(5,2)],6)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 2 = 3 - 1
[1,6,4,2,3,5] => [1,6,2,3,5,4] => ([(0,3),(0,4),(4,5),(5,1),(5,2)],6)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 2 = 3 - 1
[1,6,4,2,5,3] => [1,6,2,5,3,4] => ([(0,3),(0,5),(4,2),(5,1),(5,4)],6)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 2 = 3 - 1
[1,6,4,3,2,5] => [1,6,2,5,3,4] => ([(0,3),(0,5),(4,2),(5,1),(5,4)],6)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 2 = 3 - 1
[1,6,4,3,5,2] => [1,6,2,3,5,4] => ([(0,3),(0,4),(4,5),(5,1),(5,2)],6)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 2 = 3 - 1
[2,1,6,3,5,4] => [1,6,2,3,5,4] => ([(0,3),(0,4),(4,5),(5,1),(5,2)],6)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 2 = 3 - 1
[2,1,6,4,3,5] => [1,6,2,3,5,4] => ([(0,3),(0,4),(4,5),(5,1),(5,2)],6)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 2 = 3 - 1
[2,3,5,1,6,4] => [1,6,2,3,5,4] => ([(0,3),(0,4),(4,5),(5,1),(5,2)],6)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 2 = 3 - 1
[2,3,5,4,1,6] => [1,6,2,3,5,4] => ([(0,3),(0,4),(4,5),(5,1),(5,2)],6)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 2 = 3 - 1
[2,4,1,5,6,3] => [1,5,6,2,4,3] => ([(0,4),(0,5),(4,3),(5,1),(5,2)],6)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 2 = 3 - 1
[2,4,3,1,5,6] => [1,5,6,2,4,3] => ([(0,4),(0,5),(4,3),(5,1),(5,2)],6)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 2 = 3 - 1
[2,4,5,1,6,3] => [1,6,2,4,5,3] => ([(0,3),(0,5),(4,2),(5,1),(5,4)],6)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 2 = 3 - 1
[2,4,5,3,1,6] => [1,6,2,4,5,3] => ([(0,3),(0,5),(4,2),(5,1),(5,4)],6)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 2 = 3 - 1
[2,5,1,6,3,4] => [1,6,2,5,3,4] => ([(0,3),(0,5),(4,2),(5,1),(5,4)],6)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 2 = 3 - 1
[2,5,1,6,4,3] => [1,6,2,5,3,4] => ([(0,3),(0,5),(4,2),(5,1),(5,4)],6)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 2 = 3 - 1
[2,5,3,1,6,4] => [1,6,2,5,3,4] => ([(0,3),(0,5),(4,2),(5,1),(5,4)],6)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 2 = 3 - 1
[2,5,3,4,1,6] => [1,6,2,5,3,4] => ([(0,3),(0,5),(4,2),(5,1),(5,4)],6)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 2 = 3 - 1
[2,5,4,1,6,3] => [1,6,2,5,3,4] => ([(0,3),(0,5),(4,2),(5,1),(5,4)],6)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 2 = 3 - 1
[2,5,4,3,1,6] => [1,6,2,5,3,4] => ([(0,3),(0,5),(4,2),(5,1),(5,4)],6)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 2 = 3 - 1
[2,6,3,5,4,1] => [1,2,6,3,5,4] => ([(0,5),(4,2),(4,3),(5,1),(5,4)],6)
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2 = 3 - 1
[2,6,4,3,5,1] => [1,2,6,3,5,4] => ([(0,5),(4,2),(4,3),(5,1),(5,4)],6)
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2 = 3 - 1
[3,1,5,6,2,4] => [1,5,6,2,4,3] => ([(0,4),(0,5),(4,3),(5,1),(5,2)],6)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 2 = 3 - 1
[3,1,6,2,4,5] => [1,6,2,4,5,3] => ([(0,3),(0,5),(4,2),(5,1),(5,4)],6)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 2 = 3 - 1
[3,1,6,2,5,4] => [1,6,2,5,3,4] => ([(0,3),(0,5),(4,2),(5,1),(5,4)],6)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 2 = 3 - 1
[3,1,6,4,2,5] => [1,6,2,5,3,4] => ([(0,3),(0,5),(4,2),(5,1),(5,4)],6)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 2 = 3 - 1
[3,2,4,1,5,6] => [1,5,6,2,4,3] => ([(0,4),(0,5),(4,3),(5,1),(5,2)],6)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 2 = 3 - 1
[3,2,4,5,1,6] => [1,6,2,4,5,3] => ([(0,3),(0,5),(4,2),(5,1),(5,4)],6)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 2 = 3 - 1
[3,2,5,1,6,4] => [1,6,2,5,3,4] => ([(0,3),(0,5),(4,2),(5,1),(5,4)],6)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 2 = 3 - 1
[3,2,5,4,1,6] => [1,6,2,5,3,4] => ([(0,3),(0,5),(4,2),(5,1),(5,4)],6)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 2 = 3 - 1
[3,4,1,6,2,5] => [1,6,2,5,3,4] => ([(0,3),(0,5),(4,2),(5,1),(5,4)],6)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 2 = 3 - 1
[3,4,2,5,1,6] => [1,6,2,5,3,4] => ([(0,3),(0,5),(4,2),(5,1),(5,4)],6)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 2 = 3 - 1
[3,5,1,2,6,4] => [1,2,6,3,5,4] => ([(0,5),(4,2),(4,3),(5,1),(5,4)],6)
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2 = 3 - 1
[3,5,1,6,4,2] => [1,6,2,3,5,4] => ([(0,3),(0,4),(4,5),(5,1),(5,2)],6)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 2 = 3 - 1
Description
The hat guessing number of a graph.
Suppose that each vertex of a graph corresponds to a player, wearing a hat whose color is arbitrarily chosen from a set of q possible colors. Each player can see the hat colors of his neighbors, but not his own hat color. All of the players are asked to guess their own hat colors simultaneously, according to a predetermined guessing strategy and the hat colors they see, where no communication between them is allowed. The hat guessing number HG(G) of a graph G is the largest integer q such that there exists a guessing strategy guaranteeing at least one correct guess for any hat assignment of q possible colors.
Because it suffices that a single player guesses correctly, the hat guessing number of a graph is the maximum of the hat guessing numbers of its connected components.
Matching statistic: St000801
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00223: Permutations —runsort⟶ Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
St000801: Permutations ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 100%
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
St000801: Permutations ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 100%
Values
[1,3,2,5,4] => [1,3,2,5,4] => [3,4,2,5,1] => 0 = 3 - 3
[1,4,2,5,3] => [1,4,2,5,3] => [3,4,5,1,2] => 1 = 4 - 3
[1,4,3,2,5] => [1,4,2,5,3] => [3,4,5,1,2] => 1 = 4 - 3
[1,5,2,4,3] => [1,5,2,4,3] => [3,2,5,4,1] => 0 = 3 - 3
[1,5,3,2,4] => [1,5,2,4,3] => [3,2,5,4,1] => 0 = 3 - 3
[2,4,1,5,3] => [1,5,2,4,3] => [3,2,5,4,1] => 0 = 3 - 3
[2,4,3,1,5] => [1,5,2,4,3] => [3,2,5,4,1] => 0 = 3 - 3
[2,5,1,4,3] => [1,4,2,5,3] => [3,4,5,1,2] => 1 = 4 - 3
[2,5,3,1,4] => [1,4,2,5,3] => [3,4,5,1,2] => 1 = 4 - 3
[2,5,4,1,3] => [1,3,2,5,4] => [3,4,2,5,1] => 0 = 3 - 3
[3,1,4,2,5] => [1,4,2,5,3] => [3,4,5,1,2] => 1 = 4 - 3
[3,1,5,2,4] => [1,5,2,4,3] => [3,2,5,4,1] => 0 = 3 - 3
[3,2,4,1,5] => [1,5,2,4,3] => [3,2,5,4,1] => 0 = 3 - 3
[3,2,5,1,4] => [1,4,2,5,3] => [3,4,5,1,2] => 1 = 4 - 3
[4,1,3,2,5] => [1,3,2,5,4] => [3,4,2,5,1] => 0 = 3 - 3
[4,2,5,1,3] => [1,3,2,5,4] => [3,4,2,5,1] => 0 = 3 - 3
[1,2,4,3,6,5] => [1,2,4,3,6,5] => [3,4,5,2,6,1] => 0 = 3 - 3
[1,2,5,3,6,4] => [1,2,5,3,6,4] => [3,4,5,6,1,2] => 1 = 4 - 3
[1,2,5,4,3,6] => [1,2,5,3,6,4] => [3,4,5,6,1,2] => 1 = 4 - 3
[1,2,6,3,5,4] => [1,2,6,3,5,4] => [3,4,2,6,5,1] => 0 = 3 - 3
[1,2,6,4,3,5] => [1,2,6,3,5,4] => [3,4,2,6,5,1] => 0 = 3 - 3
[1,3,2,4,6,5] => [1,3,2,4,6,5] => [3,4,2,5,6,1] => 0 = 3 - 3
[1,3,2,5,4,6] => [1,3,2,5,4,6] => [3,4,2,5,1,6] => 0 = 3 - 3
[1,3,2,5,6,4] => [1,3,2,5,6,4] => [3,4,2,6,1,5] => 0 = 3 - 3
[1,3,2,6,4,5] => [1,3,2,6,4,5] => [3,4,2,1,6,5] => 0 = 3 - 3
[1,3,2,6,5,4] => [1,3,2,6,4,5] => [3,4,2,1,6,5] => 0 = 3 - 3
[1,3,4,2,6,5] => [1,3,4,2,6,5] => [3,5,2,4,6,1] => 0 = 3 - 3
[1,3,5,2,6,4] => [1,3,5,2,6,4] => [3,5,2,6,1,4] => 0 = 3 - 3
[1,3,5,4,2,6] => [1,3,5,2,6,4] => [3,5,2,6,1,4] => 0 = 3 - 3
[1,3,6,2,5,4] => [1,3,6,2,5,4] => [3,2,5,6,4,1] => 0 = 3 - 3
[1,3,6,4,2,5] => [1,3,6,2,5,4] => [3,2,5,6,4,1] => 0 = 3 - 3
[1,4,2,3,6,5] => [1,4,2,3,6,5] => [3,2,5,4,6,1] => 0 = 3 - 3
[1,4,2,5,3,6] => [1,4,2,5,3,6] => [3,4,5,1,2,6] => 1 = 4 - 3
[1,4,2,5,6,3] => [1,4,2,5,6,3] => [3,4,6,1,2,5] => 1 = 4 - 3
[1,4,2,6,3,5] => [1,4,2,6,3,5] => [3,4,1,5,6,2] => 1 = 4 - 3
[1,4,2,6,5,3] => [1,4,2,6,3,5] => [3,4,1,5,6,2] => 1 = 4 - 3
[1,4,3,2,5,6] => [1,4,2,5,6,3] => [3,4,6,1,2,5] => 1 = 4 - 3
[1,4,3,2,6,5] => [1,4,2,6,3,5] => [3,4,1,5,6,2] => 1 = 4 - 3
[1,4,3,5,2,6] => [1,4,2,6,3,5] => [3,4,1,5,6,2] => 1 = 4 - 3
[1,4,3,6,2,5] => [1,4,2,5,3,6] => [3,4,5,1,2,6] => 1 = 4 - 3
[1,4,3,6,5,2] => [1,4,2,3,6,5] => [3,2,5,4,6,1] => 0 = 3 - 3
[1,4,5,2,6,3] => [1,4,5,2,6,3] => [3,5,6,1,2,4] => 1 = 4 - 3
[1,4,5,3,2,6] => [1,4,5,2,6,3] => [3,5,6,1,2,4] => 1 = 4 - 3
[1,4,6,2,5,3] => [1,4,6,2,5,3] => [3,2,6,4,5,1] => 0 = 3 - 3
[1,4,6,3,2,5] => [1,4,6,2,5,3] => [3,2,6,4,5,1] => 0 = 3 - 3
[1,5,2,3,6,4] => [1,5,2,3,6,4] => [3,2,5,6,1,4] => 0 = 3 - 3
[1,5,2,4,3,6] => [1,5,2,4,3,6] => [3,2,5,4,1,6] => 0 = 3 - 3
[1,5,2,4,6,3] => [1,5,2,4,6,3] => [3,2,6,4,1,5] => 0 = 3 - 3
[1,5,2,6,3,4] => [1,5,2,6,3,4] => [3,4,1,6,2,5] => 1 = 4 - 3
[1,5,2,6,4,3] => [1,5,2,6,3,4] => [3,4,1,6,2,5] => 1 = 4 - 3
[1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => [3,4,5,6,2,7,1] => ? = 3 - 3
[1,2,3,6,4,7,5] => [1,2,3,6,4,7,5] => [3,4,5,6,7,1,2] => ? = 4 - 3
[1,2,3,6,5,4,7] => [1,2,3,6,4,7,5] => [3,4,5,6,7,1,2] => ? = 4 - 3
[1,2,3,7,4,6,5] => [1,2,3,7,4,6,5] => [3,4,5,2,7,6,1] => ? = 3 - 3
[1,2,3,7,5,4,6] => [1,2,3,7,4,6,5] => [3,4,5,2,7,6,1] => ? = 3 - 3
[1,2,4,3,5,7,6] => [1,2,4,3,5,7,6] => [3,4,5,2,6,7,1] => ? = 3 - 3
[1,2,4,3,6,5,7] => [1,2,4,3,6,5,7] => [3,4,5,2,6,1,7] => ? = 3 - 3
[1,2,4,3,6,7,5] => [1,2,4,3,6,7,5] => [3,4,5,2,7,1,6] => ? = 3 - 3
[1,2,4,3,7,5,6] => [1,2,4,3,7,5,6] => [3,4,5,2,1,7,6] => ? = 3 - 3
[1,2,4,3,7,6,5] => [1,2,4,3,7,5,6] => [3,4,5,2,1,7,6] => ? = 3 - 3
[1,2,4,5,3,7,6] => [1,2,4,5,3,7,6] => [3,4,6,2,5,7,1] => ? = 3 - 3
[1,2,4,6,3,7,5] => [1,2,4,6,3,7,5] => [3,4,6,2,7,1,5] => ? = 3 - 3
[1,2,4,6,5,3,7] => [1,2,4,6,3,7,5] => [3,4,6,2,7,1,5] => ? = 3 - 3
[1,2,4,7,3,6,5] => [1,2,4,7,3,6,5] => [3,4,2,6,7,5,1] => ? = 3 - 3
[1,2,4,7,5,3,6] => [1,2,4,7,3,6,5] => [3,4,2,6,7,5,1] => ? = 3 - 3
[1,2,5,3,4,7,6] => [1,2,5,3,4,7,6] => [3,4,2,6,5,7,1] => ? = 3 - 3
[1,2,5,3,6,4,7] => [1,2,5,3,6,4,7] => [3,4,5,6,1,2,7] => ? = 4 - 3
[1,2,5,3,6,7,4] => [1,2,5,3,6,7,4] => [3,4,5,7,1,2,6] => ? = 4 - 3
[1,2,5,3,7,4,6] => [1,2,5,3,7,4,6] => [3,4,5,1,6,7,2] => ? = 4 - 3
[1,2,5,3,7,6,4] => [1,2,5,3,7,4,6] => [3,4,5,1,6,7,2] => ? = 4 - 3
[1,2,5,4,3,6,7] => [1,2,5,3,6,7,4] => [3,4,5,7,1,2,6] => ? = 4 - 3
[1,2,5,4,3,7,6] => [1,2,5,3,7,4,6] => [3,4,5,1,6,7,2] => ? = 4 - 3
[1,2,5,4,6,3,7] => [1,2,5,3,7,4,6] => [3,4,5,1,6,7,2] => ? = 4 - 3
[1,2,5,4,7,3,6] => [1,2,5,3,6,4,7] => [3,4,5,6,1,2,7] => ? = 4 - 3
[1,2,5,4,7,6,3] => [1,2,5,3,4,7,6] => [3,4,2,6,5,7,1] => ? = 3 - 3
[1,2,5,6,3,7,4] => [1,2,5,6,3,7,4] => [3,4,6,7,1,2,5] => ? = 4 - 3
[1,2,5,6,4,3,7] => [1,2,5,6,3,7,4] => [3,4,6,7,1,2,5] => ? = 4 - 3
[1,2,5,7,3,6,4] => [1,2,5,7,3,6,4] => [3,4,2,7,5,6,1] => ? = 3 - 3
[1,2,5,7,4,3,6] => [1,2,5,7,3,6,4] => [3,4,2,7,5,6,1] => ? = 3 - 3
[1,2,6,3,4,7,5] => [1,2,6,3,4,7,5] => [3,4,2,6,7,1,5] => ? = 3 - 3
[1,2,6,3,5,4,7] => [1,2,6,3,5,4,7] => [3,4,2,6,5,1,7] => ? = 3 - 3
[1,2,6,3,5,7,4] => [1,2,6,3,5,7,4] => [3,4,2,7,5,1,6] => ? = 3 - 3
[1,2,6,3,7,4,5] => [1,2,6,3,7,4,5] => [3,4,5,1,7,2,6] => ? = 4 - 3
[1,2,6,3,7,5,4] => [1,2,6,3,7,4,5] => [3,4,5,1,7,2,6] => ? = 4 - 3
[1,2,6,4,3,5,7] => [1,2,6,3,5,7,4] => [3,4,2,7,5,1,6] => ? = 3 - 3
[1,2,6,4,3,7,5] => [1,2,6,3,7,4,5] => [3,4,5,1,7,2,6] => ? = 4 - 3
[1,2,6,4,5,3,7] => [1,2,6,3,7,4,5] => [3,4,5,1,7,2,6] => ? = 4 - 3
[1,2,6,4,7,3,5] => [1,2,6,3,5,4,7] => [3,4,2,6,5,1,7] => ? = 3 - 3
[1,2,6,4,7,5,3] => [1,2,6,3,4,7,5] => [3,4,2,6,7,1,5] => ? = 3 - 3
[1,2,6,5,3,4,7] => [1,2,6,3,4,7,5] => [3,4,2,6,7,1,5] => ? = 3 - 3
[1,2,6,5,3,7,4] => [1,2,6,3,7,4,5] => [3,4,5,1,7,2,6] => ? = 4 - 3
[1,2,6,5,4,3,7] => [1,2,6,3,7,4,5] => [3,4,5,1,7,2,6] => ? = 4 - 3
[1,2,6,5,4,7,3] => [1,2,6,3,4,7,5] => [3,4,2,6,7,1,5] => ? = 3 - 3
[1,2,6,7,3,5,4] => [1,2,6,7,3,5,4] => [3,4,2,7,6,1,5] => ? = 3 - 3
[1,2,6,7,4,3,5] => [1,2,6,7,3,5,4] => [3,4,2,7,6,1,5] => ? = 3 - 3
[1,2,7,3,4,6,5] => [1,2,7,3,4,6,5] => [3,4,2,5,7,6,1] => ? = 3 - 3
[1,2,7,3,5,4,6] => [1,2,7,3,5,4,6] => [3,4,2,6,1,7,5] => ? = 3 - 3
[1,2,7,3,5,6,4] => [1,2,7,3,5,6,4] => [3,4,2,7,1,6,5] => ? = 3 - 3
[1,2,7,3,6,4,5] => [1,2,7,3,6,4,5] => [3,4,2,1,7,6,5] => ? = 3 - 3
[1,2,7,3,6,5,4] => [1,2,7,3,6,4,5] => [3,4,2,1,7,6,5] => ? = 3 - 3
Description
The number of occurrences of the vincular pattern |312 in a permutation.
This is the number of occurrences of the pattern (3,1,2), such that the letter matched by 3 is the first entry of the permutation.
Matching statistic: St000654
Mp00223: Permutations —runsort⟶ Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000654: Permutations ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 100%
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000654: Permutations ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 100%
Values
[1,3,2,5,4] => [1,3,2,5,4] => [3,4,2,5,1] => [5,3,1,2,4] => 1 = 3 - 2
[1,4,2,5,3] => [1,4,2,5,3] => [3,4,5,1,2] => [4,5,1,2,3] => 2 = 4 - 2
[1,4,3,2,5] => [1,4,2,5,3] => [3,4,5,1,2] => [4,5,1,2,3] => 2 = 4 - 2
[1,5,2,4,3] => [1,5,2,4,3] => [3,2,5,4,1] => [5,2,1,4,3] => 1 = 3 - 2
[1,5,3,2,4] => [1,5,2,4,3] => [3,2,5,4,1] => [5,2,1,4,3] => 1 = 3 - 2
[2,4,1,5,3] => [1,5,2,4,3] => [3,2,5,4,1] => [5,2,1,4,3] => 1 = 3 - 2
[2,4,3,1,5] => [1,5,2,4,3] => [3,2,5,4,1] => [5,2,1,4,3] => 1 = 3 - 2
[2,5,1,4,3] => [1,4,2,5,3] => [3,4,5,1,2] => [4,5,1,2,3] => 2 = 4 - 2
[2,5,3,1,4] => [1,4,2,5,3] => [3,4,5,1,2] => [4,5,1,2,3] => 2 = 4 - 2
[2,5,4,1,3] => [1,3,2,5,4] => [3,4,2,5,1] => [5,3,1,2,4] => 1 = 3 - 2
[3,1,4,2,5] => [1,4,2,5,3] => [3,4,5,1,2] => [4,5,1,2,3] => 2 = 4 - 2
[3,1,5,2,4] => [1,5,2,4,3] => [3,2,5,4,1] => [5,2,1,4,3] => 1 = 3 - 2
[3,2,4,1,5] => [1,5,2,4,3] => [3,2,5,4,1] => [5,2,1,4,3] => 1 = 3 - 2
[3,2,5,1,4] => [1,4,2,5,3] => [3,4,5,1,2] => [4,5,1,2,3] => 2 = 4 - 2
[4,1,3,2,5] => [1,3,2,5,4] => [3,4,2,5,1] => [5,3,1,2,4] => 1 = 3 - 2
[4,2,5,1,3] => [1,3,2,5,4] => [3,4,2,5,1] => [5,3,1,2,4] => 1 = 3 - 2
[1,2,4,3,6,5] => [1,2,4,3,6,5] => [3,4,5,2,6,1] => [6,4,1,2,3,5] => 1 = 3 - 2
[1,2,5,3,6,4] => [1,2,5,3,6,4] => [3,4,5,6,1,2] => [5,6,1,2,3,4] => 2 = 4 - 2
[1,2,5,4,3,6] => [1,2,5,3,6,4] => [3,4,5,6,1,2] => [5,6,1,2,3,4] => 2 = 4 - 2
[1,2,6,3,5,4] => [1,2,6,3,5,4] => [3,4,2,6,5,1] => [6,3,1,2,5,4] => 1 = 3 - 2
[1,2,6,4,3,5] => [1,2,6,3,5,4] => [3,4,2,6,5,1] => [6,3,1,2,5,4] => 1 = 3 - 2
[1,3,2,4,6,5] => [1,3,2,4,6,5] => [3,4,2,5,6,1] => [6,3,1,2,4,5] => 1 = 3 - 2
[1,3,2,5,4,6] => [1,3,2,5,4,6] => [3,4,2,5,1,6] => [5,3,1,2,4,6] => 1 = 3 - 2
[1,3,2,5,6,4] => [1,3,2,5,6,4] => [3,4,2,6,1,5] => [5,3,1,2,6,4] => 1 = 3 - 2
[1,3,2,6,4,5] => [1,3,2,6,4,5] => [3,4,2,1,6,5] => [4,3,1,2,6,5] => 1 = 3 - 2
[1,3,2,6,5,4] => [1,3,2,6,4,5] => [3,4,2,1,6,5] => [4,3,1,2,6,5] => 1 = 3 - 2
[1,3,4,2,6,5] => [1,3,4,2,6,5] => [3,5,2,4,6,1] => [6,3,1,4,2,5] => 1 = 3 - 2
[1,3,5,2,6,4] => [1,3,5,2,6,4] => [3,5,2,6,1,4] => [5,3,1,6,2,4] => 1 = 3 - 2
[1,3,5,4,2,6] => [1,3,5,2,6,4] => [3,5,2,6,1,4] => [5,3,1,6,2,4] => 1 = 3 - 2
[1,3,6,2,5,4] => [1,3,6,2,5,4] => [3,2,5,6,4,1] => [6,2,1,5,3,4] => 1 = 3 - 2
[1,3,6,4,2,5] => [1,3,6,2,5,4] => [3,2,5,6,4,1] => [6,2,1,5,3,4] => 1 = 3 - 2
[1,4,2,3,6,5] => [1,4,2,3,6,5] => [3,2,5,4,6,1] => [6,2,1,4,3,5] => 1 = 3 - 2
[1,4,2,5,3,6] => [1,4,2,5,3,6] => [3,4,5,1,2,6] => [4,5,1,2,3,6] => 2 = 4 - 2
[1,4,2,5,6,3] => [1,4,2,5,6,3] => [3,4,6,1,2,5] => [4,5,1,2,6,3] => 2 = 4 - 2
[1,4,2,6,3,5] => [1,4,2,6,3,5] => [3,4,1,5,6,2] => [3,6,1,2,4,5] => 2 = 4 - 2
[1,4,2,6,5,3] => [1,4,2,6,3,5] => [3,4,1,5,6,2] => [3,6,1,2,4,5] => 2 = 4 - 2
[1,4,3,2,5,6] => [1,4,2,5,6,3] => [3,4,6,1,2,5] => [4,5,1,2,6,3] => 2 = 4 - 2
[1,4,3,2,6,5] => [1,4,2,6,3,5] => [3,4,1,5,6,2] => [3,6,1,2,4,5] => 2 = 4 - 2
[1,4,3,5,2,6] => [1,4,2,6,3,5] => [3,4,1,5,6,2] => [3,6,1,2,4,5] => 2 = 4 - 2
[1,4,3,6,2,5] => [1,4,2,5,3,6] => [3,4,5,1,2,6] => [4,5,1,2,3,6] => 2 = 4 - 2
[1,4,3,6,5,2] => [1,4,2,3,6,5] => [3,2,5,4,6,1] => [6,2,1,4,3,5] => 1 = 3 - 2
[1,4,5,2,6,3] => [1,4,5,2,6,3] => [3,5,6,1,2,4] => [4,5,1,6,2,3] => 2 = 4 - 2
[1,4,5,3,2,6] => [1,4,5,2,6,3] => [3,5,6,1,2,4] => [4,5,1,6,2,3] => 2 = 4 - 2
[1,4,6,2,5,3] => [1,4,6,2,5,3] => [3,2,6,4,5,1] => [6,2,1,4,5,3] => 1 = 3 - 2
[1,4,6,3,2,5] => [1,4,6,2,5,3] => [3,2,6,4,5,1] => [6,2,1,4,5,3] => 1 = 3 - 2
[1,5,2,3,6,4] => [1,5,2,3,6,4] => [3,2,5,6,1,4] => [5,2,1,6,3,4] => 1 = 3 - 2
[1,5,2,4,3,6] => [1,5,2,4,3,6] => [3,2,5,4,1,6] => [5,2,1,4,3,6] => 1 = 3 - 2
[1,5,2,4,6,3] => [1,5,2,4,6,3] => [3,2,6,4,1,5] => [5,2,1,4,6,3] => 1 = 3 - 2
[1,5,2,6,3,4] => [1,5,2,6,3,4] => [3,4,1,6,2,5] => [3,5,1,2,6,4] => 2 = 4 - 2
[1,5,2,6,4,3] => [1,5,2,6,3,4] => [3,4,1,6,2,5] => [3,5,1,2,6,4] => 2 = 4 - 2
[1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => [3,4,5,6,2,7,1] => [7,5,1,2,3,4,6] => ? = 3 - 2
[1,2,3,6,4,7,5] => [1,2,3,6,4,7,5] => [3,4,5,6,7,1,2] => [6,7,1,2,3,4,5] => ? = 4 - 2
[1,2,3,6,5,4,7] => [1,2,3,6,4,7,5] => [3,4,5,6,7,1,2] => [6,7,1,2,3,4,5] => ? = 4 - 2
[1,2,3,7,4,6,5] => [1,2,3,7,4,6,5] => [3,4,5,2,7,6,1] => [7,4,1,2,3,6,5] => ? = 3 - 2
[1,2,3,7,5,4,6] => [1,2,3,7,4,6,5] => [3,4,5,2,7,6,1] => [7,4,1,2,3,6,5] => ? = 3 - 2
[1,2,4,3,5,7,6] => [1,2,4,3,5,7,6] => [3,4,5,2,6,7,1] => [7,4,1,2,3,5,6] => ? = 3 - 2
[1,2,4,3,6,5,7] => [1,2,4,3,6,5,7] => [3,4,5,2,6,1,7] => [6,4,1,2,3,5,7] => ? = 3 - 2
[1,2,4,3,6,7,5] => [1,2,4,3,6,7,5] => [3,4,5,2,7,1,6] => [6,4,1,2,3,7,5] => ? = 3 - 2
[1,2,4,3,7,5,6] => [1,2,4,3,7,5,6] => [3,4,5,2,1,7,6] => [5,4,1,2,3,7,6] => ? = 3 - 2
[1,2,4,3,7,6,5] => [1,2,4,3,7,5,6] => [3,4,5,2,1,7,6] => [5,4,1,2,3,7,6] => ? = 3 - 2
[1,2,4,5,3,7,6] => [1,2,4,5,3,7,6] => [3,4,6,2,5,7,1] => [7,4,1,2,5,3,6] => ? = 3 - 2
[1,2,4,6,3,7,5] => [1,2,4,6,3,7,5] => [3,4,6,2,7,1,5] => [6,4,1,2,7,3,5] => ? = 3 - 2
[1,2,4,6,5,3,7] => [1,2,4,6,3,7,5] => [3,4,6,2,7,1,5] => [6,4,1,2,7,3,5] => ? = 3 - 2
[1,2,4,7,3,6,5] => [1,2,4,7,3,6,5] => [3,4,2,6,7,5,1] => [7,3,1,2,6,4,5] => ? = 3 - 2
[1,2,4,7,5,3,6] => [1,2,4,7,3,6,5] => [3,4,2,6,7,5,1] => [7,3,1,2,6,4,5] => ? = 3 - 2
[1,2,5,3,4,7,6] => [1,2,5,3,4,7,6] => [3,4,2,6,5,7,1] => [7,3,1,2,5,4,6] => ? = 3 - 2
[1,2,5,3,6,4,7] => [1,2,5,3,6,4,7] => [3,4,5,6,1,2,7] => [5,6,1,2,3,4,7] => ? = 4 - 2
[1,2,5,3,6,7,4] => [1,2,5,3,6,7,4] => [3,4,5,7,1,2,6] => [5,6,1,2,3,7,4] => ? = 4 - 2
[1,2,5,3,7,4,6] => [1,2,5,3,7,4,6] => [3,4,5,1,6,7,2] => [4,7,1,2,3,5,6] => ? = 4 - 2
[1,2,5,3,7,6,4] => [1,2,5,3,7,4,6] => [3,4,5,1,6,7,2] => [4,7,1,2,3,5,6] => ? = 4 - 2
[1,2,5,4,3,6,7] => [1,2,5,3,6,7,4] => [3,4,5,7,1,2,6] => [5,6,1,2,3,7,4] => ? = 4 - 2
[1,2,5,4,3,7,6] => [1,2,5,3,7,4,6] => [3,4,5,1,6,7,2] => [4,7,1,2,3,5,6] => ? = 4 - 2
[1,2,5,4,6,3,7] => [1,2,5,3,7,4,6] => [3,4,5,1,6,7,2] => [4,7,1,2,3,5,6] => ? = 4 - 2
[1,2,5,4,7,3,6] => [1,2,5,3,6,4,7] => [3,4,5,6,1,2,7] => [5,6,1,2,3,4,7] => ? = 4 - 2
[1,2,5,4,7,6,3] => [1,2,5,3,4,7,6] => [3,4,2,6,5,7,1] => [7,3,1,2,5,4,6] => ? = 3 - 2
[1,2,5,6,3,7,4] => [1,2,5,6,3,7,4] => [3,4,6,7,1,2,5] => [5,6,1,2,7,3,4] => ? = 4 - 2
[1,2,5,6,4,3,7] => [1,2,5,6,3,7,4] => [3,4,6,7,1,2,5] => [5,6,1,2,7,3,4] => ? = 4 - 2
[1,2,5,7,3,6,4] => [1,2,5,7,3,6,4] => [3,4,2,7,5,6,1] => [7,3,1,2,5,6,4] => ? = 3 - 2
[1,2,5,7,4,3,6] => [1,2,5,7,3,6,4] => [3,4,2,7,5,6,1] => [7,3,1,2,5,6,4] => ? = 3 - 2
[1,2,6,3,4,7,5] => [1,2,6,3,4,7,5] => [3,4,2,6,7,1,5] => [6,3,1,2,7,4,5] => ? = 3 - 2
[1,2,6,3,5,4,7] => [1,2,6,3,5,4,7] => [3,4,2,6,5,1,7] => [6,3,1,2,5,4,7] => ? = 3 - 2
[1,2,6,3,5,7,4] => [1,2,6,3,5,7,4] => [3,4,2,7,5,1,6] => [6,3,1,2,5,7,4] => ? = 3 - 2
[1,2,6,3,7,4,5] => [1,2,6,3,7,4,5] => [3,4,5,1,7,2,6] => [4,6,1,2,3,7,5] => ? = 4 - 2
[1,2,6,3,7,5,4] => [1,2,6,3,7,4,5] => [3,4,5,1,7,2,6] => [4,6,1,2,3,7,5] => ? = 4 - 2
[1,2,6,4,3,5,7] => [1,2,6,3,5,7,4] => [3,4,2,7,5,1,6] => [6,3,1,2,5,7,4] => ? = 3 - 2
[1,2,6,4,3,7,5] => [1,2,6,3,7,4,5] => [3,4,5,1,7,2,6] => [4,6,1,2,3,7,5] => ? = 4 - 2
[1,2,6,4,5,3,7] => [1,2,6,3,7,4,5] => [3,4,5,1,7,2,6] => [4,6,1,2,3,7,5] => ? = 4 - 2
[1,2,6,4,7,3,5] => [1,2,6,3,5,4,7] => [3,4,2,6,5,1,7] => [6,3,1,2,5,4,7] => ? = 3 - 2
[1,2,6,4,7,5,3] => [1,2,6,3,4,7,5] => [3,4,2,6,7,1,5] => [6,3,1,2,7,4,5] => ? = 3 - 2
[1,2,6,5,3,4,7] => [1,2,6,3,4,7,5] => [3,4,2,6,7,1,5] => [6,3,1,2,7,4,5] => ? = 3 - 2
[1,2,6,5,3,7,4] => [1,2,6,3,7,4,5] => [3,4,5,1,7,2,6] => [4,6,1,2,3,7,5] => ? = 4 - 2
[1,2,6,5,4,3,7] => [1,2,6,3,7,4,5] => [3,4,5,1,7,2,6] => [4,6,1,2,3,7,5] => ? = 4 - 2
[1,2,6,5,4,7,3] => [1,2,6,3,4,7,5] => [3,4,2,6,7,1,5] => [6,3,1,2,7,4,5] => ? = 3 - 2
[1,2,6,7,3,5,4] => [1,2,6,7,3,5,4] => [3,4,2,7,6,1,5] => [6,3,1,2,7,5,4] => ? = 3 - 2
[1,2,6,7,4,3,5] => [1,2,6,7,3,5,4] => [3,4,2,7,6,1,5] => [6,3,1,2,7,5,4] => ? = 3 - 2
[1,2,7,3,4,6,5] => [1,2,7,3,4,6,5] => [3,4,2,5,7,6,1] => [7,3,1,2,4,6,5] => ? = 3 - 2
[1,2,7,3,5,4,6] => [1,2,7,3,5,4,6] => [3,4,2,6,1,7,5] => [5,3,1,2,7,4,6] => ? = 3 - 2
[1,2,7,3,5,6,4] => [1,2,7,3,5,6,4] => [3,4,2,7,1,6,5] => [5,3,1,2,7,6,4] => ? = 3 - 2
[1,2,7,3,6,4,5] => [1,2,7,3,6,4,5] => [3,4,2,1,7,6,5] => [4,3,1,2,7,6,5] => ? = 3 - 2
[1,2,7,3,6,5,4] => [1,2,7,3,6,4,5] => [3,4,2,1,7,6,5] => [4,3,1,2,7,6,5] => ? = 3 - 2
Description
The first descent of a permutation.
For a permutation π of {1,…,n}, this is the smallest index 0<i≤n such that π(i)>π(i+1) where one considers π(n+1)=0.
Matching statistic: St000803
Mp00223: Permutations —runsort⟶ Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000803: Permutations ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 100%
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000803: Permutations ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 100%
Values
[1,3,2,5,4] => [1,3,2,5,4] => [3,4,2,5,1] => [3,2,4,1,5] => 0 = 3 - 3
[1,4,2,5,3] => [1,4,2,5,3] => [3,4,5,1,2] => [3,2,1,5,4] => 1 = 4 - 3
[1,4,3,2,5] => [1,4,2,5,3] => [3,4,5,1,2] => [3,2,1,5,4] => 1 = 4 - 3
[1,5,2,4,3] => [1,5,2,4,3] => [3,2,5,4,1] => [3,4,1,2,5] => 0 = 3 - 3
[1,5,3,2,4] => [1,5,2,4,3] => [3,2,5,4,1] => [3,4,1,2,5] => 0 = 3 - 3
[2,4,1,5,3] => [1,5,2,4,3] => [3,2,5,4,1] => [3,4,1,2,5] => 0 = 3 - 3
[2,4,3,1,5] => [1,5,2,4,3] => [3,2,5,4,1] => [3,4,1,2,5] => 0 = 3 - 3
[2,5,1,4,3] => [1,4,2,5,3] => [3,4,5,1,2] => [3,2,1,5,4] => 1 = 4 - 3
[2,5,3,1,4] => [1,4,2,5,3] => [3,4,5,1,2] => [3,2,1,5,4] => 1 = 4 - 3
[2,5,4,1,3] => [1,3,2,5,4] => [3,4,2,5,1] => [3,2,4,1,5] => 0 = 3 - 3
[3,1,4,2,5] => [1,4,2,5,3] => [3,4,5,1,2] => [3,2,1,5,4] => 1 = 4 - 3
[3,1,5,2,4] => [1,5,2,4,3] => [3,2,5,4,1] => [3,4,1,2,5] => 0 = 3 - 3
[3,2,4,1,5] => [1,5,2,4,3] => [3,2,5,4,1] => [3,4,1,2,5] => 0 = 3 - 3
[3,2,5,1,4] => [1,4,2,5,3] => [3,4,5,1,2] => [3,2,1,5,4] => 1 = 4 - 3
[4,1,3,2,5] => [1,3,2,5,4] => [3,4,2,5,1] => [3,2,4,1,5] => 0 = 3 - 3
[4,2,5,1,3] => [1,3,2,5,4] => [3,4,2,5,1] => [3,2,4,1,5] => 0 = 3 - 3
[1,2,4,3,6,5] => [1,2,4,3,6,5] => [3,4,5,2,6,1] => [4,3,2,5,1,6] => 0 = 3 - 3
[1,2,5,3,6,4] => [1,2,5,3,6,4] => [3,4,5,6,1,2] => [4,3,2,1,6,5] => 1 = 4 - 3
[1,2,5,4,3,6] => [1,2,5,3,6,4] => [3,4,5,6,1,2] => [4,3,2,1,6,5] => 1 = 4 - 3
[1,2,6,3,5,4] => [1,2,6,3,5,4] => [3,4,2,6,5,1] => [4,3,5,1,2,6] => 0 = 3 - 3
[1,2,6,4,3,5] => [1,2,6,3,5,4] => [3,4,2,6,5,1] => [4,3,5,1,2,6] => 0 = 3 - 3
[1,3,2,4,6,5] => [1,3,2,4,6,5] => [3,4,2,5,6,1] => [4,3,5,2,1,6] => 0 = 3 - 3
[1,3,2,5,4,6] => [1,3,2,5,4,6] => [3,4,2,5,1,6] => [4,3,5,2,6,1] => 0 = 3 - 3
[1,3,2,5,6,4] => [1,3,2,5,6,4] => [3,4,2,6,1,5] => [4,3,5,1,6,2] => 0 = 3 - 3
[1,3,2,6,4,5] => [1,3,2,6,4,5] => [3,4,2,1,6,5] => [4,3,5,6,1,2] => 0 = 3 - 3
[1,3,2,6,5,4] => [1,3,2,6,4,5] => [3,4,2,1,6,5] => [4,3,5,6,1,2] => 0 = 3 - 3
[1,3,4,2,6,5] => [1,3,4,2,6,5] => [3,5,2,4,6,1] => [4,2,5,3,1,6] => 0 = 3 - 3
[1,3,5,2,6,4] => [1,3,5,2,6,4] => [3,5,2,6,1,4] => [4,2,5,1,6,3] => 0 = 3 - 3
[1,3,5,4,2,6] => [1,3,5,2,6,4] => [3,5,2,6,1,4] => [4,2,5,1,6,3] => 0 = 3 - 3
[1,3,6,2,5,4] => [1,3,6,2,5,4] => [3,2,5,6,4,1] => [4,5,2,1,3,6] => 0 = 3 - 3
[1,3,6,4,2,5] => [1,3,6,2,5,4] => [3,2,5,6,4,1] => [4,5,2,1,3,6] => 0 = 3 - 3
[1,4,2,3,6,5] => [1,4,2,3,6,5] => [3,2,5,4,6,1] => [4,5,2,3,1,6] => 0 = 3 - 3
[1,4,2,5,3,6] => [1,4,2,5,3,6] => [3,4,5,1,2,6] => [4,3,2,6,5,1] => 1 = 4 - 3
[1,4,2,5,6,3] => [1,4,2,5,6,3] => [3,4,6,1,2,5] => [4,3,1,6,5,2] => 1 = 4 - 3
[1,4,2,6,3,5] => [1,4,2,6,3,5] => [3,4,1,5,6,2] => [4,3,6,2,1,5] => 1 = 4 - 3
[1,4,2,6,5,3] => [1,4,2,6,3,5] => [3,4,1,5,6,2] => [4,3,6,2,1,5] => 1 = 4 - 3
[1,4,3,2,5,6] => [1,4,2,5,6,3] => [3,4,6,1,2,5] => [4,3,1,6,5,2] => 1 = 4 - 3
[1,4,3,2,6,5] => [1,4,2,6,3,5] => [3,4,1,5,6,2] => [4,3,6,2,1,5] => 1 = 4 - 3
[1,4,3,5,2,6] => [1,4,2,6,3,5] => [3,4,1,5,6,2] => [4,3,6,2,1,5] => 1 = 4 - 3
[1,4,3,6,2,5] => [1,4,2,5,3,6] => [3,4,5,1,2,6] => [4,3,2,6,5,1] => 1 = 4 - 3
[1,4,3,6,5,2] => [1,4,2,3,6,5] => [3,2,5,4,6,1] => [4,5,2,3,1,6] => 0 = 3 - 3
[1,4,5,2,6,3] => [1,4,5,2,6,3] => [3,5,6,1,2,4] => [4,2,1,6,5,3] => 1 = 4 - 3
[1,4,5,3,2,6] => [1,4,5,2,6,3] => [3,5,6,1,2,4] => [4,2,1,6,5,3] => 1 = 4 - 3
[1,4,6,2,5,3] => [1,4,6,2,5,3] => [3,2,6,4,5,1] => [4,5,1,3,2,6] => 0 = 3 - 3
[1,4,6,3,2,5] => [1,4,6,2,5,3] => [3,2,6,4,5,1] => [4,5,1,3,2,6] => 0 = 3 - 3
[1,5,2,3,6,4] => [1,5,2,3,6,4] => [3,2,5,6,1,4] => [4,5,2,1,6,3] => 0 = 3 - 3
[1,5,2,4,3,6] => [1,5,2,4,3,6] => [3,2,5,4,1,6] => [4,5,2,3,6,1] => 0 = 3 - 3
[1,5,2,4,6,3] => [1,5,2,4,6,3] => [3,2,6,4,1,5] => [4,5,1,3,6,2] => 0 = 3 - 3
[1,5,2,6,3,4] => [1,5,2,6,3,4] => [3,4,1,6,2,5] => [4,3,6,1,5,2] => 1 = 4 - 3
[1,5,2,6,4,3] => [1,5,2,6,3,4] => [3,4,1,6,2,5] => [4,3,6,1,5,2] => 1 = 4 - 3
[1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => [3,4,5,6,2,7,1] => [5,4,3,2,6,1,7] => ? = 3 - 3
[1,2,3,6,4,7,5] => [1,2,3,6,4,7,5] => [3,4,5,6,7,1,2] => [5,4,3,2,1,7,6] => ? = 4 - 3
[1,2,3,6,5,4,7] => [1,2,3,6,4,7,5] => [3,4,5,6,7,1,2] => [5,4,3,2,1,7,6] => ? = 4 - 3
[1,2,3,7,4,6,5] => [1,2,3,7,4,6,5] => [3,4,5,2,7,6,1] => [5,4,3,6,1,2,7] => ? = 3 - 3
[1,2,3,7,5,4,6] => [1,2,3,7,4,6,5] => [3,4,5,2,7,6,1] => [5,4,3,6,1,2,7] => ? = 3 - 3
[1,2,4,3,5,7,6] => [1,2,4,3,5,7,6] => [3,4,5,2,6,7,1] => [5,4,3,6,2,1,7] => ? = 3 - 3
[1,2,4,3,6,5,7] => [1,2,4,3,6,5,7] => [3,4,5,2,6,1,7] => [5,4,3,6,2,7,1] => ? = 3 - 3
[1,2,4,3,6,7,5] => [1,2,4,3,6,7,5] => [3,4,5,2,7,1,6] => [5,4,3,6,1,7,2] => ? = 3 - 3
[1,2,4,3,7,5,6] => [1,2,4,3,7,5,6] => [3,4,5,2,1,7,6] => [5,4,3,6,7,1,2] => ? = 3 - 3
[1,2,4,3,7,6,5] => [1,2,4,3,7,5,6] => [3,4,5,2,1,7,6] => [5,4,3,6,7,1,2] => ? = 3 - 3
[1,2,4,5,3,7,6] => [1,2,4,5,3,7,6] => [3,4,6,2,5,7,1] => [5,4,2,6,3,1,7] => ? = 3 - 3
[1,2,4,6,3,7,5] => [1,2,4,6,3,7,5] => [3,4,6,2,7,1,5] => [5,4,2,6,1,7,3] => ? = 3 - 3
[1,2,4,6,5,3,7] => [1,2,4,6,3,7,5] => [3,4,6,2,7,1,5] => [5,4,2,6,1,7,3] => ? = 3 - 3
[1,2,4,7,3,6,5] => [1,2,4,7,3,6,5] => [3,4,2,6,7,5,1] => [5,4,6,2,1,3,7] => ? = 3 - 3
[1,2,4,7,5,3,6] => [1,2,4,7,3,6,5] => [3,4,2,6,7,5,1] => [5,4,6,2,1,3,7] => ? = 3 - 3
[1,2,5,3,4,7,6] => [1,2,5,3,4,7,6] => [3,4,2,6,5,7,1] => [5,4,6,2,3,1,7] => ? = 3 - 3
[1,2,5,3,6,4,7] => [1,2,5,3,6,4,7] => [3,4,5,6,1,2,7] => [5,4,3,2,7,6,1] => ? = 4 - 3
[1,2,5,3,6,7,4] => [1,2,5,3,6,7,4] => [3,4,5,7,1,2,6] => [5,4,3,1,7,6,2] => ? = 4 - 3
[1,2,5,3,7,4,6] => [1,2,5,3,7,4,6] => [3,4,5,1,6,7,2] => [5,4,3,7,2,1,6] => ? = 4 - 3
[1,2,5,3,7,6,4] => [1,2,5,3,7,4,6] => [3,4,5,1,6,7,2] => [5,4,3,7,2,1,6] => ? = 4 - 3
[1,2,5,4,3,6,7] => [1,2,5,3,6,7,4] => [3,4,5,7,1,2,6] => [5,4,3,1,7,6,2] => ? = 4 - 3
[1,2,5,4,3,7,6] => [1,2,5,3,7,4,6] => [3,4,5,1,6,7,2] => [5,4,3,7,2,1,6] => ? = 4 - 3
[1,2,5,4,6,3,7] => [1,2,5,3,7,4,6] => [3,4,5,1,6,7,2] => [5,4,3,7,2,1,6] => ? = 4 - 3
[1,2,5,4,7,3,6] => [1,2,5,3,6,4,7] => [3,4,5,6,1,2,7] => [5,4,3,2,7,6,1] => ? = 4 - 3
[1,2,5,4,7,6,3] => [1,2,5,3,4,7,6] => [3,4,2,6,5,7,1] => [5,4,6,2,3,1,7] => ? = 3 - 3
[1,2,5,6,3,7,4] => [1,2,5,6,3,7,4] => [3,4,6,7,1,2,5] => [5,4,2,1,7,6,3] => ? = 4 - 3
[1,2,5,6,4,3,7] => [1,2,5,6,3,7,4] => [3,4,6,7,1,2,5] => [5,4,2,1,7,6,3] => ? = 4 - 3
[1,2,5,7,3,6,4] => [1,2,5,7,3,6,4] => [3,4,2,7,5,6,1] => [5,4,6,1,3,2,7] => ? = 3 - 3
[1,2,5,7,4,3,6] => [1,2,5,7,3,6,4] => [3,4,2,7,5,6,1] => [5,4,6,1,3,2,7] => ? = 3 - 3
[1,2,6,3,4,7,5] => [1,2,6,3,4,7,5] => [3,4,2,6,7,1,5] => [5,4,6,2,1,7,3] => ? = 3 - 3
[1,2,6,3,5,4,7] => [1,2,6,3,5,4,7] => [3,4,2,6,5,1,7] => [5,4,6,2,3,7,1] => ? = 3 - 3
[1,2,6,3,5,7,4] => [1,2,6,3,5,7,4] => [3,4,2,7,5,1,6] => [5,4,6,1,3,7,2] => ? = 3 - 3
[1,2,6,3,7,4,5] => [1,2,6,3,7,4,5] => [3,4,5,1,7,2,6] => [5,4,3,7,1,6,2] => ? = 4 - 3
[1,2,6,3,7,5,4] => [1,2,6,3,7,4,5] => [3,4,5,1,7,2,6] => [5,4,3,7,1,6,2] => ? = 4 - 3
[1,2,6,4,3,5,7] => [1,2,6,3,5,7,4] => [3,4,2,7,5,1,6] => [5,4,6,1,3,7,2] => ? = 3 - 3
[1,2,6,4,3,7,5] => [1,2,6,3,7,4,5] => [3,4,5,1,7,2,6] => [5,4,3,7,1,6,2] => ? = 4 - 3
[1,2,6,4,5,3,7] => [1,2,6,3,7,4,5] => [3,4,5,1,7,2,6] => [5,4,3,7,1,6,2] => ? = 4 - 3
[1,2,6,4,7,3,5] => [1,2,6,3,5,4,7] => [3,4,2,6,5,1,7] => [5,4,6,2,3,7,1] => ? = 3 - 3
[1,2,6,4,7,5,3] => [1,2,6,3,4,7,5] => [3,4,2,6,7,1,5] => [5,4,6,2,1,7,3] => ? = 3 - 3
[1,2,6,5,3,4,7] => [1,2,6,3,4,7,5] => [3,4,2,6,7,1,5] => [5,4,6,2,1,7,3] => ? = 3 - 3
[1,2,6,5,3,7,4] => [1,2,6,3,7,4,5] => [3,4,5,1,7,2,6] => [5,4,3,7,1,6,2] => ? = 4 - 3
[1,2,6,5,4,3,7] => [1,2,6,3,7,4,5] => [3,4,5,1,7,2,6] => [5,4,3,7,1,6,2] => ? = 4 - 3
[1,2,6,5,4,7,3] => [1,2,6,3,4,7,5] => [3,4,2,6,7,1,5] => [5,4,6,2,1,7,3] => ? = 3 - 3
[1,2,6,7,3,5,4] => [1,2,6,7,3,5,4] => [3,4,2,7,6,1,5] => [5,4,6,1,2,7,3] => ? = 3 - 3
[1,2,6,7,4,3,5] => [1,2,6,7,3,5,4] => [3,4,2,7,6,1,5] => [5,4,6,1,2,7,3] => ? = 3 - 3
[1,2,7,3,4,6,5] => [1,2,7,3,4,6,5] => [3,4,2,5,7,6,1] => [5,4,6,3,1,2,7] => ? = 3 - 3
[1,2,7,3,5,4,6] => [1,2,7,3,5,4,6] => [3,4,2,6,1,7,5] => [5,4,6,2,7,1,3] => ? = 3 - 3
[1,2,7,3,5,6,4] => [1,2,7,3,5,6,4] => [3,4,2,7,1,6,5] => [5,4,6,1,7,2,3] => ? = 3 - 3
[1,2,7,3,6,4,5] => [1,2,7,3,6,4,5] => [3,4,2,1,7,6,5] => [5,4,6,7,1,2,3] => ? = 3 - 3
[1,2,7,3,6,5,4] => [1,2,7,3,6,4,5] => [3,4,2,1,7,6,5] => [5,4,6,7,1,2,3] => ? = 3 - 3
Description
The number of occurrences of the vincular pattern |132 in a permutation.
This is the number of occurrences of the pattern (1,3,2), such that the letter matched by 1 is the first entry of the permutation.
Matching statistic: St001906
Mp00223: Permutations —runsort⟶ Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St001906: Permutations ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 100%
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St001906: Permutations ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 100%
Values
[1,3,2,5,4] => [1,3,2,5,4] => [3,4,2,5,1] => [3,5,2,4,1] => 0 = 3 - 3
[1,4,2,5,3] => [1,4,2,5,3] => [3,4,5,1,2] => [3,5,4,1,2] => 1 = 4 - 3
[1,4,3,2,5] => [1,4,2,5,3] => [3,4,5,1,2] => [3,5,4,1,2] => 1 = 4 - 3
[1,5,2,4,3] => [1,5,2,4,3] => [3,2,5,4,1] => [3,2,5,4,1] => 0 = 3 - 3
[1,5,3,2,4] => [1,5,2,4,3] => [3,2,5,4,1] => [3,2,5,4,1] => 0 = 3 - 3
[2,4,1,5,3] => [1,5,2,4,3] => [3,2,5,4,1] => [3,2,5,4,1] => 0 = 3 - 3
[2,4,3,1,5] => [1,5,2,4,3] => [3,2,5,4,1] => [3,2,5,4,1] => 0 = 3 - 3
[2,5,1,4,3] => [1,4,2,5,3] => [3,4,5,1,2] => [3,5,4,1,2] => 1 = 4 - 3
[2,5,3,1,4] => [1,4,2,5,3] => [3,4,5,1,2] => [3,5,4,1,2] => 1 = 4 - 3
[2,5,4,1,3] => [1,3,2,5,4] => [3,4,2,5,1] => [3,5,2,4,1] => 0 = 3 - 3
[3,1,4,2,5] => [1,4,2,5,3] => [3,4,5,1,2] => [3,5,4,1,2] => 1 = 4 - 3
[3,1,5,2,4] => [1,5,2,4,3] => [3,2,5,4,1] => [3,2,5,4,1] => 0 = 3 - 3
[3,2,4,1,5] => [1,5,2,4,3] => [3,2,5,4,1] => [3,2,5,4,1] => 0 = 3 - 3
[3,2,5,1,4] => [1,4,2,5,3] => [3,4,5,1,2] => [3,5,4,1,2] => 1 = 4 - 3
[4,1,3,2,5] => [1,3,2,5,4] => [3,4,2,5,1] => [3,5,2,4,1] => 0 = 3 - 3
[4,2,5,1,3] => [1,3,2,5,4] => [3,4,2,5,1] => [3,5,2,4,1] => 0 = 3 - 3
[1,2,4,3,6,5] => [1,2,4,3,6,5] => [3,4,5,2,6,1] => [3,6,5,2,4,1] => 0 = 3 - 3
[1,2,5,3,6,4] => [1,2,5,3,6,4] => [3,4,5,6,1,2] => [3,6,5,4,1,2] => 1 = 4 - 3
[1,2,5,4,3,6] => [1,2,5,3,6,4] => [3,4,5,6,1,2] => [3,6,5,4,1,2] => 1 = 4 - 3
[1,2,6,3,5,4] => [1,2,6,3,5,4] => [3,4,2,6,5,1] => [3,6,2,5,4,1] => 0 = 3 - 3
[1,2,6,4,3,5] => [1,2,6,3,5,4] => [3,4,2,6,5,1] => [3,6,2,5,4,1] => 0 = 3 - 3
[1,3,2,4,6,5] => [1,3,2,4,6,5] => [3,4,2,5,6,1] => [3,6,2,5,4,1] => 0 = 3 - 3
[1,3,2,5,4,6] => [1,3,2,5,4,6] => [3,4,2,5,1,6] => [3,6,2,5,1,4] => 0 = 3 - 3
[1,3,2,5,6,4] => [1,3,2,5,6,4] => [3,4,2,6,1,5] => [3,6,2,5,1,4] => 0 = 3 - 3
[1,3,2,6,4,5] => [1,3,2,6,4,5] => [3,4,2,1,6,5] => [3,6,2,1,5,4] => 0 = 3 - 3
[1,3,2,6,5,4] => [1,3,2,6,4,5] => [3,4,2,1,6,5] => [3,6,2,1,5,4] => 0 = 3 - 3
[1,3,4,2,6,5] => [1,3,4,2,6,5] => [3,5,2,4,6,1] => [3,6,2,5,4,1] => 0 = 3 - 3
[1,3,5,2,6,4] => [1,3,5,2,6,4] => [3,5,2,6,1,4] => [3,6,2,5,1,4] => 0 = 3 - 3
[1,3,5,4,2,6] => [1,3,5,2,6,4] => [3,5,2,6,1,4] => [3,6,2,5,1,4] => 0 = 3 - 3
[1,3,6,2,5,4] => [1,3,6,2,5,4] => [3,2,5,6,4,1] => [3,2,6,5,4,1] => 0 = 3 - 3
[1,3,6,4,2,5] => [1,3,6,2,5,4] => [3,2,5,6,4,1] => [3,2,6,5,4,1] => 0 = 3 - 3
[1,4,2,3,6,5] => [1,4,2,3,6,5] => [3,2,5,4,6,1] => [3,2,6,5,4,1] => 0 = 3 - 3
[1,4,2,5,3,6] => [1,4,2,5,3,6] => [3,4,5,1,2,6] => [3,6,5,1,4,2] => 1 = 4 - 3
[1,4,2,5,6,3] => [1,4,2,5,6,3] => [3,4,6,1,2,5] => [3,6,5,1,4,2] => 1 = 4 - 3
[1,4,2,6,3,5] => [1,4,2,6,3,5] => [3,4,1,5,6,2] => [3,6,1,5,4,2] => 1 = 4 - 3
[1,4,2,6,5,3] => [1,4,2,6,3,5] => [3,4,1,5,6,2] => [3,6,1,5,4,2] => 1 = 4 - 3
[1,4,3,2,5,6] => [1,4,2,5,6,3] => [3,4,6,1,2,5] => [3,6,5,1,4,2] => 1 = 4 - 3
[1,4,3,2,6,5] => [1,4,2,6,3,5] => [3,4,1,5,6,2] => [3,6,1,5,4,2] => 1 = 4 - 3
[1,4,3,5,2,6] => [1,4,2,6,3,5] => [3,4,1,5,6,2] => [3,6,1,5,4,2] => 1 = 4 - 3
[1,4,3,6,2,5] => [1,4,2,5,3,6] => [3,4,5,1,2,6] => [3,6,5,1,4,2] => 1 = 4 - 3
[1,4,3,6,5,2] => [1,4,2,3,6,5] => [3,2,5,4,6,1] => [3,2,6,5,4,1] => 0 = 3 - 3
[1,4,5,2,6,3] => [1,4,5,2,6,3] => [3,5,6,1,2,4] => [3,6,5,1,4,2] => 1 = 4 - 3
[1,4,5,3,2,6] => [1,4,5,2,6,3] => [3,5,6,1,2,4] => [3,6,5,1,4,2] => 1 = 4 - 3
[1,4,6,2,5,3] => [1,4,6,2,5,3] => [3,2,6,4,5,1] => [3,2,6,5,4,1] => 0 = 3 - 3
[1,4,6,3,2,5] => [1,4,6,2,5,3] => [3,2,6,4,5,1] => [3,2,6,5,4,1] => 0 = 3 - 3
[1,5,2,3,6,4] => [1,5,2,3,6,4] => [3,2,5,6,1,4] => [3,2,6,5,1,4] => 0 = 3 - 3
[1,5,2,4,3,6] => [1,5,2,4,3,6] => [3,2,5,4,1,6] => [3,2,6,5,1,4] => 0 = 3 - 3
[1,5,2,4,6,3] => [1,5,2,4,6,3] => [3,2,6,4,1,5] => [3,2,6,5,1,4] => 0 = 3 - 3
[1,5,2,6,3,4] => [1,5,2,6,3,4] => [3,4,1,6,2,5] => [3,6,1,5,4,2] => 1 = 4 - 3
[1,5,2,6,4,3] => [1,5,2,6,3,4] => [3,4,1,6,2,5] => [3,6,1,5,4,2] => 1 = 4 - 3
[1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => [3,4,5,6,2,7,1] => [3,7,6,5,2,4,1] => ? = 3 - 3
[1,2,3,6,4,7,5] => [1,2,3,6,4,7,5] => [3,4,5,6,7,1,2] => [3,7,6,5,4,1,2] => ? = 4 - 3
[1,2,3,6,5,4,7] => [1,2,3,6,4,7,5] => [3,4,5,6,7,1,2] => [3,7,6,5,4,1,2] => ? = 4 - 3
[1,2,3,7,4,6,5] => [1,2,3,7,4,6,5] => [3,4,5,2,7,6,1] => [3,7,6,2,5,4,1] => ? = 3 - 3
[1,2,3,7,5,4,6] => [1,2,3,7,4,6,5] => [3,4,5,2,7,6,1] => [3,7,6,2,5,4,1] => ? = 3 - 3
[1,2,4,3,5,7,6] => [1,2,4,3,5,7,6] => [3,4,5,2,6,7,1] => [3,7,6,2,5,4,1] => ? = 3 - 3
[1,2,4,3,6,5,7] => [1,2,4,3,6,5,7] => [3,4,5,2,6,1,7] => [3,7,6,2,5,1,4] => ? = 3 - 3
[1,2,4,3,6,7,5] => [1,2,4,3,6,7,5] => [3,4,5,2,7,1,6] => [3,7,6,2,5,1,4] => ? = 3 - 3
[1,2,4,3,7,5,6] => [1,2,4,3,7,5,6] => [3,4,5,2,1,7,6] => [3,7,6,2,1,5,4] => ? = 3 - 3
[1,2,4,3,7,6,5] => [1,2,4,3,7,5,6] => [3,4,5,2,1,7,6] => [3,7,6,2,1,5,4] => ? = 3 - 3
[1,2,4,5,3,7,6] => [1,2,4,5,3,7,6] => [3,4,6,2,5,7,1] => [3,7,6,2,5,4,1] => ? = 3 - 3
[1,2,4,6,3,7,5] => [1,2,4,6,3,7,5] => [3,4,6,2,7,1,5] => [3,7,6,2,5,1,4] => ? = 3 - 3
[1,2,4,6,5,3,7] => [1,2,4,6,3,7,5] => [3,4,6,2,7,1,5] => [3,7,6,2,5,1,4] => ? = 3 - 3
[1,2,4,7,3,6,5] => [1,2,4,7,3,6,5] => [3,4,2,6,7,5,1] => [3,7,2,6,5,4,1] => ? = 3 - 3
[1,2,4,7,5,3,6] => [1,2,4,7,3,6,5] => [3,4,2,6,7,5,1] => [3,7,2,6,5,4,1] => ? = 3 - 3
[1,2,5,3,4,7,6] => [1,2,5,3,4,7,6] => [3,4,2,6,5,7,1] => [3,7,2,6,5,4,1] => ? = 3 - 3
[1,2,5,3,6,4,7] => [1,2,5,3,6,4,7] => [3,4,5,6,1,2,7] => [3,7,6,5,1,4,2] => ? = 4 - 3
[1,2,5,3,6,7,4] => [1,2,5,3,6,7,4] => [3,4,5,7,1,2,6] => [3,7,6,5,1,4,2] => ? = 4 - 3
[1,2,5,3,7,4,6] => [1,2,5,3,7,4,6] => [3,4,5,1,6,7,2] => [3,7,6,1,5,4,2] => ? = 4 - 3
[1,2,5,3,7,6,4] => [1,2,5,3,7,4,6] => [3,4,5,1,6,7,2] => [3,7,6,1,5,4,2] => ? = 4 - 3
[1,2,5,4,3,6,7] => [1,2,5,3,6,7,4] => [3,4,5,7,1,2,6] => [3,7,6,5,1,4,2] => ? = 4 - 3
[1,2,5,4,3,7,6] => [1,2,5,3,7,4,6] => [3,4,5,1,6,7,2] => [3,7,6,1,5,4,2] => ? = 4 - 3
[1,2,5,4,6,3,7] => [1,2,5,3,7,4,6] => [3,4,5,1,6,7,2] => [3,7,6,1,5,4,2] => ? = 4 - 3
[1,2,5,4,7,3,6] => [1,2,5,3,6,4,7] => [3,4,5,6,1,2,7] => [3,7,6,5,1,4,2] => ? = 4 - 3
[1,2,5,4,7,6,3] => [1,2,5,3,4,7,6] => [3,4,2,6,5,7,1] => [3,7,2,6,5,4,1] => ? = 3 - 3
[1,2,5,6,3,7,4] => [1,2,5,6,3,7,4] => [3,4,6,7,1,2,5] => [3,7,6,5,1,4,2] => ? = 4 - 3
[1,2,5,6,4,3,7] => [1,2,5,6,3,7,4] => [3,4,6,7,1,2,5] => [3,7,6,5,1,4,2] => ? = 4 - 3
[1,2,5,7,3,6,4] => [1,2,5,7,3,6,4] => [3,4,2,7,5,6,1] => [3,7,2,6,5,4,1] => ? = 3 - 3
[1,2,5,7,4,3,6] => [1,2,5,7,3,6,4] => [3,4,2,7,5,6,1] => [3,7,2,6,5,4,1] => ? = 3 - 3
[1,2,6,3,4,7,5] => [1,2,6,3,4,7,5] => [3,4,2,6,7,1,5] => [3,7,2,6,5,1,4] => ? = 3 - 3
[1,2,6,3,5,4,7] => [1,2,6,3,5,4,7] => [3,4,2,6,5,1,7] => [3,7,2,6,5,1,4] => ? = 3 - 3
[1,2,6,3,5,7,4] => [1,2,6,3,5,7,4] => [3,4,2,7,5,1,6] => [3,7,2,6,5,1,4] => ? = 3 - 3
[1,2,6,3,7,4,5] => [1,2,6,3,7,4,5] => [3,4,5,1,7,2,6] => [3,7,6,1,5,4,2] => ? = 4 - 3
[1,2,6,3,7,5,4] => [1,2,6,3,7,4,5] => [3,4,5,1,7,2,6] => [3,7,6,1,5,4,2] => ? = 4 - 3
[1,2,6,4,3,5,7] => [1,2,6,3,5,7,4] => [3,4,2,7,5,1,6] => [3,7,2,6,5,1,4] => ? = 3 - 3
[1,2,6,4,3,7,5] => [1,2,6,3,7,4,5] => [3,4,5,1,7,2,6] => [3,7,6,1,5,4,2] => ? = 4 - 3
[1,2,6,4,5,3,7] => [1,2,6,3,7,4,5] => [3,4,5,1,7,2,6] => [3,7,6,1,5,4,2] => ? = 4 - 3
[1,2,6,4,7,3,5] => [1,2,6,3,5,4,7] => [3,4,2,6,5,1,7] => [3,7,2,6,5,1,4] => ? = 3 - 3
[1,2,6,4,7,5,3] => [1,2,6,3,4,7,5] => [3,4,2,6,7,1,5] => [3,7,2,6,5,1,4] => ? = 3 - 3
[1,2,6,5,3,4,7] => [1,2,6,3,4,7,5] => [3,4,2,6,7,1,5] => [3,7,2,6,5,1,4] => ? = 3 - 3
[1,2,6,5,3,7,4] => [1,2,6,3,7,4,5] => [3,4,5,1,7,2,6] => [3,7,6,1,5,4,2] => ? = 4 - 3
[1,2,6,5,4,3,7] => [1,2,6,3,7,4,5] => [3,4,5,1,7,2,6] => [3,7,6,1,5,4,2] => ? = 4 - 3
[1,2,6,5,4,7,3] => [1,2,6,3,4,7,5] => [3,4,2,6,7,1,5] => [3,7,2,6,5,1,4] => ? = 3 - 3
[1,2,6,7,3,5,4] => [1,2,6,7,3,5,4] => [3,4,2,7,6,1,5] => [3,7,2,6,5,1,4] => ? = 3 - 3
[1,2,6,7,4,3,5] => [1,2,6,7,3,5,4] => [3,4,2,7,6,1,5] => [3,7,2,6,5,1,4] => ? = 3 - 3
[1,2,7,3,4,6,5] => [1,2,7,3,4,6,5] => [3,4,2,5,7,6,1] => [3,7,2,6,5,4,1] => ? = 3 - 3
[1,2,7,3,5,4,6] => [1,2,7,3,5,4,6] => [3,4,2,6,1,7,5] => [3,7,2,6,1,5,4] => ? = 3 - 3
[1,2,7,3,5,6,4] => [1,2,7,3,5,6,4] => [3,4,2,7,1,6,5] => [3,7,2,6,1,5,4] => ? = 3 - 3
[1,2,7,3,6,4,5] => [1,2,7,3,6,4,5] => [3,4,2,1,7,6,5] => [3,7,2,1,6,5,4] => ? = 3 - 3
[1,2,7,3,6,5,4] => [1,2,7,3,6,4,5] => [3,4,2,1,7,6,5] => [3,7,2,1,6,5,4] => ? = 3 - 3
Description
Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation.
Let π be a permutation. Its total displacement [[St000830]] is D(π)=∑i|π(i)−i|, and its absolute length [[St000216]] is the minimal number T(π) of transpositions whose product is π. Finally, let I(π) be the number of inversions [[St000018]] of π.
This statistic equals (D(π)−T(π)−I(π))/2.
Diaconis and Graham [1] proved that this statistic is always nonnegative.
Matching statistic: St000741
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00223: Permutations —runsort⟶ Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000741: Graphs ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 50%
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000741: Graphs ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 50%
Values
[1,3,2,5,4] => [1,3,2,5,4] => [3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,4,2,5,3] => [1,4,2,5,3] => [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ? = 4 - 1
[1,4,3,2,5] => [1,4,2,5,3] => [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ? = 4 - 1
[1,5,2,4,3] => [1,5,2,4,3] => [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,5,3,2,4] => [1,5,2,4,3] => [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 2 = 3 - 1
[2,4,1,5,3] => [1,5,2,4,3] => [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 2 = 3 - 1
[2,4,3,1,5] => [1,5,2,4,3] => [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 2 = 3 - 1
[2,5,1,4,3] => [1,4,2,5,3] => [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ? = 4 - 1
[2,5,3,1,4] => [1,4,2,5,3] => [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ? = 4 - 1
[2,5,4,1,3] => [1,3,2,5,4] => [3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[3,1,4,2,5] => [1,4,2,5,3] => [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ? = 4 - 1
[3,1,5,2,4] => [1,5,2,4,3] => [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 2 = 3 - 1
[3,2,4,1,5] => [1,5,2,4,3] => [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 2 = 3 - 1
[3,2,5,1,4] => [1,4,2,5,3] => [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ? = 4 - 1
[4,1,3,2,5] => [1,3,2,5,4] => [3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[4,2,5,1,3] => [1,3,2,5,4] => [3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,2,4,3,6,5] => [1,2,4,3,6,5] => [3,4,5,2,6,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 1
[1,2,5,3,6,4] => [1,2,5,3,6,4] => [3,4,5,6,1,2] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 4 - 1
[1,2,5,4,3,6] => [1,2,5,3,6,4] => [3,4,5,6,1,2] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 4 - 1
[1,2,6,3,5,4] => [1,2,6,3,5,4] => [3,4,2,6,5,1] => ([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 3 - 1
[1,2,6,4,3,5] => [1,2,6,3,5,4] => [3,4,2,6,5,1] => ([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 3 - 1
[1,3,2,4,6,5] => [1,3,2,4,6,5] => [3,4,2,5,6,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 3 - 1
[1,3,2,5,4,6] => [1,3,2,5,4,6] => [3,4,2,5,1,6] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 3 - 1
[1,3,2,5,6,4] => [1,3,2,5,6,4] => [3,4,2,6,1,5] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6)
=> 2 = 3 - 1
[1,3,2,6,4,5] => [1,3,2,6,4,5] => [3,4,2,1,6,5] => ([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 3 - 1
[1,3,2,6,5,4] => [1,3,2,6,4,5] => [3,4,2,1,6,5] => ([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 3 - 1
[1,3,4,2,6,5] => [1,3,4,2,6,5] => [3,5,2,4,6,1] => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 3 - 1
[1,3,5,2,6,4] => [1,3,5,2,6,4] => [3,5,2,6,1,4] => ([(0,1),(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 3 - 1
[1,3,5,4,2,6] => [1,3,5,2,6,4] => [3,5,2,6,1,4] => ([(0,1),(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 3 - 1
[1,3,6,2,5,4] => [1,3,6,2,5,4] => [3,2,5,6,4,1] => ([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 3 - 1
[1,3,6,4,2,5] => [1,3,6,2,5,4] => [3,2,5,6,4,1] => ([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 3 - 1
[1,4,2,3,6,5] => [1,4,2,3,6,5] => [3,2,5,4,6,1] => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> 2 = 3 - 1
[1,4,2,5,3,6] => [1,4,2,5,3,6] => [3,4,5,1,2,6] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 4 - 1
[1,4,2,5,6,3] => [1,4,2,5,6,3] => [3,4,6,1,2,5] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ? = 4 - 1
[1,4,2,6,3,5] => [1,4,2,6,3,5] => [3,4,1,5,6,2] => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ? = 4 - 1
[1,4,2,6,5,3] => [1,4,2,6,3,5] => [3,4,1,5,6,2] => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ? = 4 - 1
[1,4,3,2,5,6] => [1,4,2,5,6,3] => [3,4,6,1,2,5] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ? = 4 - 1
[1,4,3,2,6,5] => [1,4,2,6,3,5] => [3,4,1,5,6,2] => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ? = 4 - 1
[1,4,3,5,2,6] => [1,4,2,6,3,5] => [3,4,1,5,6,2] => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ? = 4 - 1
[1,4,3,6,2,5] => [1,4,2,5,3,6] => [3,4,5,1,2,6] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 4 - 1
[1,4,3,6,5,2] => [1,4,2,3,6,5] => [3,2,5,4,6,1] => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> 2 = 3 - 1
[1,4,5,2,6,3] => [1,4,5,2,6,3] => [3,5,6,1,2,4] => ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 4 - 1
[1,4,5,3,2,6] => [1,4,5,2,6,3] => [3,5,6,1,2,4] => ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 4 - 1
[1,4,6,2,5,3] => [1,4,6,2,5,3] => [3,2,6,4,5,1] => ([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 3 - 1
[1,4,6,3,2,5] => [1,4,6,2,5,3] => [3,2,6,4,5,1] => ([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 3 - 1
[1,5,2,3,6,4] => [1,5,2,3,6,4] => [3,2,5,6,1,4] => ([(0,3),(0,4),(1,2),(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 3 - 1
[1,5,2,4,3,6] => [1,5,2,4,3,6] => [3,2,5,4,1,6] => ([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> 2 = 3 - 1
[1,5,2,4,6,3] => [1,5,2,4,6,3] => [3,2,6,4,1,5] => ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 3 - 1
[1,5,2,6,3,4] => [1,5,2,6,3,4] => [3,4,1,6,2,5] => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> ? = 4 - 1
[1,5,2,6,4,3] => [1,5,2,6,3,4] => [3,4,1,6,2,5] => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> ? = 4 - 1
[1,5,3,2,4,6] => [1,5,2,4,6,3] => [3,2,6,4,1,5] => ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 3 - 1
[1,5,3,2,6,4] => [1,5,2,6,3,4] => [3,4,1,6,2,5] => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> ? = 4 - 1
[1,5,3,4,2,6] => [1,5,2,6,3,4] => [3,4,1,6,2,5] => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> ? = 4 - 1
[1,5,3,6,2,4] => [1,5,2,4,3,6] => [3,2,5,4,1,6] => ([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> 2 = 3 - 1
[1,5,3,6,4,2] => [1,5,2,3,6,4] => [3,2,5,6,1,4] => ([(0,3),(0,4),(1,2),(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 3 - 1
[1,5,4,2,3,6] => [1,5,2,3,6,4] => [3,2,5,6,1,4] => ([(0,3),(0,4),(1,2),(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 3 - 1
[1,5,4,2,6,3] => [1,5,2,6,3,4] => [3,4,1,6,2,5] => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> ? = 4 - 1
[1,5,4,3,2,6] => [1,5,2,6,3,4] => [3,4,1,6,2,5] => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> ? = 4 - 1
[1,5,4,3,6,2] => [1,5,2,3,6,4] => [3,2,5,6,1,4] => ([(0,3),(0,4),(1,2),(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 3 - 1
[1,5,6,2,4,3] => [1,5,6,2,4,3] => [3,2,6,5,1,4] => ([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> 2 = 3 - 1
[1,5,6,3,2,4] => [1,5,6,2,4,3] => [3,2,6,5,1,4] => ([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> 2 = 3 - 1
[1,6,2,3,5,4] => [1,6,2,3,5,4] => [3,2,4,6,5,1] => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> 2 = 3 - 1
[1,6,2,4,3,5] => [1,6,2,4,3,5] => [3,2,5,1,6,4] => ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
=> 2 = 3 - 1
[1,6,2,4,5,3] => [1,6,2,4,5,3] => [3,2,6,1,5,4] => ([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(4,5)],6)
=> 2 = 3 - 1
[1,6,2,5,3,4] => [1,6,2,5,3,4] => [3,2,1,6,5,4] => ([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
=> 2 = 3 - 1
[1,6,2,5,4,3] => [1,6,2,5,3,4] => [3,2,1,6,5,4] => ([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
=> 2 = 3 - 1
[1,6,3,2,4,5] => [1,6,2,4,5,3] => [3,2,6,1,5,4] => ([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(4,5)],6)
=> 2 = 3 - 1
[1,6,3,2,5,4] => [1,6,2,5,3,4] => [3,2,1,6,5,4] => ([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
=> 2 = 3 - 1
[1,6,3,4,2,5] => [1,6,2,5,3,4] => [3,2,1,6,5,4] => ([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
=> 2 = 3 - 1
[1,6,3,5,2,4] => [1,6,2,4,3,5] => [3,2,5,1,6,4] => ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
=> 2 = 3 - 1
[1,6,3,5,4,2] => [1,6,2,3,5,4] => [3,2,4,6,5,1] => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> 2 = 3 - 1
[1,6,4,2,3,5] => [1,6,2,3,5,4] => [3,2,4,6,5,1] => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> 2 = 3 - 1
[1,6,4,2,5,3] => [1,6,2,5,3,4] => [3,2,1,6,5,4] => ([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
=> 2 = 3 - 1
[1,6,4,3,2,5] => [1,6,2,5,3,4] => [3,2,1,6,5,4] => ([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
=> 2 = 3 - 1
[1,6,4,3,5,2] => [1,6,2,3,5,4] => [3,2,4,6,5,1] => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> 2 = 3 - 1
[2,4,3,6,5,1] => [1,2,4,3,6,5] => [3,4,5,2,6,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 1
[2,5,1,4,3,6] => [1,4,2,5,3,6] => [3,4,5,1,2,6] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 4 - 1
[2,5,3,6,1,4] => [1,4,2,5,3,6] => [3,4,5,1,2,6] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 4 - 1
[2,5,3,6,4,1] => [1,2,5,3,6,4] => [3,4,5,6,1,2] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 4 - 1
[2,5,4,3,6,1] => [1,2,5,3,6,4] => [3,4,5,6,1,2] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 4 - 1
[2,5,6,1,4,3] => [1,4,2,5,6,3] => [3,4,6,1,2,5] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ? = 4 - 1
[2,5,6,3,1,4] => [1,4,2,5,6,3] => [3,4,6,1,2,5] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ? = 4 - 1
[2,6,1,4,3,5] => [1,4,2,6,3,5] => [3,4,1,5,6,2] => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ? = 4 - 1
[2,6,1,4,5,3] => [1,4,5,2,6,3] => [3,5,6,1,2,4] => ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 4 - 1
[2,6,1,5,3,4] => [1,5,2,6,3,4] => [3,4,1,6,2,5] => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> ? = 4 - 1
[2,6,1,5,4,3] => [1,5,2,6,3,4] => [3,4,1,6,2,5] => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> ? = 4 - 1
[2,6,3,1,4,5] => [1,4,5,2,6,3] => [3,5,6,1,2,4] => ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 4 - 1
[2,6,3,1,5,4] => [1,5,2,6,3,4] => [3,4,1,6,2,5] => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> ? = 4 - 1
[2,6,3,4,1,5] => [1,5,2,6,3,4] => [3,4,1,6,2,5] => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> ? = 4 - 1
[2,6,3,5,1,4] => [1,4,2,6,3,5] => [3,4,1,5,6,2] => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ? = 4 - 1
[2,6,4,1,5,3] => [1,5,2,6,3,4] => [3,4,1,6,2,5] => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> ? = 4 - 1
[2,6,4,3,1,5] => [1,5,2,6,3,4] => [3,4,1,6,2,5] => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> ? = 4 - 1
[2,6,5,1,4,3] => [1,4,2,6,3,5] => [3,4,1,5,6,2] => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ? = 4 - 1
[2,6,5,3,1,4] => [1,4,2,6,3,5] => [3,4,1,5,6,2] => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ? = 4 - 1
[3,1,4,2,5,6] => [1,4,2,5,6,3] => [3,4,6,1,2,5] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ? = 4 - 1
[3,1,4,2,6,5] => [1,4,2,6,3,5] => [3,4,1,5,6,2] => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ? = 4 - 1
[3,1,4,5,2,6] => [1,4,5,2,6,3] => [3,5,6,1,2,4] => ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 4 - 1
[3,1,5,2,6,4] => [1,5,2,6,3,4] => [3,4,1,6,2,5] => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> ? = 4 - 1
[3,1,5,4,2,6] => [1,5,2,6,3,4] => [3,4,1,6,2,5] => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> ? = 4 - 1
[3,2,5,6,1,4] => [1,4,2,5,6,3] => [3,4,6,1,2,5] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ? = 4 - 1
Description
The Colin de Verdière graph invariant.
Matching statistic: St001060
Mp00223: Permutations —runsort⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00154: Graphs —core⟶ Graphs
St001060: Graphs ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 50%
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00154: Graphs —core⟶ Graphs
St001060: Graphs ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 50%
Values
[1,3,2,5,4] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ([(0,1)],2)
=> ? = 3
[1,4,2,5,3] => [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ? = 4
[1,4,3,2,5] => [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ? = 4
[1,5,2,4,3] => [1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,5,3,2,4] => [1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[2,4,1,5,3] => [1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[2,4,3,1,5] => [1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[2,5,1,4,3] => [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ? = 4
[2,5,3,1,4] => [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ? = 4
[2,5,4,1,3] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ([(0,1)],2)
=> ? = 3
[3,1,4,2,5] => [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ? = 4
[3,1,5,2,4] => [1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[3,2,4,1,5] => [1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[3,2,5,1,4] => [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ? = 4
[4,1,3,2,5] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ([(0,1)],2)
=> ? = 3
[4,2,5,1,3] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ([(0,1)],2)
=> ? = 3
[1,2,4,3,6,5] => [1,2,4,3,6,5] => ([(2,5),(3,4)],6)
=> ([(0,1)],2)
=> ? = 3
[1,2,5,3,6,4] => [1,2,5,3,6,4] => ([(2,5),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> ? = 4
[1,2,5,4,3,6] => [1,2,5,3,6,4] => ([(2,5),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> ? = 4
[1,2,6,3,5,4] => [1,2,6,3,5,4] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,2,6,4,3,5] => [1,2,6,3,5,4] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,3,2,4,6,5] => [1,3,2,4,6,5] => ([(2,5),(3,4)],6)
=> ([(0,1)],2)
=> ? = 3
[1,3,2,5,4,6] => [1,3,2,5,4,6] => ([(2,5),(3,4)],6)
=> ([(0,1)],2)
=> ? = 3
[1,3,2,5,6,4] => [1,3,2,5,6,4] => ([(1,2),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? = 3
[1,3,2,6,4,5] => [1,3,2,6,4,5] => ([(1,2),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? = 3
[1,3,2,6,5,4] => [1,3,2,6,4,5] => ([(1,2),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? = 3
[1,3,4,2,6,5] => [1,3,4,2,6,5] => ([(1,2),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? = 3
[1,3,5,2,6,4] => [1,3,5,2,6,4] => ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> ? = 3
[1,3,5,4,2,6] => [1,3,5,2,6,4] => ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> ? = 3
[1,3,6,2,5,4] => [1,3,6,2,5,4] => ([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,3,6,4,2,5] => [1,3,6,2,5,4] => ([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,4,2,3,6,5] => [1,4,2,3,6,5] => ([(1,2),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? = 3
[1,4,2,5,3,6] => [1,4,2,5,3,6] => ([(2,5),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> ? = 4
[1,4,2,5,6,3] => [1,4,2,5,6,3] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> ? = 4
[1,4,2,6,3,5] => [1,4,2,6,3,5] => ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> ? = 4
[1,4,2,6,5,3] => [1,4,2,6,3,5] => ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> ? = 4
[1,4,3,2,5,6] => [1,4,2,5,6,3] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> ? = 4
[1,4,3,2,6,5] => [1,4,2,6,3,5] => ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> ? = 4
[1,4,3,5,2,6] => [1,4,2,6,3,5] => ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> ? = 4
[1,4,3,6,2,5] => [1,4,2,5,3,6] => ([(2,5),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> ? = 4
[1,4,3,6,5,2] => [1,4,2,3,6,5] => ([(1,2),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? = 3
[1,4,5,2,6,3] => [1,4,5,2,6,3] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? = 4
[1,4,5,3,2,6] => [1,4,5,2,6,3] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? = 4
[1,4,6,2,5,3] => [1,4,6,2,5,3] => ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,4,6,3,2,5] => [1,4,6,2,5,3] => ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,5,2,3,6,4] => [1,5,2,3,6,4] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> ? = 3
[1,5,2,4,3,6] => [1,5,2,4,3,6] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,5,2,4,6,3] => [1,5,2,4,6,3] => ([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,5,2,6,3,4] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? = 4
[1,5,2,6,4,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? = 4
[1,5,3,2,4,6] => [1,5,2,4,6,3] => ([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,5,3,2,6,4] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? = 4
[1,5,3,4,2,6] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? = 4
[1,5,3,6,2,4] => [1,5,2,4,3,6] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,5,3,6,4,2] => [1,5,2,3,6,4] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> ? = 3
[1,5,4,2,3,6] => [1,5,2,3,6,4] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> ? = 3
[1,5,4,2,6,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? = 4
[1,5,4,3,2,6] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? = 4
[1,5,4,3,6,2] => [1,5,2,3,6,4] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> ? = 3
[1,5,6,2,4,3] => [1,5,6,2,4,3] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,5,6,3,2,4] => [1,5,6,2,4,3] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,6,2,3,5,4] => [1,6,2,3,5,4] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,6,2,4,3,5] => [1,6,2,4,3,5] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,6,2,4,5,3] => [1,6,2,4,5,3] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,6,2,5,3,4] => [1,6,2,5,3,4] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,6,2,5,4,3] => [1,6,2,5,3,4] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,6,3,2,4,5] => [1,6,2,4,5,3] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,6,3,2,5,4] => [1,6,2,5,3,4] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,6,3,4,2,5] => [1,6,2,5,3,4] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,6,3,5,2,4] => [1,6,2,4,3,5] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,6,3,5,4,2] => [1,6,2,3,5,4] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,6,4,2,3,5] => [1,6,2,3,5,4] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,6,4,2,5,3] => [1,6,2,5,3,4] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,6,4,3,2,5] => [1,6,2,5,3,4] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,6,4,3,5,2] => [1,6,2,3,5,4] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,6,5,2,4,3] => [1,6,2,4,3,5] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,6,5,3,2,4] => [1,6,2,4,3,5] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[2,1,4,3,6,5] => [1,4,2,3,6,5] => ([(1,2),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? = 3
[2,1,5,3,6,4] => [1,5,2,3,6,4] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> ? = 3
[2,1,5,4,3,6] => [1,5,2,3,6,4] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> ? = 3
[2,1,6,3,5,4] => [1,6,2,3,5,4] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[2,1,6,4,3,5] => [1,6,2,3,5,4] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[2,3,5,1,6,4] => [1,6,2,3,5,4] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[2,3,5,4,1,6] => [1,6,2,3,5,4] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[2,3,6,1,5,4] => [1,5,2,3,6,4] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> ? = 3
[2,3,6,4,1,5] => [1,5,2,3,6,4] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> ? = 3
[2,3,6,5,1,4] => [1,4,2,3,6,5] => ([(1,2),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? = 3
[2,4,1,5,3,6] => [1,5,2,4,3,6] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[2,4,1,5,6,3] => [1,5,6,2,4,3] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[2,4,1,6,3,5] => [1,6,2,4,3,5] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[2,4,1,6,5,3] => [1,6,2,4,3,5] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[2,4,3,1,5,6] => [1,5,6,2,4,3] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[2,4,3,1,6,5] => [1,6,2,4,3,5] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[2,4,3,5,1,6] => [1,6,2,4,3,5] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[2,4,3,6,1,5] => [1,5,2,4,3,6] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[2,4,3,6,5,1] => [1,2,4,3,6,5] => ([(2,5),(3,4)],6)
=> ([(0,1)],2)
=> ? = 3
[2,4,5,1,6,3] => [1,6,2,4,5,3] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[2,4,5,3,1,6] => [1,6,2,4,5,3] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[2,4,6,1,5,3] => [1,5,2,4,6,3] => ([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[2,4,6,3,1,5] => [1,5,2,4,6,3] => ([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
Description
The distinguishing index of a graph.
This is the smallest number of colours such that there is a colouring of the edges which is not preserved by any automorphism.
If the graph has a connected component which is a single edge, or at least two isolated vertices, this statistic is undefined.
Matching statistic: St001570
Mp00223: Permutations —runsort⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00154: Graphs —core⟶ Graphs
St001570: Graphs ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 50%
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00154: Graphs —core⟶ Graphs
St001570: Graphs ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 50%
Values
[1,3,2,5,4] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ([(0,1)],2)
=> ? = 3 - 3
[1,4,2,5,3] => [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ? = 4 - 3
[1,4,3,2,5] => [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ? = 4 - 3
[1,5,2,4,3] => [1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 3 - 3
[1,5,3,2,4] => [1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 3 - 3
[2,4,1,5,3] => [1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 3 - 3
[2,4,3,1,5] => [1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 3 - 3
[2,5,1,4,3] => [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ? = 4 - 3
[2,5,3,1,4] => [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ? = 4 - 3
[2,5,4,1,3] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ([(0,1)],2)
=> ? = 3 - 3
[3,1,4,2,5] => [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ? = 4 - 3
[3,1,5,2,4] => [1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 3 - 3
[3,2,4,1,5] => [1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 3 - 3
[3,2,5,1,4] => [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ? = 4 - 3
[4,1,3,2,5] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ([(0,1)],2)
=> ? = 3 - 3
[4,2,5,1,3] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ([(0,1)],2)
=> ? = 3 - 3
[1,2,4,3,6,5] => [1,2,4,3,6,5] => ([(2,5),(3,4)],6)
=> ([(0,1)],2)
=> ? = 3 - 3
[1,2,5,3,6,4] => [1,2,5,3,6,4] => ([(2,5),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> ? = 4 - 3
[1,2,5,4,3,6] => [1,2,5,3,6,4] => ([(2,5),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> ? = 4 - 3
[1,2,6,3,5,4] => [1,2,6,3,5,4] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 3 - 3
[1,2,6,4,3,5] => [1,2,6,3,5,4] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 3 - 3
[1,3,2,4,6,5] => [1,3,2,4,6,5] => ([(2,5),(3,4)],6)
=> ([(0,1)],2)
=> ? = 3 - 3
[1,3,2,5,4,6] => [1,3,2,5,4,6] => ([(2,5),(3,4)],6)
=> ([(0,1)],2)
=> ? = 3 - 3
[1,3,2,5,6,4] => [1,3,2,5,6,4] => ([(1,2),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? = 3 - 3
[1,3,2,6,4,5] => [1,3,2,6,4,5] => ([(1,2),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? = 3 - 3
[1,3,2,6,5,4] => [1,3,2,6,4,5] => ([(1,2),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? = 3 - 3
[1,3,4,2,6,5] => [1,3,4,2,6,5] => ([(1,2),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? = 3 - 3
[1,3,5,2,6,4] => [1,3,5,2,6,4] => ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> ? = 3 - 3
[1,3,5,4,2,6] => [1,3,5,2,6,4] => ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> ? = 3 - 3
[1,3,6,2,5,4] => [1,3,6,2,5,4] => ([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 3 - 3
[1,3,6,4,2,5] => [1,3,6,2,5,4] => ([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 3 - 3
[1,4,2,3,6,5] => [1,4,2,3,6,5] => ([(1,2),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? = 3 - 3
[1,4,2,5,3,6] => [1,4,2,5,3,6] => ([(2,5),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> ? = 4 - 3
[1,4,2,5,6,3] => [1,4,2,5,6,3] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> ? = 4 - 3
[1,4,2,6,3,5] => [1,4,2,6,3,5] => ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> ? = 4 - 3
[1,4,2,6,5,3] => [1,4,2,6,3,5] => ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> ? = 4 - 3
[1,4,3,2,5,6] => [1,4,2,5,6,3] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> ? = 4 - 3
[1,4,3,2,6,5] => [1,4,2,6,3,5] => ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> ? = 4 - 3
[1,4,3,5,2,6] => [1,4,2,6,3,5] => ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> ? = 4 - 3
[1,4,3,6,2,5] => [1,4,2,5,3,6] => ([(2,5),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> ? = 4 - 3
[1,4,3,6,5,2] => [1,4,2,3,6,5] => ([(1,2),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? = 3 - 3
[1,4,5,2,6,3] => [1,4,5,2,6,3] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? = 4 - 3
[1,4,5,3,2,6] => [1,4,5,2,6,3] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? = 4 - 3
[1,4,6,2,5,3] => [1,4,6,2,5,3] => ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 3 - 3
[1,4,6,3,2,5] => [1,4,6,2,5,3] => ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 3 - 3
[1,5,2,3,6,4] => [1,5,2,3,6,4] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> ? = 3 - 3
[1,5,2,4,3,6] => [1,5,2,4,3,6] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 3 - 3
[1,5,2,4,6,3] => [1,5,2,4,6,3] => ([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 3 - 3
[1,5,2,6,3,4] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? = 4 - 3
[1,5,2,6,4,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? = 4 - 3
[1,5,3,2,4,6] => [1,5,2,4,6,3] => ([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 3 - 3
[1,5,3,2,6,4] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? = 4 - 3
[1,5,3,4,2,6] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? = 4 - 3
[1,5,3,6,2,4] => [1,5,2,4,3,6] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 3 - 3
[1,5,3,6,4,2] => [1,5,2,3,6,4] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> ? = 3 - 3
[1,5,4,2,3,6] => [1,5,2,3,6,4] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> ? = 3 - 3
[1,5,4,2,6,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? = 4 - 3
[1,5,4,3,2,6] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? = 4 - 3
[1,5,4,3,6,2] => [1,5,2,3,6,4] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> ? = 3 - 3
[1,5,6,2,4,3] => [1,5,6,2,4,3] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 3 - 3
[1,5,6,3,2,4] => [1,5,6,2,4,3] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 3 - 3
[1,6,2,3,5,4] => [1,6,2,3,5,4] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 3 - 3
[1,6,2,4,3,5] => [1,6,2,4,3,5] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 3 - 3
[1,6,2,4,5,3] => [1,6,2,4,5,3] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 3 - 3
[1,6,2,5,3,4] => [1,6,2,5,3,4] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 3 - 3
[1,6,2,5,4,3] => [1,6,2,5,3,4] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 3 - 3
[1,6,3,2,4,5] => [1,6,2,4,5,3] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 3 - 3
[1,6,3,2,5,4] => [1,6,2,5,3,4] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 3 - 3
[1,6,3,4,2,5] => [1,6,2,5,3,4] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 3 - 3
[1,6,3,5,2,4] => [1,6,2,4,3,5] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 3 - 3
[1,6,3,5,4,2] => [1,6,2,3,5,4] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 3 - 3
[1,6,4,2,3,5] => [1,6,2,3,5,4] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 3 - 3
[1,6,4,2,5,3] => [1,6,2,5,3,4] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 3 - 3
[1,6,4,3,2,5] => [1,6,2,5,3,4] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 3 - 3
[1,6,4,3,5,2] => [1,6,2,3,5,4] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 3 - 3
[1,6,5,2,4,3] => [1,6,2,4,3,5] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 3 - 3
[1,6,5,3,2,4] => [1,6,2,4,3,5] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 3 - 3
[2,1,4,3,6,5] => [1,4,2,3,6,5] => ([(1,2),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? = 3 - 3
[2,1,5,3,6,4] => [1,5,2,3,6,4] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> ? = 3 - 3
[2,1,5,4,3,6] => [1,5,2,3,6,4] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> ? = 3 - 3
[2,1,6,3,5,4] => [1,6,2,3,5,4] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 3 - 3
[2,1,6,4,3,5] => [1,6,2,3,5,4] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 3 - 3
[2,3,5,1,6,4] => [1,6,2,3,5,4] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 3 - 3
[2,3,5,4,1,6] => [1,6,2,3,5,4] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 3 - 3
[2,3,6,1,5,4] => [1,5,2,3,6,4] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> ? = 3 - 3
[2,3,6,4,1,5] => [1,5,2,3,6,4] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> ? = 3 - 3
[2,3,6,5,1,4] => [1,4,2,3,6,5] => ([(1,2),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? = 3 - 3
[2,4,1,5,3,6] => [1,5,2,4,3,6] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 3 - 3
[2,4,1,5,6,3] => [1,5,6,2,4,3] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 3 - 3
[2,4,1,6,3,5] => [1,6,2,4,3,5] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 3 - 3
[2,4,1,6,5,3] => [1,6,2,4,3,5] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 3 - 3
[2,4,3,1,5,6] => [1,5,6,2,4,3] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 3 - 3
[2,4,3,1,6,5] => [1,6,2,4,3,5] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 3 - 3
[2,4,3,5,1,6] => [1,6,2,4,3,5] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 3 - 3
[2,4,3,6,1,5] => [1,5,2,4,3,6] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 3 - 3
[2,4,3,6,5,1] => [1,2,4,3,6,5] => ([(2,5),(3,4)],6)
=> ([(0,1)],2)
=> ? = 3 - 3
[2,4,5,1,6,3] => [1,6,2,4,5,3] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 3 - 3
[2,4,5,3,1,6] => [1,6,2,4,5,3] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 3 - 3
[2,4,6,1,5,3] => [1,5,2,4,6,3] => ([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 3 - 3
[2,4,6,3,1,5] => [1,5,2,4,6,3] => ([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 3 - 3
Description
The minimal number of edges to add to make a graph Hamiltonian.
A graph is Hamiltonian if it contains a cycle as a subgraph, which contains all vertices.
The following 38 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000298The order dimension or Dushnik-Miller dimension of a poset. St000307The number of rowmotion orbits of a poset. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St001632The number of indecomposable injective modules I with dimExt1(I,A)=1 for the incidence algebra A of a poset. St000632The jump number of the poset. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001651The Frankl number of a lattice. St000454The largest eigenvalue of a graph if it is integral. St001644The dimension of a graph. St000422The energy of a graph, if it is integral. St000888The maximal sum of entries on a diagonal of an alternating sign matrix. St000892The maximal number of nonzero entries on a diagonal of an alternating sign matrix. St000100The number of linear extensions of a poset. St000007The number of saliances of the permutation. St000214The number of adjacencies of a permutation. St000215The number of adjacencies of a permutation, zero appended. St000366The number of double descents of a permutation. St001115The number of even descents of a permutation. St001645The pebbling number of a connected graph. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St000068The number of minimal elements in a poset. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000909The number of maximal chains of maximal size in a poset. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St001616The number of neutral elements in a lattice. St001720The minimal length of a chain of small intervals in a lattice. St001613The binary logarithm of the size of the center of a lattice. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001820The size of the image of the pop stack sorting operator. St001881The number of factors of a lattice as a Cartesian product of lattices. St001846The number of elements which do not have a complement in the lattice.
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