Your data matches 4 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000259
Mapping
Mp00223: Permutations —runsort⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000259: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => ([],1)
 => 0
[1,3,2] => [1,3,2] => [3,1,2] => ([(0,2),(1,2)],3)
 => 2
[2,1,3] => [1,3,2] => [3,1,2] => ([(0,2),(1,2)],3)
 => 2
[1,2,4,3] => [1,2,4,3] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 2
[1,3,4,2] => [1,3,4,2] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 2
[2,1,3,4] => [1,3,4,2] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 2
[2,4,3,1] => [1,2,4,3] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 2
[3,1,2,4] => [1,2,4,3] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 2
[3,2,4,1] => [1,2,4,3] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 2
[1,2,3,5,4] => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
[1,2,4,5,3] => [1,2,4,5,3] => [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 2
[1,3,2,5,4] => [1,3,2,5,4] => [3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 3
[1,3,4,5,2] => [1,3,4,5,2] => [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 2
[1,3,5,2,4] => [1,3,5,2,4] => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,3,5,4,2] => [1,3,5,2,4] => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,4,5,2,3] => [1,4,5,2,3] => [4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 3
[1,4,5,3,2] => [1,4,5,2,3] => [4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 3
[1,5,2,4,3] => [1,5,2,4,3] => [5,1,4,2,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,5,3,2,4] => [1,5,2,4,3] => [5,1,4,2,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[2,1,3,4,5] => [1,3,4,5,2] => [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 2
[2,1,3,5,4] => [1,3,5,2,4] => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[2,1,4,5,3] => [1,4,5,2,3] => [4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 3
[2,3,1,4,5] => [1,4,5,2,3] => [4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 3
[2,3,5,4,1] => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
[2,4,1,3,5] => [1,3,5,2,4] => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[2,4,1,5,3] => [1,5,2,4,3] => [5,1,4,2,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[2,4,3,1,5] => [1,5,2,4,3] => [5,1,4,2,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[2,4,5,3,1] => [1,2,4,5,3] => [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 2
[2,5,4,1,3] => [1,3,2,5,4] => [3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 3
[3,1,2,4,5] => [1,2,4,5,3] => [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 2
[3,1,4,5,2] => [1,4,5,2,3] => [4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 3
[3,1,5,2,4] => [1,5,2,4,3] => [5,1,4,2,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[3,2,1,4,5] => [1,4,5,2,3] => [4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 3
[3,2,4,1,5] => [1,5,2,4,3] => [5,1,4,2,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[3,2,4,5,1] => [1,2,4,5,3] => [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 2
[3,5,4,1,2] => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
[3,5,4,2,1] => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
[4,1,2,3,5] => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
[4,1,3,2,5] => [1,3,2,5,4] => [3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 3
[4,1,3,5,2] => [1,3,5,2,4] => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[4,2,1,3,5] => [1,3,5,2,4] => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[4,2,3,5,1] => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
[4,2,5,1,3] => [1,3,2,5,4] => [3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 3
[4,3,5,1,2] => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
[4,3,5,2,1] => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
[1,2,3,4,6,5] => [1,2,3,4,6,5] => [6,1,2,3,4,5] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 2
[1,2,3,5,6,4] => [1,2,3,5,6,4] => [5,6,1,2,3,4] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => 2
[1,2,4,3,6,5] => [1,2,4,3,6,5] => [4,6,1,2,3,5] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => 3
[1,2,4,5,6,3] => [1,2,4,5,6,3] => [4,5,6,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => 2
[1,2,4,6,3,5] => [1,2,4,6,3,5] => [6,4,1,2,3,5] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
Description
The diameter of a connected graph. This is the greatest distance between any pair of vertices.
Matching statistic: St000741
Mapping
Mp00223: Permutations —runsort⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000741: Graphs ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 50%
Values
[1] => [1] => [1] => ([],1)
 => 0
[1,3,2] => [1,3,2] => [3,1,2] => ([(0,2),(1,2)],3)
 => 2
[2,1,3] => [1,3,2] => [3,1,2] => ([(0,2),(1,2)],3)
 => 2
[1,2,4,3] => [1,2,4,3] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 2
[1,3,4,2] => [1,3,4,2] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ? = 2
[2,1,3,4] => [1,3,4,2] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ? = 2
[2,4,3,1] => [1,2,4,3] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 2
[3,1,2,4] => [1,2,4,3] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 2
[3,2,4,1] => [1,2,4,3] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 2
[1,2,3,5,4] => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
[1,2,4,5,3] => [1,2,4,5,3] => [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ? = 2
[1,3,2,5,4] => [1,3,2,5,4] => [3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ? = 3
[1,3,4,5,2] => [1,3,4,5,2] => [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ? = 2
[1,3,5,2,4] => [1,3,5,2,4] => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,3,5,4,2] => [1,3,5,2,4] => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,4,5,2,3] => [1,4,5,2,3] => [4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ? = 3
[1,4,5,3,2] => [1,4,5,2,3] => [4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ? = 3
[1,5,2,4,3] => [1,5,2,4,3] => [5,1,4,2,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,5,3,2,4] => [1,5,2,4,3] => [5,1,4,2,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[2,1,3,4,5] => [1,3,4,5,2] => [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ? = 2
[2,1,3,5,4] => [1,3,5,2,4] => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[2,1,4,5,3] => [1,4,5,2,3] => [4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ? = 3
[2,3,1,4,5] => [1,4,5,2,3] => [4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ? = 3
[2,3,5,4,1] => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
[2,4,1,3,5] => [1,3,5,2,4] => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[2,4,1,5,3] => [1,5,2,4,3] => [5,1,4,2,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[2,4,3,1,5] => [1,5,2,4,3] => [5,1,4,2,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[2,4,5,3,1] => [1,2,4,5,3] => [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ? = 2
[2,5,4,1,3] => [1,3,2,5,4] => [3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ? = 3
[3,1,2,4,5] => [1,2,4,5,3] => [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ? = 2
[3,1,4,5,2] => [1,4,5,2,3] => [4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ? = 3
[3,1,5,2,4] => [1,5,2,4,3] => [5,1,4,2,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[3,2,1,4,5] => [1,4,5,2,3] => [4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ? = 3
[3,2,4,1,5] => [1,5,2,4,3] => [5,1,4,2,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[3,2,4,5,1] => [1,2,4,5,3] => [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ? = 2
[3,5,4,1,2] => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
[3,5,4,2,1] => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
[4,1,2,3,5] => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
[4,1,3,2,5] => [1,3,2,5,4] => [3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ? = 3
[4,1,3,5,2] => [1,3,5,2,4] => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[4,2,1,3,5] => [1,3,5,2,4] => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[4,2,3,5,1] => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
[4,2,5,1,3] => [1,3,2,5,4] => [3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ? = 3
[4,3,5,1,2] => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
[4,3,5,2,1] => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
[1,2,3,4,6,5] => [1,2,3,4,6,5] => [6,1,2,3,4,5] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 2
[1,2,3,5,6,4] => [1,2,3,5,6,4] => [5,6,1,2,3,4] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ? = 2
[1,2,4,3,6,5] => [1,2,4,3,6,5] => [4,6,1,2,3,5] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ? = 3
[1,2,4,5,6,3] => [1,2,4,5,6,3] => [4,5,6,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => ? = 2
[1,2,4,6,3,5] => [1,2,4,6,3,5] => [6,4,1,2,3,5] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2
[1,2,4,6,5,3] => [1,2,4,6,3,5] => [6,4,1,2,3,5] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2
[1,2,5,6,3,4] => [1,2,5,6,3,4] => [5,1,6,2,3,4] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ? = 3
[1,2,5,6,4,3] => [1,2,5,6,3,4] => [5,1,6,2,3,4] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ? = 3
[1,2,6,3,5,4] => [1,2,6,3,5,4] => [6,1,5,2,3,4] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2
[1,2,6,4,3,5] => [1,2,6,3,5,4] => [6,1,5,2,3,4] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2
[1,3,2,4,6,5] => [1,3,2,4,6,5] => [3,6,1,2,4,5] => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ? = 3
[1,3,2,5,6,4] => [1,3,2,5,6,4] => [3,5,6,1,2,4] => ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ? = 3
[1,3,2,6,4,5] => [1,3,2,6,4,5] => [6,3,1,2,4,5] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
[1,3,2,6,5,4] => [1,3,2,6,4,5] => [6,3,1,2,4,5] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
[1,3,4,2,6,5] => [1,3,4,2,6,5] => [3,4,6,1,2,5] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ? = 3
[1,3,4,5,6,2] => [1,3,4,5,6,2] => [3,4,5,6,1,2] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ? = 2
[1,3,4,6,2,5] => [1,3,4,6,2,5] => [6,3,4,1,2,5] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ? = 2
[1,3,4,6,5,2] => [1,3,4,6,2,5] => [6,3,4,1,2,5] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ? = 2
[1,3,5,2,6,4] => [1,3,5,2,6,4] => [5,3,6,1,2,4] => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ? = 2
[1,3,5,4,2,6] => [1,3,5,2,6,4] => [5,3,6,1,2,4] => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ? = 2
[1,3,5,6,2,4] => [1,3,5,6,2,4] => [5,6,3,1,2,4] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ? = 2
[1,3,5,6,4,2] => [1,3,5,6,2,4] => [5,6,3,1,2,4] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ? = 2
[1,3,6,2,4,5] => [1,3,6,2,4,5] => [6,1,3,2,4,5] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
[1,3,6,2,5,4] => [1,3,6,2,5,4] => [3,6,5,1,2,4] => ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3
[1,3,6,4,2,5] => [1,3,6,2,5,4] => [3,6,5,1,2,4] => ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3
[1,3,6,4,5,2] => [1,3,6,2,4,5] => [6,1,3,2,4,5] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
[1,3,6,5,2,4] => [1,3,6,2,4,5] => [6,1,3,2,4,5] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
[1,3,6,5,4,2] => [1,3,6,2,4,5] => [6,1,3,2,4,5] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
[1,4,2,6,3,5] => [1,4,2,6,3,5] => [4,1,6,2,3,5] => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ? = 4
[1,4,2,6,5,3] => [1,4,2,6,3,5] => [4,1,6,2,3,5] => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ? = 4
[1,4,3,2,6,5] => [1,4,2,6,3,5] => [4,1,6,2,3,5] => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ? = 4
[1,4,3,5,2,6] => [1,4,2,6,3,5] => [4,1,6,2,3,5] => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ? = 4
[1,4,5,2,6,3] => [1,4,5,2,6,3] => [4,1,5,6,2,3] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ? = 3
[1,4,5,3,2,6] => [1,4,5,2,6,3] => [4,1,5,6,2,3] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ? = 3
[1,4,5,6,2,3] => [1,4,5,6,2,3] => [4,5,1,6,2,3] => ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ? = 3
[1,4,5,6,3,2] => [1,4,5,6,2,3] => [4,5,1,6,2,3] => ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ? = 3
[1,4,6,2,5,3] => [1,4,6,2,5,3] => [4,6,1,5,2,3] => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ? = 2
[1,4,6,3,2,5] => [1,4,6,2,5,3] => [4,6,1,5,2,3] => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ? = 2
[1,5,2,4,6,3] => [1,5,2,4,6,3] => [5,1,4,6,2,3] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ? = 3
[1,6,2,3,5,4] => [1,6,2,3,5,4] => [6,1,2,5,3,4] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
[1,6,2,4,3,5] => [1,6,2,4,3,5] => [6,1,2,4,3,5] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
[1,6,3,5,2,4] => [1,6,2,4,3,5] => [6,1,2,4,3,5] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
[1,6,3,5,4,2] => [1,6,2,3,5,4] => [6,1,2,5,3,4] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
[1,6,4,2,3,5] => [1,6,2,3,5,4] => [6,1,2,5,3,4] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
[1,6,4,3,5,2] => [1,6,2,3,5,4] => [6,1,2,5,3,4] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
[1,6,5,2,4,3] => [1,6,2,4,3,5] => [6,1,2,4,3,5] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
[1,6,5,3,2,4] => [1,6,2,4,3,5] => [6,1,2,4,3,5] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
[2,1,3,6,4,5] => [1,3,6,2,4,5] => [6,1,3,2,4,5] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
[2,1,3,6,5,4] => [1,3,6,2,4,5] => [6,1,3,2,4,5] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
[2,1,6,3,5,4] => [1,6,2,3,5,4] => [6,1,2,5,3,4] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
[2,1,6,4,3,5] => [1,6,2,3,5,4] => [6,1,2,5,3,4] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
[2,3,4,6,5,1] => [1,2,3,4,6,5] => [6,1,2,3,4,5] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 2
[2,3,5,1,6,4] => [1,6,2,3,5,4] => [6,1,2,5,3,4] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
[2,3,5,4,1,6] => [1,6,2,3,5,4] => [6,1,2,5,3,4] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
[2,4,1,3,6,5] => [1,3,6,2,4,5] => [6,1,3,2,4,5] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
Description
The Colin de Verdière graph invariant.
Matching statistic: St001811
Mapping
Mp00223: Permutations —runsort⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
St001811: Permutations ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 50%
Values
[1] => [1] => [1] => [1] => ? = 0 - 2
[1,3,2] => [1,3,2] => [3,1,2] => [2,3,1] => 0 = 2 - 2
[2,1,3] => [1,3,2] => [3,1,2] => [2,3,1] => 0 = 2 - 2
[1,2,4,3] => [1,2,4,3] => [4,3,1,2] => [3,4,2,1] => 0 = 2 - 2
[1,3,4,2] => [1,3,4,2] => [4,2,1,3] => [3,2,4,1] => 0 = 2 - 2
[2,1,3,4] => [1,3,4,2] => [4,2,1,3] => [3,2,4,1] => 0 = 2 - 2
[2,4,3,1] => [1,2,4,3] => [4,3,1,2] => [3,4,2,1] => 0 = 2 - 2
[3,1,2,4] => [1,2,4,3] => [4,3,1,2] => [3,4,2,1] => 0 = 2 - 2
[3,2,4,1] => [1,2,4,3] => [4,3,1,2] => [3,4,2,1] => 0 = 2 - 2
[1,2,3,5,4] => [1,2,3,5,4] => [5,4,3,1,2] => [4,5,3,2,1] => 0 = 2 - 2
[1,2,4,5,3] => [1,2,4,5,3] => [5,4,2,1,3] => [4,3,5,2,1] => 0 = 2 - 2
[1,3,2,5,4] => [1,3,2,5,4] => [5,3,4,1,2] => [4,2,5,3,1] => 1 = 3 - 2
[1,3,4,5,2] => [1,3,4,5,2] => [5,3,2,1,4] => [4,3,2,5,1] => 0 = 2 - 2
[1,3,5,2,4] => [1,3,5,2,4] => [5,3,1,4,2] => [5,3,2,4,1] => 0 = 2 - 2
[1,3,5,4,2] => [1,3,5,2,4] => [5,3,1,4,2] => [5,3,2,4,1] => 0 = 2 - 2
[1,4,5,2,3] => [1,4,5,2,3] => [5,2,1,4,3] => [2,5,3,4,1] => 1 = 3 - 2
[1,4,5,3,2] => [1,4,5,2,3] => [5,2,1,4,3] => [2,5,3,4,1] => 1 = 3 - 2
[1,5,2,4,3] => [1,5,2,4,3] => [5,1,4,2,3] => [4,5,2,3,1] => 0 = 2 - 2
[1,5,3,2,4] => [1,5,2,4,3] => [5,1,4,2,3] => [4,5,2,3,1] => 0 = 2 - 2
[2,1,3,4,5] => [1,3,4,5,2] => [5,3,2,1,4] => [4,3,2,5,1] => 0 = 2 - 2
[2,1,3,5,4] => [1,3,5,2,4] => [5,3,1,4,2] => [5,3,2,4,1] => 0 = 2 - 2
[2,1,4,5,3] => [1,4,5,2,3] => [5,2,1,4,3] => [2,5,3,4,1] => 1 = 3 - 2
[2,3,1,4,5] => [1,4,5,2,3] => [5,2,1,4,3] => [2,5,3,4,1] => 1 = 3 - 2
[2,3,5,4,1] => [1,2,3,5,4] => [5,4,3,1,2] => [4,5,3,2,1] => 0 = 2 - 2
[2,4,1,3,5] => [1,3,5,2,4] => [5,3,1,4,2] => [5,3,2,4,1] => 0 = 2 - 2
[2,4,1,5,3] => [1,5,2,4,3] => [5,1,4,2,3] => [4,5,2,3,1] => 0 = 2 - 2
[2,4,3,1,5] => [1,5,2,4,3] => [5,1,4,2,3] => [4,5,2,3,1] => 0 = 2 - 2
[2,4,5,3,1] => [1,2,4,5,3] => [5,4,2,1,3] => [4,3,5,2,1] => 0 = 2 - 2
[2,5,4,1,3] => [1,3,2,5,4] => [5,3,4,1,2] => [4,2,5,3,1] => 1 = 3 - 2
[3,1,2,4,5] => [1,2,4,5,3] => [5,4,2,1,3] => [4,3,5,2,1] => 0 = 2 - 2
[3,1,4,5,2] => [1,4,5,2,3] => [5,2,1,4,3] => [2,5,3,4,1] => 1 = 3 - 2
[3,1,5,2,4] => [1,5,2,4,3] => [5,1,4,2,3] => [4,5,2,3,1] => 0 = 2 - 2
[3,2,1,4,5] => [1,4,5,2,3] => [5,2,1,4,3] => [2,5,3,4,1] => 1 = 3 - 2
[3,2,4,1,5] => [1,5,2,4,3] => [5,1,4,2,3] => [4,5,2,3,1] => 0 = 2 - 2
[3,2,4,5,1] => [1,2,4,5,3] => [5,4,2,1,3] => [4,3,5,2,1] => 0 = 2 - 2
[3,5,4,1,2] => [1,2,3,5,4] => [5,4,3,1,2] => [4,5,3,2,1] => 0 = 2 - 2
[3,5,4,2,1] => [1,2,3,5,4] => [5,4,3,1,2] => [4,5,3,2,1] => 0 = 2 - 2
[4,1,2,3,5] => [1,2,3,5,4] => [5,4,3,1,2] => [4,5,3,2,1] => 0 = 2 - 2
[4,1,3,2,5] => [1,3,2,5,4] => [5,3,4,1,2] => [4,2,5,3,1] => 1 = 3 - 2
[4,1,3,5,2] => [1,3,5,2,4] => [5,3,1,4,2] => [5,3,2,4,1] => 0 = 2 - 2
[4,2,1,3,5] => [1,3,5,2,4] => [5,3,1,4,2] => [5,3,2,4,1] => 0 = 2 - 2
[4,2,3,5,1] => [1,2,3,5,4] => [5,4,3,1,2] => [4,5,3,2,1] => 0 = 2 - 2
[4,2,5,1,3] => [1,3,2,5,4] => [5,3,4,1,2] => [4,2,5,3,1] => 1 = 3 - 2
[4,3,5,1,2] => [1,2,3,5,4] => [5,4,3,1,2] => [4,5,3,2,1] => 0 = 2 - 2
[4,3,5,2,1] => [1,2,3,5,4] => [5,4,3,1,2] => [4,5,3,2,1] => 0 = 2 - 2
[1,2,3,4,6,5] => [1,2,3,4,6,5] => [6,5,4,3,1,2] => [5,6,4,3,2,1] => ? = 2 - 2
[1,2,3,5,6,4] => [1,2,3,5,6,4] => [6,5,4,2,1,3] => [5,4,6,3,2,1] => ? = 2 - 2
[1,2,4,3,6,5] => [1,2,4,3,6,5] => [6,5,3,4,1,2] => [5,3,6,4,2,1] => ? = 3 - 2
[1,2,4,5,6,3] => [1,2,4,5,6,3] => [6,5,3,2,1,4] => [5,4,3,6,2,1] => ? = 2 - 2
[1,2,4,6,3,5] => [1,2,4,6,3,5] => [6,5,3,1,4,2] => [6,4,3,5,2,1] => ? = 2 - 2
[1,2,4,6,5,3] => [1,2,4,6,3,5] => [6,5,3,1,4,2] => [6,4,3,5,2,1] => ? = 2 - 2
[1,2,5,6,3,4] => [1,2,5,6,3,4] => [6,5,2,1,4,3] => [3,6,4,5,2,1] => ? = 3 - 2
[1,2,5,6,4,3] => [1,2,5,6,3,4] => [6,5,2,1,4,3] => [3,6,4,5,2,1] => ? = 3 - 2
[1,2,6,3,5,4] => [1,2,6,3,5,4] => [6,5,1,4,2,3] => [5,6,3,4,2,1] => ? = 2 - 2
[1,2,6,4,3,5] => [1,2,6,3,5,4] => [6,5,1,4,2,3] => [5,6,3,4,2,1] => ? = 2 - 2
[1,3,2,4,6,5] => [1,3,2,4,6,5] => [6,4,5,3,1,2] => [5,2,6,4,3,1] => ? = 3 - 2
[1,3,2,5,6,4] => [1,3,2,5,6,4] => [6,4,5,2,1,3] => [5,4,2,6,3,1] => ? = 3 - 2
[1,3,2,6,4,5] => [1,3,2,6,4,5] => [6,4,5,1,3,2] => [6,4,2,5,3,1] => ? = 2 - 2
[1,3,2,6,5,4] => [1,3,2,6,4,5] => [6,4,5,1,3,2] => [6,4,2,5,3,1] => ? = 2 - 2
[1,3,4,2,6,5] => [1,3,4,2,6,5] => [6,4,3,5,1,2] => [5,3,2,6,4,1] => ? = 3 - 2
[1,3,4,5,6,2] => [1,3,4,5,6,2] => [6,4,3,2,1,5] => [5,4,3,2,6,1] => ? = 2 - 2
[1,3,4,6,2,5] => [1,3,4,6,2,5] => [6,4,3,1,5,2] => [6,4,3,2,5,1] => ? = 2 - 2
[1,3,4,6,5,2] => [1,3,4,6,2,5] => [6,4,3,1,5,2] => [6,4,3,2,5,1] => ? = 2 - 2
[1,3,5,2,6,4] => [1,3,5,2,6,4] => [6,4,2,5,1,3] => [3,5,2,6,4,1] => ? = 2 - 2
[1,3,5,4,2,6] => [1,3,5,2,6,4] => [6,4,2,5,1,3] => [3,5,2,6,4,1] => ? = 2 - 2
[1,3,5,6,2,4] => [1,3,5,6,2,4] => [6,4,2,1,5,3] => [3,6,4,2,5,1] => ? = 2 - 2
[1,3,5,6,4,2] => [1,3,5,6,2,4] => [6,4,2,1,5,3] => [3,6,4,2,5,1] => ? = 2 - 2
[1,3,6,2,4,5] => [1,3,6,2,4,5] => [6,4,1,5,3,2] => [6,5,3,2,4,1] => ? = 2 - 2
[1,3,6,2,5,4] => [1,3,6,2,5,4] => [6,4,1,5,2,3] => [5,6,3,2,4,1] => ? = 3 - 2
[1,3,6,4,2,5] => [1,3,6,2,5,4] => [6,4,1,5,2,3] => [5,6,3,2,4,1] => ? = 3 - 2
[1,3,6,4,5,2] => [1,3,6,2,4,5] => [6,4,1,5,3,2] => [6,5,3,2,4,1] => ? = 2 - 2
[1,3,6,5,2,4] => [1,3,6,2,4,5] => [6,4,1,5,3,2] => [6,5,3,2,4,1] => ? = 2 - 2
[1,3,6,5,4,2] => [1,3,6,2,4,5] => [6,4,1,5,3,2] => [6,5,3,2,4,1] => ? = 2 - 2
[1,4,2,6,3,5] => [1,4,2,6,3,5] => [6,3,5,1,4,2] => [2,6,4,5,3,1] => ? = 4 - 2
[1,4,2,6,5,3] => [1,4,2,6,3,5] => [6,3,5,1,4,2] => [2,6,4,5,3,1] => ? = 4 - 2
[1,4,3,2,6,5] => [1,4,2,6,3,5] => [6,3,5,1,4,2] => [2,6,4,5,3,1] => ? = 4 - 2
[1,4,3,5,2,6] => [1,4,2,6,3,5] => [6,3,5,1,4,2] => [2,6,4,5,3,1] => ? = 4 - 2
[1,4,5,2,6,3] => [1,4,5,2,6,3] => [6,3,2,5,1,4] => [3,2,5,6,4,1] => ? = 3 - 2
[1,4,5,3,2,6] => [1,4,5,2,6,3] => [6,3,2,5,1,4] => [3,2,5,6,4,1] => ? = 3 - 2
[1,4,5,6,2,3] => [1,4,5,6,2,3] => [6,3,2,1,5,4] => [3,2,6,4,5,1] => ? = 3 - 2
[1,4,5,6,3,2] => [1,4,5,6,2,3] => [6,3,2,1,5,4] => [3,2,6,4,5,1] => ? = 3 - 2
[1,4,6,2,5,3] => [1,4,6,2,5,3] => [6,3,1,5,2,4] => [5,2,6,3,4,1] => ? = 2 - 2
[1,4,6,3,2,5] => [1,4,6,2,5,3] => [6,3,1,5,2,4] => [5,2,6,3,4,1] => ? = 2 - 2
[1,5,2,4,6,3] => [1,5,2,4,6,3] => [6,2,5,3,1,4] => [5,2,4,6,3,1] => ? = 3 - 2
[1,5,2,6,3,4] => [1,5,2,6,3,4] => [6,2,5,1,4,3] => [6,2,4,5,3,1] => ? = 3 - 2
[1,5,2,6,4,3] => [1,5,2,6,3,4] => [6,2,5,1,4,3] => [6,2,4,5,3,1] => ? = 3 - 2
[1,5,3,2,4,6] => [1,5,2,4,6,3] => [6,2,5,3,1,4] => [5,2,4,6,3,1] => ? = 3 - 2
[1,5,3,2,6,4] => [1,5,2,6,3,4] => [6,2,5,1,4,3] => [6,2,4,5,3,1] => ? = 3 - 2
[1,5,3,4,2,6] => [1,5,2,6,3,4] => [6,2,5,1,4,3] => [6,2,4,5,3,1] => ? = 3 - 2
[1,5,4,2,6,3] => [1,5,2,6,3,4] => [6,2,5,1,4,3] => [6,2,4,5,3,1] => ? = 3 - 2
[1,5,4,3,2,6] => [1,5,2,6,3,4] => [6,2,5,1,4,3] => [6,2,4,5,3,1] => ? = 3 - 2
[1,6,2,3,5,4] => [1,6,2,3,5,4] => [6,1,5,4,2,3] => [5,6,4,2,3,1] => ? = 2 - 2
[1,6,2,4,3,5] => [1,6,2,4,3,5] => [6,1,5,3,4,2] => [4,6,5,2,3,1] => ? = 2 - 2
[1,6,3,5,2,4] => [1,6,2,4,3,5] => [6,1,5,3,4,2] => [4,6,5,2,3,1] => ? = 2 - 2
Description
The Castelnuovo-Mumford regularity of a permutation. The ''Castelnuovo-Mumford regularity'' of a permutation $\sigma$ is the ''Castelnuovo-Mumford regularity'' of the ''matrix Schubert variety'' $X_\sigma$. Equivalently, it is the difference between the degrees of the ''Grothendieck polynomial'' and the ''Schubert polynomial'' for $\sigma$. It can be computed by subtracting the ''Coxeter length'' [[St000018]] from the ''Rajchgot index'' [[St001759]].
Matching statistic: St001948
Mapping
Mp00223: Permutations —runsort⟶ Permutations
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
St001948: Permutations ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 50%
Values
[1] => [1] => [1] => [1] => ? = 0 - 2
[1,3,2] => [1,3,2] => [2,1,3] => [2,1,3] => 0 = 2 - 2
[2,1,3] => [1,3,2] => [2,1,3] => [2,1,3] => 0 = 2 - 2
[1,2,4,3] => [1,2,4,3] => [2,3,1,4] => [3,2,1,4] => 0 = 2 - 2
[1,3,4,2] => [1,3,4,2] => [2,4,1,3] => [4,2,1,3] => 0 = 2 - 2
[2,1,3,4] => [1,3,4,2] => [2,4,1,3] => [4,2,1,3] => 0 = 2 - 2
[2,4,3,1] => [1,2,4,3] => [2,3,1,4] => [3,2,1,4] => 0 = 2 - 2
[3,1,2,4] => [1,2,4,3] => [2,3,1,4] => [3,2,1,4] => 0 = 2 - 2
[3,2,4,1] => [1,2,4,3] => [2,3,1,4] => [3,2,1,4] => 0 = 2 - 2
[1,2,3,5,4] => [1,2,3,5,4] => [2,3,4,1,5] => [4,3,2,1,5] => 0 = 2 - 2
[1,2,4,5,3] => [1,2,4,5,3] => [2,3,5,1,4] => [5,3,2,1,4] => 0 = 2 - 2
[1,3,2,5,4] => [1,3,2,5,4] => [2,4,3,1,5] => [3,4,2,1,5] => 1 = 3 - 2
[1,3,4,5,2] => [1,3,4,5,2] => [2,4,5,1,3] => [5,2,1,4,3] => 0 = 2 - 2
[1,3,5,2,4] => [1,3,5,2,4] => [2,4,1,5,3] => [5,4,2,1,3] => 0 = 2 - 2
[1,3,5,4,2] => [1,3,5,2,4] => [2,4,1,5,3] => [5,4,2,1,3] => 0 = 2 - 2
[1,4,5,2,3] => [1,4,5,2,3] => [2,5,1,4,3] => [4,5,2,1,3] => 1 = 3 - 2
[1,4,5,3,2] => [1,4,5,2,3] => [2,5,1,4,3] => [4,5,2,1,3] => 1 = 3 - 2
[1,5,2,4,3] => [1,5,2,4,3] => [2,1,4,3,5] => [2,1,4,3,5] => 0 = 2 - 2
[1,5,3,2,4] => [1,5,2,4,3] => [2,1,4,3,5] => [2,1,4,3,5] => 0 = 2 - 2
[2,1,3,4,5] => [1,3,4,5,2] => [2,4,5,1,3] => [5,2,1,4,3] => 0 = 2 - 2
[2,1,3,5,4] => [1,3,5,2,4] => [2,4,1,5,3] => [5,4,2,1,3] => 0 = 2 - 2
[2,1,4,5,3] => [1,4,5,2,3] => [2,5,1,4,3] => [4,5,2,1,3] => 1 = 3 - 2
[2,3,1,4,5] => [1,4,5,2,3] => [2,5,1,4,3] => [4,5,2,1,3] => 1 = 3 - 2
[2,3,5,4,1] => [1,2,3,5,4] => [2,3,4,1,5] => [4,3,2,1,5] => 0 = 2 - 2
[2,4,1,3,5] => [1,3,5,2,4] => [2,4,1,5,3] => [5,4,2,1,3] => 0 = 2 - 2
[2,4,1,5,3] => [1,5,2,4,3] => [2,1,4,3,5] => [2,1,4,3,5] => 0 = 2 - 2
[2,4,3,1,5] => [1,5,2,4,3] => [2,1,4,3,5] => [2,1,4,3,5] => 0 = 2 - 2
[2,4,5,3,1] => [1,2,4,5,3] => [2,3,5,1,4] => [5,3,2,1,4] => 0 = 2 - 2
[2,5,4,1,3] => [1,3,2,5,4] => [2,4,3,1,5] => [3,4,2,1,5] => 1 = 3 - 2
[3,1,2,4,5] => [1,2,4,5,3] => [2,3,5,1,4] => [5,3,2,1,4] => 0 = 2 - 2
[3,1,4,5,2] => [1,4,5,2,3] => [2,5,1,4,3] => [4,5,2,1,3] => 1 = 3 - 2
[3,1,5,2,4] => [1,5,2,4,3] => [2,1,4,3,5] => [2,1,4,3,5] => 0 = 2 - 2
[3,2,1,4,5] => [1,4,5,2,3] => [2,5,1,4,3] => [4,5,2,1,3] => 1 = 3 - 2
[3,2,4,1,5] => [1,5,2,4,3] => [2,1,4,3,5] => [2,1,4,3,5] => 0 = 2 - 2
[3,2,4,5,1] => [1,2,4,5,3] => [2,3,5,1,4] => [5,3,2,1,4] => 0 = 2 - 2
[3,5,4,1,2] => [1,2,3,5,4] => [2,3,4,1,5] => [4,3,2,1,5] => 0 = 2 - 2
[3,5,4,2,1] => [1,2,3,5,4] => [2,3,4,1,5] => [4,3,2,1,5] => 0 = 2 - 2
[4,1,2,3,5] => [1,2,3,5,4] => [2,3,4,1,5] => [4,3,2,1,5] => 0 = 2 - 2
[4,1,3,2,5] => [1,3,2,5,4] => [2,4,3,1,5] => [3,4,2,1,5] => 1 = 3 - 2
[4,1,3,5,2] => [1,3,5,2,4] => [2,4,1,5,3] => [5,4,2,1,3] => 0 = 2 - 2
[4,2,1,3,5] => [1,3,5,2,4] => [2,4,1,5,3] => [5,4,2,1,3] => 0 = 2 - 2
[4,2,3,5,1] => [1,2,3,5,4] => [2,3,4,1,5] => [4,3,2,1,5] => 0 = 2 - 2
[4,2,5,1,3] => [1,3,2,5,4] => [2,4,3,1,5] => [3,4,2,1,5] => 1 = 3 - 2
[4,3,5,1,2] => [1,2,3,5,4] => [2,3,4,1,5] => [4,3,2,1,5] => 0 = 2 - 2
[4,3,5,2,1] => [1,2,3,5,4] => [2,3,4,1,5] => [4,3,2,1,5] => 0 = 2 - 2
[1,2,3,4,6,5] => [1,2,3,4,6,5] => [2,3,4,5,1,6] => [5,4,3,2,1,6] => ? = 2 - 2
[1,2,3,5,6,4] => [1,2,3,5,6,4] => [2,3,4,6,1,5] => [6,4,3,2,1,5] => ? = 2 - 2
[1,2,4,3,6,5] => [1,2,4,3,6,5] => [2,3,5,4,1,6] => [4,5,3,2,1,6] => ? = 3 - 2
[1,2,4,5,6,3] => [1,2,4,5,6,3] => [2,3,5,6,1,4] => [6,3,2,1,5,4] => ? = 2 - 2
[1,2,4,6,3,5] => [1,2,4,6,3,5] => [2,3,5,1,6,4] => [6,5,3,2,1,4] => ? = 2 - 2
[1,2,4,6,5,3] => [1,2,4,6,3,5] => [2,3,5,1,6,4] => [6,5,3,2,1,4] => ? = 2 - 2
[1,2,5,6,3,4] => [1,2,5,6,3,4] => [2,3,6,1,5,4] => [5,6,3,2,1,4] => ? = 3 - 2
[1,2,5,6,4,3] => [1,2,5,6,3,4] => [2,3,6,1,5,4] => [5,6,3,2,1,4] => ? = 3 - 2
[1,2,6,3,5,4] => [1,2,6,3,5,4] => [2,3,1,5,4,6] => [3,2,1,5,4,6] => ? = 2 - 2
[1,2,6,4,3,5] => [1,2,6,3,5,4] => [2,3,1,5,4,6] => [3,2,1,5,4,6] => ? = 2 - 2
[1,3,2,4,6,5] => [1,3,2,4,6,5] => [2,4,3,5,1,6] => [3,5,4,2,1,6] => ? = 3 - 2
[1,3,2,5,6,4] => [1,3,2,5,6,4] => [2,4,3,6,1,5] => [3,6,4,2,1,5] => ? = 3 - 2
[1,3,2,6,4,5] => [1,3,2,6,4,5] => [2,4,3,1,6,5] => [3,4,2,1,6,5] => ? = 2 - 2
[1,3,2,6,5,4] => [1,3,2,6,4,5] => [2,4,3,1,6,5] => [3,4,2,1,6,5] => ? = 2 - 2
[1,3,4,2,6,5] => [1,3,4,2,6,5] => [2,4,5,3,1,6] => [5,3,4,2,1,6] => ? = 3 - 2
[1,3,4,5,6,2] => [1,3,4,5,6,2] => [2,4,5,6,1,3] => [6,2,1,5,4,3] => ? = 2 - 2
[1,3,4,6,2,5] => [1,3,4,6,2,5] => [2,4,5,1,6,3] => [6,5,2,1,4,3] => ? = 2 - 2
[1,3,4,6,5,2] => [1,3,4,6,2,5] => [2,4,5,1,6,3] => [6,5,2,1,4,3] => ? = 2 - 2
[1,3,5,2,6,4] => [1,3,5,2,6,4] => [2,4,6,3,1,5] => [6,3,4,2,1,5] => ? = 2 - 2
[1,3,5,4,2,6] => [1,3,5,2,6,4] => [2,4,6,3,1,5] => [6,3,4,2,1,5] => ? = 2 - 2
[1,3,5,6,2,4] => [1,3,5,6,2,4] => [2,4,6,1,5,3] => [5,6,2,1,4,3] => ? = 2 - 2
[1,3,5,6,4,2] => [1,3,5,6,2,4] => [2,4,6,1,5,3] => [5,6,2,1,4,3] => ? = 2 - 2
[1,3,6,2,4,5] => [1,3,6,2,4,5] => [2,4,1,5,6,3] => [6,5,4,2,1,3] => ? = 2 - 2
[1,3,6,2,5,4] => [1,3,6,2,5,4] => [2,4,1,5,3,6] => [5,4,2,1,3,6] => ? = 3 - 2
[1,3,6,4,2,5] => [1,3,6,2,5,4] => [2,4,1,5,3,6] => [5,4,2,1,3,6] => ? = 3 - 2
[1,3,6,4,5,2] => [1,3,6,2,4,5] => [2,4,1,5,6,3] => [6,5,4,2,1,3] => ? = 2 - 2
[1,3,6,5,2,4] => [1,3,6,2,4,5] => [2,4,1,5,6,3] => [6,5,4,2,1,3] => ? = 2 - 2
[1,3,6,5,4,2] => [1,3,6,2,4,5] => [2,4,1,5,6,3] => [6,5,4,2,1,3] => ? = 2 - 2
[1,4,2,6,3,5] => [1,4,2,6,3,5] => [2,5,3,1,6,4] => [3,6,5,2,1,4] => ? = 4 - 2
[1,4,2,6,5,3] => [1,4,2,6,3,5] => [2,5,3,1,6,4] => [3,6,5,2,1,4] => ? = 4 - 2
[1,4,3,2,6,5] => [1,4,2,6,3,5] => [2,5,3,1,6,4] => [3,6,5,2,1,4] => ? = 4 - 2
[1,4,3,5,2,6] => [1,4,2,6,3,5] => [2,5,3,1,6,4] => [3,6,5,2,1,4] => ? = 4 - 2
[1,4,5,2,6,3] => [1,4,5,2,6,3] => [2,5,6,3,1,4] => [6,3,5,2,1,4] => ? = 3 - 2
[1,4,5,3,2,6] => [1,4,5,2,6,3] => [2,5,6,3,1,4] => [6,3,5,2,1,4] => ? = 3 - 2
[1,4,5,6,2,3] => [1,4,5,6,2,3] => [2,5,6,1,4,3] => [6,2,1,4,5,3] => ? = 3 - 2
[1,4,5,6,3,2] => [1,4,5,6,2,3] => [2,5,6,1,4,3] => [6,2,1,4,5,3] => ? = 3 - 2
[1,4,6,2,5,3] => [1,4,6,2,5,3] => [2,5,1,4,3,6] => [4,5,2,1,3,6] => ? = 2 - 2
[1,4,6,3,2,5] => [1,4,6,2,5,3] => [2,5,1,4,3,6] => [4,5,2,1,3,6] => ? = 2 - 2
[1,5,2,4,6,3] => [1,5,2,4,6,3] => [2,6,3,5,1,4] => [3,5,2,1,6,4] => ? = 3 - 2
[1,5,2,6,3,4] => [1,5,2,6,3,4] => [2,6,3,1,5,4] => [3,5,6,2,1,4] => ? = 3 - 2
[1,5,2,6,4,3] => [1,5,2,6,3,4] => [2,6,3,1,5,4] => [3,5,6,2,1,4] => ? = 3 - 2
[1,5,3,2,4,6] => [1,5,2,4,6,3] => [2,6,3,5,1,4] => [3,5,2,1,6,4] => ? = 3 - 2
[1,5,3,2,6,4] => [1,5,2,6,3,4] => [2,6,3,1,5,4] => [3,5,6,2,1,4] => ? = 3 - 2
[1,5,3,4,2,6] => [1,5,2,6,3,4] => [2,6,3,1,5,4] => [3,5,6,2,1,4] => ? = 3 - 2
[1,5,4,2,6,3] => [1,5,2,6,3,4] => [2,6,3,1,5,4] => [3,5,6,2,1,4] => ? = 3 - 2
[1,5,4,3,2,6] => [1,5,2,6,3,4] => [2,6,3,1,5,4] => [3,5,6,2,1,4] => ? = 3 - 2
[1,6,2,3,5,4] => [1,6,2,3,5,4] => [2,1,4,5,3,6] => [2,1,5,4,3,6] => ? = 2 - 2
[1,6,2,4,3,5] => [1,6,2,4,3,5] => [2,1,4,6,5,3] => [2,1,5,6,4,3] => ? = 2 - 2
[1,6,3,5,2,4] => [1,6,2,4,3,5] => [2,1,4,6,5,3] => [2,1,5,6,4,3] => ? = 2 - 2
Description
The number of augmented double ascents of a permutation. An augmented double ascent of a permutation $\pi$ is a double ascent of the augmented permutation $\tilde\pi$ obtained from $\pi$ by adding an initial $0$. A double ascent of $\tilde\pi$ then is a position $i$ such that $\tilde\pi(i) < \tilde\pi(i+1) < \tilde\pi(i+2)$.