Your data matches 5 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001232
Mapping
Mp00223: Permutations —runsort⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1,0]
 => 0
[1,2] => [1,2] => [1,2] => [1,0,1,0]
 => 1
[2,1] => [1,2] => [1,2] => [1,0,1,0]
 => 1
[1,3,2] => [1,3,2] => [1,3,2] => [1,0,1,1,0,0]
 => 2
[2,1,3] => [1,3,2] => [1,3,2] => [1,0,1,1,0,0]
 => 2
[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
 => 3
[1,3,4,2] => [1,3,4,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => 3
[2,1,3,4] => [1,3,4,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => 3
[2,4,1,3] => [1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
 => 3
[4,1,3,2] => [1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
 => 3
[4,2,1,3] => [1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
 => 3
[1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
 => 4
[1,3,4,2,5] => [1,3,4,2,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
 => 4
[1,3,4,5,2] => [1,3,4,5,2] => [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
 => 4
[1,3,5,2,4] => [1,3,5,2,4] => [1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
 => 5
[1,3,5,4,2] => [1,3,5,2,4] => [1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
 => 5
[2,1,3,4,5] => [1,3,4,5,2] => [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
 => 4
[2,1,3,5,4] => [1,3,5,2,4] => [1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
 => 5
[2,4,1,3,5] => [1,3,5,2,4] => [1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
 => 5
[2,5,1,3,4] => [1,3,4,2,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
 => 4
[2,5,4,1,3] => [1,3,2,5,4] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
 => 4
[4,1,3,2,5] => [1,3,2,5,4] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
 => 4
[4,1,3,5,2] => [1,3,5,2,4] => [1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
 => 5
[4,2,1,3,5] => [1,3,5,2,4] => [1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
 => 5
[4,2,5,1,3] => [1,3,2,5,4] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
 => 4
[5,1,3,4,2] => [1,3,4,2,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
 => 4
[5,2,1,3,4] => [1,3,4,2,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
 => 4
[1,3,2,5,4,6] => [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
[1,3,2,5,6,4] => [1,3,2,5,6,4] => [1,3,2,6,5,4] => [1,0,1,1,0,0,1,1,1,0,0,0]
 => 5
[1,3,4,2,6,5] => [1,3,4,2,6,5] => [1,4,3,2,6,5] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => 5
[1,3,4,5,2,6] => [1,3,4,5,2,6] => [1,5,3,4,2,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => 5
[1,3,4,5,6,2] => [1,3,4,5,6,2] => [1,6,3,4,5,2] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 5
[1,3,4,6,2,5] => [1,3,4,6,2,5] => [1,5,3,4,6,2] => [1,0,1,1,1,1,0,0,0,1,0,0]
 => 6
[1,3,4,6,5,2] => [1,3,4,6,2,5] => [1,5,3,4,6,2] => [1,0,1,1,1,1,0,0,0,1,0,0]
 => 6
[1,3,5,2,6,4] => [1,3,5,2,6,4] => [1,5,3,6,2,4] => [1,0,1,1,1,1,0,0,1,0,0,0]
 => 7
[1,3,5,4,2,6] => [1,3,5,2,6,4] => [1,5,3,6,2,4] => [1,0,1,1,1,1,0,0,1,0,0,0]
 => 7
[1,3,5,6,2,4] => [1,3,5,6,2,4] => [1,4,3,6,5,2] => [1,0,1,1,1,0,0,1,1,0,0,0]
 => 7
[1,3,5,6,4,2] => [1,3,5,6,2,4] => [1,4,3,6,5,2] => [1,0,1,1,1,0,0,1,1,0,0,0]
 => 7
[1,3,6,2,5,4] => [1,3,6,2,5,4] => [1,5,3,6,4,2] => [1,0,1,1,1,1,0,0,1,0,0,0]
 => 7
[1,3,6,4,2,5] => [1,3,6,2,5,4] => [1,5,3,6,4,2] => [1,0,1,1,1,1,0,0,1,0,0,0]
 => 7
[1,5,2,6,3,4] => [1,5,2,6,3,4] => [1,5,4,6,2,3] => [1,0,1,1,1,1,0,0,1,0,0,0]
 => 7
[1,5,2,6,4,3] => [1,5,2,6,3,4] => [1,5,4,6,2,3] => [1,0,1,1,1,1,0,0,1,0,0,0]
 => 7
[1,5,3,2,6,4] => [1,5,2,6,3,4] => [1,5,4,6,2,3] => [1,0,1,1,1,1,0,0,1,0,0,0]
 => 7
[1,5,3,4,2,6] => [1,5,2,6,3,4] => [1,5,4,6,2,3] => [1,0,1,1,1,1,0,0,1,0,0,0]
 => 7
[1,5,4,2,6,3] => [1,5,2,6,3,4] => [1,5,4,6,2,3] => [1,0,1,1,1,1,0,0,1,0,0,0]
 => 7
[1,5,4,3,2,6] => [1,5,2,6,3,4] => [1,5,4,6,2,3] => [1,0,1,1,1,1,0,0,1,0,0,0]
 => 7
[1,6,2,5,3,4] => [1,6,2,5,3,4] => [1,5,4,6,3,2] => [1,0,1,1,1,1,0,0,1,0,0,0]
 => 7
[1,6,2,5,4,3] => [1,6,2,5,3,4] => [1,5,4,6,3,2] => [1,0,1,1,1,1,0,0,1,0,0,0]
 => 7
[1,6,3,2,5,4] => [1,6,2,5,3,4] => [1,5,4,6,3,2] => [1,0,1,1,1,1,0,0,1,0,0,0]
 => 7
[1,6,3,4,2,5] => [1,6,2,5,3,4] => [1,5,4,6,3,2] => [1,0,1,1,1,1,0,0,1,0,0,0]
 => 7
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Matching statistic: St000264
Mapping
Mp00223: Permutations —runsort⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00154: Graphs —core⟶ Graphs
St000264: Graphs ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 9%
Values
[1] => [1] => ([],1)
 => ([],1)
 => ? = 0 - 4
[1,2] => [1,2] => ([],2)
 => ([],1)
 => ? = 1 - 4
[2,1] => [1,2] => ([],2)
 => ([],1)
 => ? = 1 - 4
[1,3,2] => [1,3,2] => ([(1,2)],3)
 => ([(0,1)],2)
 => ? = 2 - 4
[2,1,3] => [1,3,2] => ([(1,2)],3)
 => ([(0,1)],2)
 => ? = 2 - 4
[1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
 => ([(0,1)],2)
 => ? = 3 - 4
[1,3,4,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
 => ([(0,1)],2)
 => ? = 3 - 4
[2,1,3,4] => [1,3,4,2] => ([(1,3),(2,3)],4)
 => ([(0,1)],2)
 => ? = 3 - 4
[2,4,1,3] => [1,3,2,4] => ([(2,3)],4)
 => ([(0,1)],2)
 => ? = 3 - 4
[4,1,3,2] => [1,3,2,4] => ([(2,3)],4)
 => ([(0,1)],2)
 => ? = 3 - 4
[4,2,1,3] => [1,3,2,4] => ([(2,3)],4)
 => ([(0,1)],2)
 => ? = 3 - 4
[1,3,2,5,4] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
 => ([(0,1)],2)
 => ? = 4 - 4
[1,3,4,2,5] => [1,3,4,2,5] => ([(2,4),(3,4)],5)
 => ([(0,1)],2)
 => ? = 4 - 4
[1,3,4,5,2] => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => ? = 4 - 4
[1,3,5,2,4] => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
 => ([(0,1)],2)
 => ? = 5 - 4
[1,3,5,4,2] => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
 => ([(0,1)],2)
 => ? = 5 - 4
[2,1,3,4,5] => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => ? = 4 - 4
[2,1,3,5,4] => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
 => ([(0,1)],2)
 => ? = 5 - 4
[2,4,1,3,5] => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
 => ([(0,1)],2)
 => ? = 5 - 4
[2,5,1,3,4] => [1,3,4,2,5] => ([(2,4),(3,4)],5)
 => ([(0,1)],2)
 => ? = 4 - 4
[2,5,4,1,3] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
 => ([(0,1)],2)
 => ? = 4 - 4
[4,1,3,2,5] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
 => ([(0,1)],2)
 => ? = 4 - 4
[4,1,3,5,2] => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
 => ([(0,1)],2)
 => ? = 5 - 4
[4,2,1,3,5] => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
 => ([(0,1)],2)
 => ? = 5 - 4
[4,2,5,1,3] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
 => ([(0,1)],2)
 => ? = 4 - 4
[5,1,3,4,2] => [1,3,4,2,5] => ([(2,4),(3,4)],5)
 => ([(0,1)],2)
 => ? = 4 - 4
[5,2,1,3,4] => [1,3,4,2,5] => ([(2,4),(3,4)],5)
 => ([(0,1)],2)
 => ? = 4 - 4
[1,3,2,5,4,6] => [1,3,2,5,4,6] => ([(2,5),(3,4)],6)
 => ([(0,1)],2)
 => ? = 5 - 4
[1,3,2,5,6,4] => [1,3,2,5,6,4] => ([(1,2),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => ? = 5 - 4
[1,3,4,2,6,5] => [1,3,4,2,6,5] => ([(1,2),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => ? = 5 - 4
[1,3,4,5,2,6] => [1,3,4,5,2,6] => ([(2,5),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => ? = 5 - 4
[1,3,4,5,6,2] => [1,3,4,5,6,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => ? = 5 - 4
[1,3,4,6,2,5] => [1,3,4,6,2,5] => ([(1,5),(2,5),(3,4),(4,5)],6)
 => ([(0,1)],2)
 => ? = 6 - 4
[1,3,4,6,5,2] => [1,3,4,6,2,5] => ([(1,5),(2,5),(3,4),(4,5)],6)
 => ([(0,1)],2)
 => ? = 6 - 4
[1,3,5,2,6,4] => [1,3,5,2,6,4] => ([(1,5),(2,4),(3,4),(3,5)],6)
 => ([(0,1)],2)
 => ? = 7 - 4
[1,3,5,4,2,6] => [1,3,5,2,6,4] => ([(1,5),(2,4),(3,4),(3,5)],6)
 => ([(0,1)],2)
 => ? = 7 - 4
[1,3,5,6,2,4] => [1,3,5,6,2,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => ? = 7 - 4
[1,3,5,6,4,2] => [1,3,5,6,2,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => ? = 7 - 4
[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 = 7 - 4
[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 = 7 - 4
[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)
 => ? = 7 - 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)
 => ? = 7 - 4
[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)
 => ? = 7 - 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)
 => ? = 7 - 4
[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)
 => ? = 7 - 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)
 => ? = 7 - 4
[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 = 7 - 4
[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 = 7 - 4
[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 = 7 - 4
[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 = 7 - 4
[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 = 7 - 4
[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 = 7 - 4
[2,1,3,4,5,6] => [1,3,4,5,6,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => ? = 5 - 4
[2,1,3,4,6,5] => [1,3,4,6,2,5] => ([(1,5),(2,5),(3,4),(4,5)],6)
 => ([(0,1)],2)
 => ? = 6 - 4
[2,1,3,5,6,4] => [1,3,5,6,2,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => ? = 7 - 4
[2,4,1,3,5,6] => [1,3,5,6,2,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => ? = 7 - 4
[2,5,1,3,4,6] => [1,3,4,6,2,5] => ([(1,5),(2,5),(3,4),(4,5)],6)
 => ([(0,1)],2)
 => ? = 6 - 4
[2,5,1,3,6,4] => [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 = 7 - 4
[2,5,1,6,3,4] => [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 = 7 - 4
[2,5,1,6,4,3] => [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 = 7 - 4
[2,5,3,1,6,4] => [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 = 7 - 4
[2,5,3,4,1,6] => [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 = 7 - 4
[2,5,4,1,3,6] => [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 = 7 - 4
[2,5,4,1,6,3] => [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 = 7 - 4
[2,5,4,3,1,6] => [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 = 7 - 4
[2,5,4,6,1,3] => [1,3,2,5,4,6] => ([(2,5),(3,4)],6)
 => ([(0,1)],2)
 => ? = 5 - 4
[3,1,6,2,5,4] => [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 = 7 - 4
[3,1,6,4,2,5] => [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 = 7 - 4
[3,2,5,1,6,4] => [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 = 7 - 4
[3,2,5,4,1,6] => [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 = 7 - 4
[3,4,1,6,2,5] => [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 = 7 - 4
[3,4,2,5,1,6] => [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 = 7 - 4
[4,1,3,6,2,5] => [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 = 7 - 4
[4,1,6,2,5,3] => [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 = 7 - 4
[4,1,6,3,2,5] => [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 = 7 - 4
[4,2,5,1,3,6] => [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 = 7 - 4
[4,2,5,1,6,3] => [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 = 7 - 4
[4,2,5,3,1,6] => [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 = 7 - 4
[4,3,1,6,2,5] => [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 = 7 - 4
[4,3,2,5,1,6] => [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 = 7 - 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: St001060
Mapping
Mp00223: Permutations —runsort⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00154: Graphs —core⟶ Graphs
St001060: Graphs ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 9%
Values
[1] => [1] => ([],1)
 => ([],1)
 => ? = 0 - 4
[1,2] => [1,2] => ([],2)
 => ([],1)
 => ? = 1 - 4
[2,1] => [1,2] => ([],2)
 => ([],1)
 => ? = 1 - 4
[1,3,2] => [1,3,2] => ([(1,2)],3)
 => ([(0,1)],2)
 => ? = 2 - 4
[2,1,3] => [1,3,2] => ([(1,2)],3)
 => ([(0,1)],2)
 => ? = 2 - 4
[1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
 => ([(0,1)],2)
 => ? = 3 - 4
[1,3,4,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
 => ([(0,1)],2)
 => ? = 3 - 4
[2,1,3,4] => [1,3,4,2] => ([(1,3),(2,3)],4)
 => ([(0,1)],2)
 => ? = 3 - 4
[2,4,1,3] => [1,3,2,4] => ([(2,3)],4)
 => ([(0,1)],2)
 => ? = 3 - 4
[4,1,3,2] => [1,3,2,4] => ([(2,3)],4)
 => ([(0,1)],2)
 => ? = 3 - 4
[4,2,1,3] => [1,3,2,4] => ([(2,3)],4)
 => ([(0,1)],2)
 => ? = 3 - 4
[1,3,2,5,4] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
 => ([(0,1)],2)
 => ? = 4 - 4
[1,3,4,2,5] => [1,3,4,2,5] => ([(2,4),(3,4)],5)
 => ([(0,1)],2)
 => ? = 4 - 4
[1,3,4,5,2] => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => ? = 4 - 4
[1,3,5,2,4] => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
 => ([(0,1)],2)
 => ? = 5 - 4
[1,3,5,4,2] => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
 => ([(0,1)],2)
 => ? = 5 - 4
[2,1,3,4,5] => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => ? = 4 - 4
[2,1,3,5,4] => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
 => ([(0,1)],2)
 => ? = 5 - 4
[2,4,1,3,5] => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
 => ([(0,1)],2)
 => ? = 5 - 4
[2,5,1,3,4] => [1,3,4,2,5] => ([(2,4),(3,4)],5)
 => ([(0,1)],2)
 => ? = 4 - 4
[2,5,4,1,3] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
 => ([(0,1)],2)
 => ? = 4 - 4
[4,1,3,2,5] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
 => ([(0,1)],2)
 => ? = 4 - 4
[4,1,3,5,2] => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
 => ([(0,1)],2)
 => ? = 5 - 4
[4,2,1,3,5] => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
 => ([(0,1)],2)
 => ? = 5 - 4
[4,2,5,1,3] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
 => ([(0,1)],2)
 => ? = 4 - 4
[5,1,3,4,2] => [1,3,4,2,5] => ([(2,4),(3,4)],5)
 => ([(0,1)],2)
 => ? = 4 - 4
[5,2,1,3,4] => [1,3,4,2,5] => ([(2,4),(3,4)],5)
 => ([(0,1)],2)
 => ? = 4 - 4
[1,3,2,5,4,6] => [1,3,2,5,4,6] => ([(2,5),(3,4)],6)
 => ([(0,1)],2)
 => ? = 5 - 4
[1,3,2,5,6,4] => [1,3,2,5,6,4] => ([(1,2),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => ? = 5 - 4
[1,3,4,2,6,5] => [1,3,4,2,6,5] => ([(1,2),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => ? = 5 - 4
[1,3,4,5,2,6] => [1,3,4,5,2,6] => ([(2,5),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => ? = 5 - 4
[1,3,4,5,6,2] => [1,3,4,5,6,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => ? = 5 - 4
[1,3,4,6,2,5] => [1,3,4,6,2,5] => ([(1,5),(2,5),(3,4),(4,5)],6)
 => ([(0,1)],2)
 => ? = 6 - 4
[1,3,4,6,5,2] => [1,3,4,6,2,5] => ([(1,5),(2,5),(3,4),(4,5)],6)
 => ([(0,1)],2)
 => ? = 6 - 4
[1,3,5,2,6,4] => [1,3,5,2,6,4] => ([(1,5),(2,4),(3,4),(3,5)],6)
 => ([(0,1)],2)
 => ? = 7 - 4
[1,3,5,4,2,6] => [1,3,5,2,6,4] => ([(1,5),(2,4),(3,4),(3,5)],6)
 => ([(0,1)],2)
 => ? = 7 - 4
[1,3,5,6,2,4] => [1,3,5,6,2,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => ? = 7 - 4
[1,3,5,6,4,2] => [1,3,5,6,2,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => ? = 7 - 4
[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 = 7 - 4
[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 = 7 - 4
[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)
 => ? = 7 - 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)
 => ? = 7 - 4
[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)
 => ? = 7 - 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)
 => ? = 7 - 4
[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)
 => ? = 7 - 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)
 => ? = 7 - 4
[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 = 7 - 4
[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 = 7 - 4
[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 = 7 - 4
[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 = 7 - 4
[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 = 7 - 4
[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 = 7 - 4
[2,1,3,4,5,6] => [1,3,4,5,6,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => ? = 5 - 4
[2,1,3,4,6,5] => [1,3,4,6,2,5] => ([(1,5),(2,5),(3,4),(4,5)],6)
 => ([(0,1)],2)
 => ? = 6 - 4
[2,1,3,5,6,4] => [1,3,5,6,2,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => ? = 7 - 4
[2,4,1,3,5,6] => [1,3,5,6,2,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => ? = 7 - 4
[2,5,1,3,4,6] => [1,3,4,6,2,5] => ([(1,5),(2,5),(3,4),(4,5)],6)
 => ([(0,1)],2)
 => ? = 6 - 4
[2,5,1,3,6,4] => [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 = 7 - 4
[2,5,1,6,3,4] => [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 = 7 - 4
[2,5,1,6,4,3] => [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 = 7 - 4
[2,5,3,1,6,4] => [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 = 7 - 4
[2,5,3,4,1,6] => [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 = 7 - 4
[2,5,4,1,3,6] => [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 = 7 - 4
[2,5,4,1,6,3] => [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 = 7 - 4
[2,5,4,3,1,6] => [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 = 7 - 4
[2,5,4,6,1,3] => [1,3,2,5,4,6] => ([(2,5),(3,4)],6)
 => ([(0,1)],2)
 => ? = 5 - 4
[3,1,6,2,5,4] => [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 = 7 - 4
[3,1,6,4,2,5] => [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 = 7 - 4
[3,2,5,1,6,4] => [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 = 7 - 4
[3,2,5,4,1,6] => [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 = 7 - 4
[3,4,1,6,2,5] => [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 = 7 - 4
[3,4,2,5,1,6] => [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 = 7 - 4
[4,1,3,6,2,5] => [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 = 7 - 4
[4,1,6,2,5,3] => [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 = 7 - 4
[4,1,6,3,2,5] => [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 = 7 - 4
[4,2,5,1,3,6] => [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 = 7 - 4
[4,2,5,1,6,3] => [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 = 7 - 4
[4,2,5,3,1,6] => [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 = 7 - 4
[4,3,1,6,2,5] => [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 = 7 - 4
[4,3,2,5,1,6] => [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 = 7 - 4
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
Mapping
Mp00223: Permutations —runsort⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00154: Graphs —core⟶ Graphs
St001570: Graphs ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 9%
Values
[1] => [1] => ([],1)
 => ([],1)
 => ? = 0 - 7
[1,2] => [1,2] => ([],2)
 => ([],1)
 => ? = 1 - 7
[2,1] => [1,2] => ([],2)
 => ([],1)
 => ? = 1 - 7
[1,3,2] => [1,3,2] => ([(1,2)],3)
 => ([(0,1)],2)
 => ? = 2 - 7
[2,1,3] => [1,3,2] => ([(1,2)],3)
 => ([(0,1)],2)
 => ? = 2 - 7
[1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
 => ([(0,1)],2)
 => ? = 3 - 7
[1,3,4,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
 => ([(0,1)],2)
 => ? = 3 - 7
[2,1,3,4] => [1,3,4,2] => ([(1,3),(2,3)],4)
 => ([(0,1)],2)
 => ? = 3 - 7
[2,4,1,3] => [1,3,2,4] => ([(2,3)],4)
 => ([(0,1)],2)
 => ? = 3 - 7
[4,1,3,2] => [1,3,2,4] => ([(2,3)],4)
 => ([(0,1)],2)
 => ? = 3 - 7
[4,2,1,3] => [1,3,2,4] => ([(2,3)],4)
 => ([(0,1)],2)
 => ? = 3 - 7
[1,3,2,5,4] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
 => ([(0,1)],2)
 => ? = 4 - 7
[1,3,4,2,5] => [1,3,4,2,5] => ([(2,4),(3,4)],5)
 => ([(0,1)],2)
 => ? = 4 - 7
[1,3,4,5,2] => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => ? = 4 - 7
[1,3,5,2,4] => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
 => ([(0,1)],2)
 => ? = 5 - 7
[1,3,5,4,2] => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
 => ([(0,1)],2)
 => ? = 5 - 7
[2,1,3,4,5] => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => ? = 4 - 7
[2,1,3,5,4] => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
 => ([(0,1)],2)
 => ? = 5 - 7
[2,4,1,3,5] => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
 => ([(0,1)],2)
 => ? = 5 - 7
[2,5,1,3,4] => [1,3,4,2,5] => ([(2,4),(3,4)],5)
 => ([(0,1)],2)
 => ? = 4 - 7
[2,5,4,1,3] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
 => ([(0,1)],2)
 => ? = 4 - 7
[4,1,3,2,5] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
 => ([(0,1)],2)
 => ? = 4 - 7
[4,1,3,5,2] => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
 => ([(0,1)],2)
 => ? = 5 - 7
[4,2,1,3,5] => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
 => ([(0,1)],2)
 => ? = 5 - 7
[4,2,5,1,3] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
 => ([(0,1)],2)
 => ? = 4 - 7
[5,1,3,4,2] => [1,3,4,2,5] => ([(2,4),(3,4)],5)
 => ([(0,1)],2)
 => ? = 4 - 7
[5,2,1,3,4] => [1,3,4,2,5] => ([(2,4),(3,4)],5)
 => ([(0,1)],2)
 => ? = 4 - 7
[1,3,2,5,4,6] => [1,3,2,5,4,6] => ([(2,5),(3,4)],6)
 => ([(0,1)],2)
 => ? = 5 - 7
[1,3,2,5,6,4] => [1,3,2,5,6,4] => ([(1,2),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => ? = 5 - 7
[1,3,4,2,6,5] => [1,3,4,2,6,5] => ([(1,2),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => ? = 5 - 7
[1,3,4,5,2,6] => [1,3,4,5,2,6] => ([(2,5),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => ? = 5 - 7
[1,3,4,5,6,2] => [1,3,4,5,6,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => ? = 5 - 7
[1,3,4,6,2,5] => [1,3,4,6,2,5] => ([(1,5),(2,5),(3,4),(4,5)],6)
 => ([(0,1)],2)
 => ? = 6 - 7
[1,3,4,6,5,2] => [1,3,4,6,2,5] => ([(1,5),(2,5),(3,4),(4,5)],6)
 => ([(0,1)],2)
 => ? = 6 - 7
[1,3,5,2,6,4] => [1,3,5,2,6,4] => ([(1,5),(2,4),(3,4),(3,5)],6)
 => ([(0,1)],2)
 => ? = 7 - 7
[1,3,5,4,2,6] => [1,3,5,2,6,4] => ([(1,5),(2,4),(3,4),(3,5)],6)
 => ([(0,1)],2)
 => ? = 7 - 7
[1,3,5,6,2,4] => [1,3,5,6,2,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => ? = 7 - 7
[1,3,5,6,4,2] => [1,3,5,6,2,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => ? = 7 - 7
[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 = 7 - 7
[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 = 7 - 7
[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)
 => ? = 7 - 7
[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)
 => ? = 7 - 7
[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)
 => ? = 7 - 7
[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)
 => ? = 7 - 7
[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)
 => ? = 7 - 7
[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)
 => ? = 7 - 7
[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 = 7 - 7
[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 = 7 - 7
[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 = 7 - 7
[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 = 7 - 7
[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 = 7 - 7
[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 = 7 - 7
[2,1,3,4,5,6] => [1,3,4,5,6,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => ? = 5 - 7
[2,1,3,4,6,5] => [1,3,4,6,2,5] => ([(1,5),(2,5),(3,4),(4,5)],6)
 => ([(0,1)],2)
 => ? = 6 - 7
[2,1,3,5,6,4] => [1,3,5,6,2,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => ? = 7 - 7
[2,4,1,3,5,6] => [1,3,5,6,2,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => ? = 7 - 7
[2,5,1,3,4,6] => [1,3,4,6,2,5] => ([(1,5),(2,5),(3,4),(4,5)],6)
 => ([(0,1)],2)
 => ? = 6 - 7
[2,5,1,3,6,4] => [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 = 7 - 7
[2,5,1,6,3,4] => [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 = 7 - 7
[2,5,1,6,4,3] => [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 = 7 - 7
[2,5,3,1,6,4] => [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 = 7 - 7
[2,5,3,4,1,6] => [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 = 7 - 7
[2,5,4,1,3,6] => [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 = 7 - 7
[2,5,4,1,6,3] => [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 = 7 - 7
[2,5,4,3,1,6] => [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 = 7 - 7
[2,5,4,6,1,3] => [1,3,2,5,4,6] => ([(2,5),(3,4)],6)
 => ([(0,1)],2)
 => ? = 5 - 7
[3,1,6,2,5,4] => [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 = 7 - 7
[3,1,6,4,2,5] => [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 = 7 - 7
[3,2,5,1,6,4] => [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 = 7 - 7
[3,2,5,4,1,6] => [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 = 7 - 7
[3,4,1,6,2,5] => [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 = 7 - 7
[3,4,2,5,1,6] => [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 = 7 - 7
[4,1,3,6,2,5] => [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 = 7 - 7
[4,1,6,2,5,3] => [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 = 7 - 7
[4,1,6,3,2,5] => [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 = 7 - 7
[4,2,5,1,3,6] => [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 = 7 - 7
[4,2,5,1,6,3] => [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 = 7 - 7
[4,2,5,3,1,6] => [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 = 7 - 7
[4,3,1,6,2,5] => [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 = 7 - 7
[4,3,2,5,1,6] => [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 = 7 - 7
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.
Matching statistic: St001003
Mapping
Mp00223: Permutations —runsort⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001003: Dyck paths ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 55%
Values
[1] => [1] => [1] => [1,0]
 => 3 = 0 + 3
[1,2] => [1,2] => [1,2] => [1,0,1,0]
 => 4 = 1 + 3
[2,1] => [1,2] => [1,2] => [1,0,1,0]
 => 4 = 1 + 3
[1,3,2] => [1,3,2] => [1,2,3] => [1,0,1,0,1,0]
 => 5 = 2 + 3
[2,1,3] => [1,3,2] => [1,2,3] => [1,0,1,0,1,0]
 => 5 = 2 + 3
[1,3,2,4] => [1,3,2,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => 6 = 3 + 3
[1,3,4,2] => [1,3,4,2] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => 6 = 3 + 3
[2,1,3,4] => [1,3,4,2] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => 6 = 3 + 3
[2,4,1,3] => [1,3,2,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => 6 = 3 + 3
[4,1,3,2] => [1,3,2,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => 6 = 3 + 3
[4,2,1,3] => [1,3,2,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => 6 = 3 + 3
[1,3,2,5,4] => [1,3,2,5,4] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => 7 = 4 + 3
[1,3,4,2,5] => [1,3,4,2,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => 7 = 4 + 3
[1,3,4,5,2] => [1,3,4,5,2] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => 7 = 4 + 3
[1,3,5,2,4] => [1,3,5,2,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
 => 8 = 5 + 3
[1,3,5,4,2] => [1,3,5,2,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
 => 8 = 5 + 3
[2,1,3,4,5] => [1,3,4,5,2] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => 7 = 4 + 3
[2,1,3,5,4] => [1,3,5,2,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
 => 8 = 5 + 3
[2,4,1,3,5] => [1,3,5,2,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
 => 8 = 5 + 3
[2,5,1,3,4] => [1,3,4,2,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => 7 = 4 + 3
[2,5,4,1,3] => [1,3,2,5,4] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => 7 = 4 + 3
[4,1,3,2,5] => [1,3,2,5,4] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => 7 = 4 + 3
[4,1,3,5,2] => [1,3,5,2,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
 => 8 = 5 + 3
[4,2,1,3,5] => [1,3,5,2,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
 => 8 = 5 + 3
[4,2,5,1,3] => [1,3,2,5,4] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => 7 = 4 + 3
[5,1,3,4,2] => [1,3,4,2,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => 7 = 4 + 3
[5,2,1,3,4] => [1,3,4,2,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => 7 = 4 + 3
[1,3,2,5,4,6] => [1,3,2,5,4,6] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 5 + 3
[1,3,2,5,6,4] => [1,3,2,5,6,4] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 5 + 3
[1,3,4,2,6,5] => [1,3,4,2,6,5] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 5 + 3
[1,3,4,5,2,6] => [1,3,4,5,2,6] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 5 + 3
[1,3,4,5,6,2] => [1,3,4,5,6,2] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 5 + 3
[1,3,4,6,2,5] => [1,3,4,6,2,5] => [1,2,3,4,6,5] => [1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 6 + 3
[1,3,4,6,5,2] => [1,3,4,6,2,5] => [1,2,3,4,6,5] => [1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 6 + 3
[1,3,5,2,6,4] => [1,3,5,2,6,4] => [1,2,3,5,6,4] => [1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 7 + 3
[1,3,5,4,2,6] => [1,3,5,2,6,4] => [1,2,3,5,6,4] => [1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 7 + 3
[1,3,5,6,2,4] => [1,3,5,6,2,4] => [1,2,3,5,4,6] => [1,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 7 + 3
[1,3,5,6,4,2] => [1,3,5,6,2,4] => [1,2,3,5,4,6] => [1,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 7 + 3
[1,3,6,2,5,4] => [1,3,6,2,5,4] => [1,2,3,6,4,5] => [1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 7 + 3
[1,3,6,4,2,5] => [1,3,6,2,5,4] => [1,2,3,6,4,5] => [1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 7 + 3
[1,5,2,6,3,4] => [1,5,2,6,3,4] => [1,2,5,3,4,6] => [1,0,1,0,1,1,1,0,0,0,1,0]
 => ? = 7 + 3
[1,5,2,6,4,3] => [1,5,2,6,3,4] => [1,2,5,3,4,6] => [1,0,1,0,1,1,1,0,0,0,1,0]
 => ? = 7 + 3
[1,5,3,2,6,4] => [1,5,2,6,3,4] => [1,2,5,3,4,6] => [1,0,1,0,1,1,1,0,0,0,1,0]
 => ? = 7 + 3
[1,5,3,4,2,6] => [1,5,2,6,3,4] => [1,2,5,3,4,6] => [1,0,1,0,1,1,1,0,0,0,1,0]
 => ? = 7 + 3
[1,5,4,2,6,3] => [1,5,2,6,3,4] => [1,2,5,3,4,6] => [1,0,1,0,1,1,1,0,0,0,1,0]
 => ? = 7 + 3
[1,5,4,3,2,6] => [1,5,2,6,3,4] => [1,2,5,3,4,6] => [1,0,1,0,1,1,1,0,0,0,1,0]
 => ? = 7 + 3
[1,6,2,5,3,4] => [1,6,2,5,3,4] => [1,2,6,4,5,3] => [1,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 7 + 3
[1,6,2,5,4,3] => [1,6,2,5,3,4] => [1,2,6,4,5,3] => [1,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 7 + 3
[1,6,3,2,5,4] => [1,6,2,5,3,4] => [1,2,6,4,5,3] => [1,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 7 + 3
[1,6,3,4,2,5] => [1,6,2,5,3,4] => [1,2,6,4,5,3] => [1,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 7 + 3
[1,6,4,2,5,3] => [1,6,2,5,3,4] => [1,2,6,4,5,3] => [1,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 7 + 3
[1,6,4,3,2,5] => [1,6,2,5,3,4] => [1,2,6,4,5,3] => [1,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 7 + 3
[2,1,3,4,5,6] => [1,3,4,5,6,2] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 5 + 3
[2,1,3,4,6,5] => [1,3,4,6,2,5] => [1,2,3,4,6,5] => [1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 6 + 3
[2,1,3,5,6,4] => [1,3,5,6,2,4] => [1,2,3,5,4,6] => [1,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 7 + 3
[2,4,1,3,5,6] => [1,3,5,6,2,4] => [1,2,3,5,4,6] => [1,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 7 + 3
[2,5,1,3,4,6] => [1,3,4,6,2,5] => [1,2,3,4,6,5] => [1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 6 + 3
[2,5,1,3,6,4] => [1,3,6,2,5,4] => [1,2,3,6,4,5] => [1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 7 + 3
[2,5,1,6,3,4] => [1,6,2,5,3,4] => [1,2,6,4,5,3] => [1,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 7 + 3
[2,5,1,6,4,3] => [1,6,2,5,3,4] => [1,2,6,4,5,3] => [1,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 7 + 3
[2,5,3,1,6,4] => [1,6,2,5,3,4] => [1,2,6,4,5,3] => [1,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 7 + 3
[2,5,3,4,1,6] => [1,6,2,5,3,4] => [1,2,6,4,5,3] => [1,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 7 + 3
[2,5,4,1,3,6] => [1,3,6,2,5,4] => [1,2,3,6,4,5] => [1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 7 + 3
[2,5,4,1,6,3] => [1,6,2,5,3,4] => [1,2,6,4,5,3] => [1,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 7 + 3
[2,5,4,3,1,6] => [1,6,2,5,3,4] => [1,2,6,4,5,3] => [1,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 7 + 3
[2,5,4,6,1,3] => [1,3,2,5,4,6] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 5 + 3
[2,5,6,4,1,3] => [1,3,2,5,6,4] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 5 + 3
[2,6,1,3,4,5] => [1,3,4,5,2,6] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 5 + 3
[2,6,1,3,5,4] => [1,3,5,2,6,4] => [1,2,3,5,6,4] => [1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 7 + 3
[2,6,1,5,3,4] => [1,5,2,6,3,4] => [1,2,5,3,4,6] => [1,0,1,0,1,1,1,0,0,0,1,0]
 => ? = 7 + 3
[2,6,1,5,4,3] => [1,5,2,6,3,4] => [1,2,5,3,4,6] => [1,0,1,0,1,1,1,0,0,0,1,0]
 => ? = 7 + 3
[2,6,3,1,5,4] => [1,5,2,6,3,4] => [1,2,5,3,4,6] => [1,0,1,0,1,1,1,0,0,0,1,0]
 => ? = 7 + 3
[2,6,3,4,1,5] => [1,5,2,6,3,4] => [1,2,5,3,4,6] => [1,0,1,0,1,1,1,0,0,0,1,0]
 => ? = 7 + 3
[2,6,4,1,3,5] => [1,3,5,2,6,4] => [1,2,3,5,6,4] => [1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 7 + 3
[2,6,4,1,5,3] => [1,5,2,6,3,4] => [1,2,5,3,4,6] => [1,0,1,0,1,1,1,0,0,0,1,0]
 => ? = 7 + 3
[2,6,4,3,1,5] => [1,5,2,6,3,4] => [1,2,5,3,4,6] => [1,0,1,0,1,1,1,0,0,0,1,0]
 => ? = 7 + 3
[2,6,5,1,3,4] => [1,3,4,2,6,5] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 5 + 3
Description
The number of indecomposable modules with projective dimension at most 1 in the Nakayama algebra corresponding to the Dyck path.