Your data matches 9 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000259
Mp00252: Permutations restrictionPermutations
Mp00159: Permutations Demazure product with inversePermutations
Mp00160: Permutations graph of inversionsGraphs
St000259: Graphs ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1,2] => [1] => [1] => ([],1)
=> 0
[2,1] => [1] => [1] => ([],1)
=> 0
[2,1,3] => [2,1] => [2,1] => ([(0,1)],2)
=> 1
[2,3,1] => [2,1] => [2,1] => ([(0,1)],2)
=> 1
[3,2,1] => [2,1] => [2,1] => ([(0,1)],2)
=> 1
[2,3,1,4] => [2,3,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[2,3,4,1] => [2,3,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[2,4,3,1] => [2,3,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[3,1,2,4] => [3,1,2] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[3,1,4,2] => [3,1,2] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[3,2,1,4] => [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[3,2,4,1] => [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[3,4,1,2] => [3,1,2] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[3,4,2,1] => [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[4,2,3,1] => [2,3,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[4,3,1,2] => [3,1,2] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[4,3,2,1] => [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[2,3,4,1,5] => [2,3,4,1] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[2,3,4,5,1] => [2,3,4,1] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[2,3,5,4,1] => [2,3,4,1] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[2,4,1,3,5] => [2,4,1,3] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[2,4,1,5,3] => [2,4,1,3] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[2,4,3,1,5] => [2,4,3,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[2,4,3,5,1] => [2,4,3,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[2,4,5,1,3] => [2,4,1,3] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[2,4,5,3,1] => [2,4,3,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[2,5,3,4,1] => [2,3,4,1] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[2,5,4,1,3] => [2,4,1,3] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[2,5,4,3,1] => [2,4,3,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[3,1,4,2,5] => [3,1,4,2] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[3,1,4,5,2] => [3,1,4,2] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[3,1,5,4,2] => [3,1,4,2] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[3,2,4,1,5] => [3,2,4,1] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[3,2,4,5,1] => [3,2,4,1] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[3,2,5,4,1] => [3,2,4,1] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[3,4,1,2,5] => [3,4,1,2] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[3,4,1,5,2] => [3,4,1,2] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[3,4,2,1,5] => [3,4,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[3,4,2,5,1] => [3,4,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[3,4,5,1,2] => [3,4,1,2] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[3,4,5,2,1] => [3,4,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[3,5,1,4,2] => [3,1,4,2] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[3,5,2,4,1] => [3,2,4,1] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[3,5,4,1,2] => [3,4,1,2] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[3,5,4,2,1] => [3,4,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[4,1,2,3,5] => [4,1,2,3] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[4,1,2,5,3] => [4,1,2,3] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[4,1,3,2,5] => [4,1,3,2] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[4,1,3,5,2] => [4,1,3,2] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[4,1,5,2,3] => [4,1,2,3] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
Description
The diameter of a connected graph. This is the greatest distance between any pair of vertices.
Mp00252: Permutations restrictionPermutations
Mp00159: Permutations Demazure product with inversePermutations
Mp00235: Permutations descent views to invisible inversion bottomsPermutations
St001569: Permutations ⟶ ℤResult quality: 6% values known / values provided: 6%distinct values known / distinct values provided: 50%
Values
[1,2] => [1] => [1] => [1] => ? = 0
[2,1] => [1] => [1] => [1] => ? = 0
[2,1,3] => [2,1] => [2,1] => [2,1] => 1
[2,3,1] => [2,1] => [2,1] => [2,1] => 1
[3,2,1] => [2,1] => [2,1] => [2,1] => 1
[2,3,1,4] => [2,3,1] => [3,2,1] => [2,3,1] => 1
[2,3,4,1] => [2,3,1] => [3,2,1] => [2,3,1] => 1
[2,4,3,1] => [2,3,1] => [3,2,1] => [2,3,1] => 1
[3,1,2,4] => [3,1,2] => [3,2,1] => [2,3,1] => 1
[3,1,4,2] => [3,1,2] => [3,2,1] => [2,3,1] => 1
[3,2,1,4] => [3,2,1] => [3,2,1] => [2,3,1] => 1
[3,2,4,1] => [3,2,1] => [3,2,1] => [2,3,1] => 1
[3,4,1,2] => [3,1,2] => [3,2,1] => [2,3,1] => 1
[3,4,2,1] => [3,2,1] => [3,2,1] => [2,3,1] => 1
[4,2,3,1] => [2,3,1] => [3,2,1] => [2,3,1] => 1
[4,3,1,2] => [3,1,2] => [3,2,1] => [2,3,1] => 1
[4,3,2,1] => [3,2,1] => [3,2,1] => [2,3,1] => 1
[2,3,4,1,5] => [2,3,4,1] => [4,2,3,1] => [3,4,2,1] => 2
[2,3,4,5,1] => [2,3,4,1] => [4,2,3,1] => [3,4,2,1] => 2
[2,3,5,4,1] => [2,3,4,1] => [4,2,3,1] => [3,4,2,1] => 2
[2,4,1,3,5] => [2,4,1,3] => [3,4,1,2] => [4,1,3,2] => 2
[2,4,1,5,3] => [2,4,1,3] => [3,4,1,2] => [4,1,3,2] => 2
[2,4,3,1,5] => [2,4,3,1] => [4,3,2,1] => [2,3,4,1] => 1
[2,4,3,5,1] => [2,4,3,1] => [4,3,2,1] => [2,3,4,1] => 1
[2,4,5,1,3] => [2,4,1,3] => [3,4,1,2] => [4,1,3,2] => 2
[2,4,5,3,1] => [2,4,3,1] => [4,3,2,1] => [2,3,4,1] => 1
[2,5,3,4,1] => [2,3,4,1] => [4,2,3,1] => [3,4,2,1] => 2
[2,5,4,1,3] => [2,4,1,3] => [3,4,1,2] => [4,1,3,2] => 2
[2,5,4,3,1] => [2,4,3,1] => [4,3,2,1] => [2,3,4,1] => 1
[3,1,4,2,5] => [3,1,4,2] => [4,2,3,1] => [3,4,2,1] => 2
[3,1,4,5,2] => [3,1,4,2] => [4,2,3,1] => [3,4,2,1] => 2
[3,1,5,4,2] => [3,1,4,2] => [4,2,3,1] => [3,4,2,1] => 2
[3,2,4,1,5] => [3,2,4,1] => [4,2,3,1] => [3,4,2,1] => 2
[3,2,4,5,1] => [3,2,4,1] => [4,2,3,1] => [3,4,2,1] => 2
[3,2,5,4,1] => [3,2,4,1] => [4,2,3,1] => [3,4,2,1] => 2
[3,4,1,2,5] => [3,4,1,2] => [4,3,2,1] => [2,3,4,1] => 1
[3,4,1,5,2] => [3,4,1,2] => [4,3,2,1] => [2,3,4,1] => 1
[3,4,2,1,5] => [3,4,2,1] => [4,3,2,1] => [2,3,4,1] => 1
[3,4,2,5,1] => [3,4,2,1] => [4,3,2,1] => [2,3,4,1] => 1
[3,4,5,1,2] => [3,4,1,2] => [4,3,2,1] => [2,3,4,1] => 1
[3,4,5,2,1] => [3,4,2,1] => [4,3,2,1] => [2,3,4,1] => 1
[3,5,1,4,2] => [3,1,4,2] => [4,2,3,1] => [3,4,2,1] => 2
[3,5,2,4,1] => [3,2,4,1] => [4,2,3,1] => [3,4,2,1] => 2
[3,5,4,1,2] => [3,4,1,2] => [4,3,2,1] => [2,3,4,1] => 1
[3,5,4,2,1] => [3,4,2,1] => [4,3,2,1] => [2,3,4,1] => 1
[4,1,2,3,5] => [4,1,2,3] => [4,2,3,1] => [3,4,2,1] => 2
[4,1,2,5,3] => [4,1,2,3] => [4,2,3,1] => [3,4,2,1] => 2
[4,1,3,2,5] => [4,1,3,2] => [4,2,3,1] => [3,4,2,1] => 2
[4,1,3,5,2] => [4,1,3,2] => [4,2,3,1] => [3,4,2,1] => 2
[4,1,5,2,3] => [4,1,2,3] => [4,2,3,1] => [3,4,2,1] => 2
[4,1,5,3,2] => [4,1,3,2] => [4,2,3,1] => [3,4,2,1] => 2
[4,2,1,3,5] => [4,2,1,3] => [4,3,2,1] => [2,3,4,1] => 1
[2,3,4,5,6,1,7] => [2,3,4,5,6,1] => [6,2,3,4,5,1] => [5,6,2,3,4,1] => ? = 2
[2,3,4,5,6,7,1] => [2,3,4,5,6,1] => [6,2,3,4,5,1] => [5,6,2,3,4,1] => ? = 2
[2,3,4,5,7,6,1] => [2,3,4,5,6,1] => [6,2,3,4,5,1] => [5,6,2,3,4,1] => ? = 2
[2,3,4,6,1,5,7] => [2,3,4,6,1,5] => [5,2,3,6,1,4] => [6,5,2,1,3,4] => ? = 2
[2,3,4,6,1,7,5] => [2,3,4,6,1,5] => [5,2,3,6,1,4] => [6,5,2,1,3,4] => ? = 2
[2,3,4,6,5,1,7] => [2,3,4,6,5,1] => [6,2,3,5,4,1] => [4,5,1,6,3,2] => ? = 2
[2,3,4,6,5,7,1] => [2,3,4,6,5,1] => [6,2,3,5,4,1] => [4,5,1,6,3,2] => ? = 2
[2,3,4,6,7,1,5] => [2,3,4,6,1,5] => [5,2,3,6,1,4] => [6,5,2,1,3,4] => ? = 2
[2,3,4,6,7,5,1] => [2,3,4,6,5,1] => [6,2,3,5,4,1] => [4,5,1,6,3,2] => ? = 2
[2,3,4,7,5,6,1] => [2,3,4,5,6,1] => [6,2,3,4,5,1] => [5,6,2,3,4,1] => ? = 2
[2,3,4,7,6,1,5] => [2,3,4,6,1,5] => [5,2,3,6,1,4] => [6,5,2,1,3,4] => ? = 2
[2,3,4,7,6,5,1] => [2,3,4,6,5,1] => [6,2,3,5,4,1] => [4,5,1,6,3,2] => ? = 2
[2,3,5,1,6,4,7] => [2,3,5,1,6,4] => [4,2,6,1,5,3] => [6,4,5,2,1,3] => ? = 3
[2,3,5,1,6,7,4] => [2,3,5,1,6,4] => [4,2,6,1,5,3] => [6,4,5,2,1,3] => ? = 3
[2,3,5,1,7,6,4] => [2,3,5,1,6,4] => [4,2,6,1,5,3] => [6,4,5,2,1,3] => ? = 3
[2,3,5,4,6,1,7] => [2,3,5,4,6,1] => [6,2,4,3,5,1] => [5,6,4,2,3,1] => ? = 2
[2,3,5,4,6,7,1] => [2,3,5,4,6,1] => [6,2,4,3,5,1] => [5,6,4,2,3,1] => ? = 2
[2,3,5,4,7,6,1] => [2,3,5,4,6,1] => [6,2,4,3,5,1] => [5,6,4,2,3,1] => ? = 2
[2,3,5,6,1,4,7] => [2,3,5,6,1,4] => [5,2,6,4,1,3] => [4,6,2,5,1,3] => ? = 2
[2,3,5,6,1,7,4] => [2,3,5,6,1,4] => [5,2,6,4,1,3] => [4,6,2,5,1,3] => ? = 2
[2,3,5,6,4,1,7] => [2,3,5,6,4,1] => [6,2,5,4,3,1] => [3,4,5,6,2,1] => ? = 2
[2,3,5,6,4,7,1] => [2,3,5,6,4,1] => [6,2,5,4,3,1] => [3,4,5,6,2,1] => ? = 2
[2,3,5,6,7,1,4] => [2,3,5,6,1,4] => [5,2,6,4,1,3] => [4,6,2,5,1,3] => ? = 2
[2,3,5,6,7,4,1] => [2,3,5,6,4,1] => [6,2,5,4,3,1] => [3,4,5,6,2,1] => ? = 2
[2,3,5,7,1,6,4] => [2,3,5,1,6,4] => [4,2,6,1,5,3] => [6,4,5,2,1,3] => ? = 3
[2,3,5,7,4,6,1] => [2,3,5,4,6,1] => [6,2,4,3,5,1] => [5,6,4,2,3,1] => ? = 2
[2,3,5,7,6,1,4] => [2,3,5,6,1,4] => [5,2,6,4,1,3] => [4,6,2,5,1,3] => ? = 2
[2,3,5,7,6,4,1] => [2,3,5,6,4,1] => [6,2,5,4,3,1] => [3,4,5,6,2,1] => ? = 2
[2,3,6,1,4,5,7] => [2,3,6,1,4,5] => [4,2,6,1,5,3] => [6,4,5,2,1,3] => ? = 3
[2,3,6,1,4,7,5] => [2,3,6,1,4,5] => [4,2,6,1,5,3] => [6,4,5,2,1,3] => ? = 3
[2,3,6,1,5,4,7] => [2,3,6,1,5,4] => [4,2,6,1,5,3] => [6,4,5,2,1,3] => ? = 3
[2,3,6,1,5,7,4] => [2,3,6,1,5,4] => [4,2,6,1,5,3] => [6,4,5,2,1,3] => ? = 3
[2,3,6,1,7,4,5] => [2,3,6,1,4,5] => [4,2,6,1,5,3] => [6,4,5,2,1,3] => ? = 3
[2,3,6,1,7,5,4] => [2,3,6,1,5,4] => [4,2,6,1,5,3] => [6,4,5,2,1,3] => ? = 3
[2,3,6,4,1,5,7] => [2,3,6,4,1,5] => [5,2,6,4,1,3] => [4,6,2,5,1,3] => ? = 2
[2,3,6,4,1,7,5] => [2,3,6,4,1,5] => [5,2,6,4,1,3] => [4,6,2,5,1,3] => ? = 2
[2,3,6,4,5,1,7] => [2,3,6,4,5,1] => [6,2,5,4,3,1] => [3,4,5,6,2,1] => ? = 2
[2,3,6,4,5,7,1] => [2,3,6,4,5,1] => [6,2,5,4,3,1] => [3,4,5,6,2,1] => ? = 2
[2,3,6,4,7,1,5] => [2,3,6,4,1,5] => [5,2,6,4,1,3] => [4,6,2,5,1,3] => ? = 2
[2,3,6,4,7,5,1] => [2,3,6,4,5,1] => [6,2,5,4,3,1] => [3,4,5,6,2,1] => ? = 2
[2,3,6,5,1,4,7] => [2,3,6,5,1,4] => [5,2,6,4,1,3] => [4,6,2,5,1,3] => ? = 2
[2,3,6,5,1,7,4] => [2,3,6,5,1,4] => [5,2,6,4,1,3] => [4,6,2,5,1,3] => ? = 2
[2,3,6,5,4,1,7] => [2,3,6,5,4,1] => [6,2,5,4,3,1] => [3,4,5,6,2,1] => ? = 2
[2,3,6,5,4,7,1] => [2,3,6,5,4,1] => [6,2,5,4,3,1] => [3,4,5,6,2,1] => ? = 2
[2,3,6,5,7,1,4] => [2,3,6,5,1,4] => [5,2,6,4,1,3] => [4,6,2,5,1,3] => ? = 2
[2,3,6,5,7,4,1] => [2,3,6,5,4,1] => [6,2,5,4,3,1] => [3,4,5,6,2,1] => ? = 2
[2,3,6,7,1,4,5] => [2,3,6,1,4,5] => [4,2,6,1,5,3] => [6,4,5,2,1,3] => ? = 3
[2,3,6,7,1,5,4] => [2,3,6,1,5,4] => [4,2,6,1,5,3] => [6,4,5,2,1,3] => ? = 3
Description
The maximal modular displacement of a permutation. This is $\max_{1\leq i \leq n} \left(\min(\pi(i)-i\pmod n, i-\pi(i)\pmod n)\right)$ for a permutation $\pi$ of $\{1,\dots,n\}$.
Mp00252: Permutations restrictionPermutations
Mp00159: Permutations Demazure product with inversePermutations
Mp00064: Permutations reversePermutations
St001207: Permutations ⟶ ℤResult quality: 1% values known / values provided: 1%distinct values known / distinct values provided: 50%
Values
[1,2] => [1] => [1] => [1] => ? = 0 - 1
[2,1] => [1] => [1] => [1] => ? = 0 - 1
[2,1,3] => [2,1] => [2,1] => [1,2] => 0 = 1 - 1
[2,3,1] => [2,1] => [2,1] => [1,2] => 0 = 1 - 1
[3,2,1] => [2,1] => [2,1] => [1,2] => 0 = 1 - 1
[2,3,1,4] => [2,3,1] => [3,2,1] => [1,2,3] => 0 = 1 - 1
[2,3,4,1] => [2,3,1] => [3,2,1] => [1,2,3] => 0 = 1 - 1
[2,4,3,1] => [2,3,1] => [3,2,1] => [1,2,3] => 0 = 1 - 1
[3,1,2,4] => [3,1,2] => [3,2,1] => [1,2,3] => 0 = 1 - 1
[3,1,4,2] => [3,1,2] => [3,2,1] => [1,2,3] => 0 = 1 - 1
[3,2,1,4] => [3,2,1] => [3,2,1] => [1,2,3] => 0 = 1 - 1
[3,2,4,1] => [3,2,1] => [3,2,1] => [1,2,3] => 0 = 1 - 1
[3,4,1,2] => [3,1,2] => [3,2,1] => [1,2,3] => 0 = 1 - 1
[3,4,2,1] => [3,2,1] => [3,2,1] => [1,2,3] => 0 = 1 - 1
[4,2,3,1] => [2,3,1] => [3,2,1] => [1,2,3] => 0 = 1 - 1
[4,3,1,2] => [3,1,2] => [3,2,1] => [1,2,3] => 0 = 1 - 1
[4,3,2,1] => [3,2,1] => [3,2,1] => [1,2,3] => 0 = 1 - 1
[2,3,4,1,5] => [2,3,4,1] => [4,2,3,1] => [1,3,2,4] => 1 = 2 - 1
[2,3,4,5,1] => [2,3,4,1] => [4,2,3,1] => [1,3,2,4] => 1 = 2 - 1
[2,3,5,4,1] => [2,3,4,1] => [4,2,3,1] => [1,3,2,4] => 1 = 2 - 1
[2,4,1,3,5] => [2,4,1,3] => [3,4,1,2] => [2,1,4,3] => 1 = 2 - 1
[2,4,1,5,3] => [2,4,1,3] => [3,4,1,2] => [2,1,4,3] => 1 = 2 - 1
[2,4,3,1,5] => [2,4,3,1] => [4,3,2,1] => [1,2,3,4] => 0 = 1 - 1
[2,4,3,5,1] => [2,4,3,1] => [4,3,2,1] => [1,2,3,4] => 0 = 1 - 1
[2,4,5,1,3] => [2,4,1,3] => [3,4,1,2] => [2,1,4,3] => 1 = 2 - 1
[2,4,5,3,1] => [2,4,3,1] => [4,3,2,1] => [1,2,3,4] => 0 = 1 - 1
[2,5,3,4,1] => [2,3,4,1] => [4,2,3,1] => [1,3,2,4] => 1 = 2 - 1
[2,5,4,1,3] => [2,4,1,3] => [3,4,1,2] => [2,1,4,3] => 1 = 2 - 1
[2,5,4,3,1] => [2,4,3,1] => [4,3,2,1] => [1,2,3,4] => 0 = 1 - 1
[3,1,4,2,5] => [3,1,4,2] => [4,2,3,1] => [1,3,2,4] => 1 = 2 - 1
[3,1,4,5,2] => [3,1,4,2] => [4,2,3,1] => [1,3,2,4] => 1 = 2 - 1
[3,1,5,4,2] => [3,1,4,2] => [4,2,3,1] => [1,3,2,4] => 1 = 2 - 1
[3,2,4,1,5] => [3,2,4,1] => [4,2,3,1] => [1,3,2,4] => 1 = 2 - 1
[3,2,4,5,1] => [3,2,4,1] => [4,2,3,1] => [1,3,2,4] => 1 = 2 - 1
[3,2,5,4,1] => [3,2,4,1] => [4,2,3,1] => [1,3,2,4] => 1 = 2 - 1
[3,4,1,2,5] => [3,4,1,2] => [4,3,2,1] => [1,2,3,4] => 0 = 1 - 1
[3,4,1,5,2] => [3,4,1,2] => [4,3,2,1] => [1,2,3,4] => 0 = 1 - 1
[3,4,2,1,5] => [3,4,2,1] => [4,3,2,1] => [1,2,3,4] => 0 = 1 - 1
[3,4,2,5,1] => [3,4,2,1] => [4,3,2,1] => [1,2,3,4] => 0 = 1 - 1
[3,4,5,1,2] => [3,4,1,2] => [4,3,2,1] => [1,2,3,4] => 0 = 1 - 1
[3,4,5,2,1] => [3,4,2,1] => [4,3,2,1] => [1,2,3,4] => 0 = 1 - 1
[3,5,1,4,2] => [3,1,4,2] => [4,2,3,1] => [1,3,2,4] => 1 = 2 - 1
[3,5,2,4,1] => [3,2,4,1] => [4,2,3,1] => [1,3,2,4] => 1 = 2 - 1
[3,5,4,1,2] => [3,4,1,2] => [4,3,2,1] => [1,2,3,4] => 0 = 1 - 1
[3,5,4,2,1] => [3,4,2,1] => [4,3,2,1] => [1,2,3,4] => 0 = 1 - 1
[4,1,2,3,5] => [4,1,2,3] => [4,2,3,1] => [1,3,2,4] => 1 = 2 - 1
[4,1,2,5,3] => [4,1,2,3] => [4,2,3,1] => [1,3,2,4] => 1 = 2 - 1
[4,1,3,2,5] => [4,1,3,2] => [4,2,3,1] => [1,3,2,4] => 1 = 2 - 1
[4,1,3,5,2] => [4,1,3,2] => [4,2,3,1] => [1,3,2,4] => 1 = 2 - 1
[4,1,5,2,3] => [4,1,2,3] => [4,2,3,1] => [1,3,2,4] => 1 = 2 - 1
[4,1,5,3,2] => [4,1,3,2] => [4,2,3,1] => [1,3,2,4] => 1 = 2 - 1
[4,2,1,3,5] => [4,2,1,3] => [4,3,2,1] => [1,2,3,4] => 0 = 1 - 1
[2,3,4,5,1,6] => [2,3,4,5,1] => [5,2,3,4,1] => [1,4,3,2,5] => ? = 2 - 1
[2,3,4,5,6,1] => [2,3,4,5,1] => [5,2,3,4,1] => [1,4,3,2,5] => ? = 2 - 1
[2,3,4,6,5,1] => [2,3,4,5,1] => [5,2,3,4,1] => [1,4,3,2,5] => ? = 2 - 1
[2,3,5,1,4,6] => [2,3,5,1,4] => [4,2,5,1,3] => [3,1,5,2,4] => ? = 2 - 1
[2,3,5,1,6,4] => [2,3,5,1,4] => [4,2,5,1,3] => [3,1,5,2,4] => ? = 2 - 1
[2,3,5,4,1,6] => [2,3,5,4,1] => [5,2,4,3,1] => [1,3,4,2,5] => ? = 2 - 1
[2,3,5,4,6,1] => [2,3,5,4,1] => [5,2,4,3,1] => [1,3,4,2,5] => ? = 2 - 1
[2,3,5,6,1,4] => [2,3,5,1,4] => [4,2,5,1,3] => [3,1,5,2,4] => ? = 2 - 1
[2,3,5,6,4,1] => [2,3,5,4,1] => [5,2,4,3,1] => [1,3,4,2,5] => ? = 2 - 1
[2,3,6,4,5,1] => [2,3,4,5,1] => [5,2,3,4,1] => [1,4,3,2,5] => ? = 2 - 1
[2,3,6,5,1,4] => [2,3,5,1,4] => [4,2,5,1,3] => [3,1,5,2,4] => ? = 2 - 1
[2,3,6,5,4,1] => [2,3,5,4,1] => [5,2,4,3,1] => [1,3,4,2,5] => ? = 2 - 1
[2,4,1,5,3,6] => [2,4,1,5,3] => [3,5,1,4,2] => [2,4,1,5,3] => ? = 2 - 1
[2,4,1,5,6,3] => [2,4,1,5,3] => [3,5,1,4,2] => [2,4,1,5,3] => ? = 2 - 1
[2,4,1,6,5,3] => [2,4,1,5,3] => [3,5,1,4,2] => [2,4,1,5,3] => ? = 2 - 1
[2,4,3,5,1,6] => [2,4,3,5,1] => [5,3,2,4,1] => [1,4,2,3,5] => ? = 2 - 1
[2,4,3,5,6,1] => [2,4,3,5,1] => [5,3,2,4,1] => [1,4,2,3,5] => ? = 2 - 1
[2,4,3,6,5,1] => [2,4,3,5,1] => [5,3,2,4,1] => [1,4,2,3,5] => ? = 2 - 1
[2,4,5,1,3,6] => [2,4,5,1,3] => [4,5,3,1,2] => [2,1,3,5,4] => ? = 2 - 1
[2,4,5,1,6,3] => [2,4,5,1,3] => [4,5,3,1,2] => [2,1,3,5,4] => ? = 2 - 1
[2,4,5,3,1,6] => [2,4,5,3,1] => [5,4,3,2,1] => [1,2,3,4,5] => ? = 1 - 1
[2,4,5,3,6,1] => [2,4,5,3,1] => [5,4,3,2,1] => [1,2,3,4,5] => ? = 1 - 1
[2,4,5,6,1,3] => [2,4,5,1,3] => [4,5,3,1,2] => [2,1,3,5,4] => ? = 2 - 1
[2,4,5,6,3,1] => [2,4,5,3,1] => [5,4,3,2,1] => [1,2,3,4,5] => ? = 1 - 1
[2,4,6,1,5,3] => [2,4,1,5,3] => [3,5,1,4,2] => [2,4,1,5,3] => ? = 2 - 1
[2,4,6,3,5,1] => [2,4,3,5,1] => [5,3,2,4,1] => [1,4,2,3,5] => ? = 2 - 1
[2,4,6,5,1,3] => [2,4,5,1,3] => [4,5,3,1,2] => [2,1,3,5,4] => ? = 2 - 1
[2,4,6,5,3,1] => [2,4,5,3,1] => [5,4,3,2,1] => [1,2,3,4,5] => ? = 1 - 1
[2,5,1,3,4,6] => [2,5,1,3,4] => [3,5,1,4,2] => [2,4,1,5,3] => ? = 2 - 1
[2,5,1,3,6,4] => [2,5,1,3,4] => [3,5,1,4,2] => [2,4,1,5,3] => ? = 2 - 1
[2,5,1,4,3,6] => [2,5,1,4,3] => [3,5,1,4,2] => [2,4,1,5,3] => ? = 2 - 1
[2,5,1,4,6,3] => [2,5,1,4,3] => [3,5,1,4,2] => [2,4,1,5,3] => ? = 2 - 1
[2,5,1,6,3,4] => [2,5,1,3,4] => [3,5,1,4,2] => [2,4,1,5,3] => ? = 2 - 1
[2,5,1,6,4,3] => [2,5,1,4,3] => [3,5,1,4,2] => [2,4,1,5,3] => ? = 2 - 1
[2,5,3,1,4,6] => [2,5,3,1,4] => [4,5,3,1,2] => [2,1,3,5,4] => ? = 2 - 1
[2,5,3,1,6,4] => [2,5,3,1,4] => [4,5,3,1,2] => [2,1,3,5,4] => ? = 2 - 1
[2,5,3,4,1,6] => [2,5,3,4,1] => [5,4,3,2,1] => [1,2,3,4,5] => ? = 1 - 1
[2,5,3,4,6,1] => [2,5,3,4,1] => [5,4,3,2,1] => [1,2,3,4,5] => ? = 1 - 1
[2,5,3,6,1,4] => [2,5,3,1,4] => [4,5,3,1,2] => [2,1,3,5,4] => ? = 2 - 1
[2,5,3,6,4,1] => [2,5,3,4,1] => [5,4,3,2,1] => [1,2,3,4,5] => ? = 1 - 1
[2,5,4,1,3,6] => [2,5,4,1,3] => [4,5,3,1,2] => [2,1,3,5,4] => ? = 2 - 1
[2,5,4,1,6,3] => [2,5,4,1,3] => [4,5,3,1,2] => [2,1,3,5,4] => ? = 2 - 1
[2,5,4,3,1,6] => [2,5,4,3,1] => [5,4,3,2,1] => [1,2,3,4,5] => ? = 1 - 1
[2,5,4,3,6,1] => [2,5,4,3,1] => [5,4,3,2,1] => [1,2,3,4,5] => ? = 1 - 1
[2,5,4,6,1,3] => [2,5,4,1,3] => [4,5,3,1,2] => [2,1,3,5,4] => ? = 2 - 1
[2,5,4,6,3,1] => [2,5,4,3,1] => [5,4,3,2,1] => [1,2,3,4,5] => ? = 1 - 1
[2,5,6,1,3,4] => [2,5,1,3,4] => [3,5,1,4,2] => [2,4,1,5,3] => ? = 2 - 1
[2,5,6,1,4,3] => [2,5,1,4,3] => [3,5,1,4,2] => [2,4,1,5,3] => ? = 2 - 1
Description
The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$.
Matching statistic: St001200
Mp00252: Permutations restrictionPermutations
Mp00159: Permutations Demazure product with inversePermutations
Mp00127: Permutations left-to-right-maxima to Dyck pathDyck paths
St001200: Dyck paths ⟶ ℤResult quality: 1% values known / values provided: 1%distinct values known / distinct values provided: 25%
Values
[1,2] => [1] => [1] => [1,0]
=> ? = 0
[2,1] => [1] => [1] => [1,0]
=> ? = 0
[2,1,3] => [2,1] => [2,1] => [1,1,0,0]
=> ? = 1
[2,3,1] => [2,1] => [2,1] => [1,1,0,0]
=> ? = 1
[3,2,1] => [2,1] => [2,1] => [1,1,0,0]
=> ? = 1
[2,3,1,4] => [2,3,1] => [3,2,1] => [1,1,1,0,0,0]
=> ? = 1
[2,3,4,1] => [2,3,1] => [3,2,1] => [1,1,1,0,0,0]
=> ? = 1
[2,4,3,1] => [2,3,1] => [3,2,1] => [1,1,1,0,0,0]
=> ? = 1
[3,1,2,4] => [3,1,2] => [3,2,1] => [1,1,1,0,0,0]
=> ? = 1
[3,1,4,2] => [3,1,2] => [3,2,1] => [1,1,1,0,0,0]
=> ? = 1
[3,2,1,4] => [3,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> ? = 1
[3,2,4,1] => [3,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> ? = 1
[3,4,1,2] => [3,1,2] => [3,2,1] => [1,1,1,0,0,0]
=> ? = 1
[3,4,2,1] => [3,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> ? = 1
[4,2,3,1] => [2,3,1] => [3,2,1] => [1,1,1,0,0,0]
=> ? = 1
[4,3,1,2] => [3,1,2] => [3,2,1] => [1,1,1,0,0,0]
=> ? = 1
[4,3,2,1] => [3,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> ? = 1
[2,3,4,1,5] => [2,3,4,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? = 2
[2,3,4,5,1] => [2,3,4,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? = 2
[2,3,5,4,1] => [2,3,4,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? = 2
[2,4,1,3,5] => [2,4,1,3] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 2
[2,4,1,5,3] => [2,4,1,3] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 2
[2,4,3,1,5] => [2,4,3,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 1
[2,4,3,5,1] => [2,4,3,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 1
[2,4,5,1,3] => [2,4,1,3] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 2
[2,4,5,3,1] => [2,4,3,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 1
[2,5,3,4,1] => [2,3,4,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? = 2
[2,5,4,1,3] => [2,4,1,3] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 2
[2,5,4,3,1] => [2,4,3,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 1
[3,1,4,2,5] => [3,1,4,2] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? = 2
[3,1,4,5,2] => [3,1,4,2] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? = 2
[3,1,5,4,2] => [3,1,4,2] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? = 2
[3,2,4,1,5] => [3,2,4,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? = 2
[3,2,4,5,1] => [3,2,4,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? = 2
[3,2,5,4,1] => [3,2,4,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? = 2
[3,4,1,2,5] => [3,4,1,2] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 1
[3,4,1,5,2] => [3,4,1,2] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 1
[3,4,2,1,5] => [3,4,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 1
[3,4,2,5,1] => [3,4,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 1
[3,4,5,1,2] => [3,4,1,2] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 1
[3,4,5,2,1] => [3,4,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 1
[3,5,1,4,2] => [3,1,4,2] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? = 2
[3,5,2,4,1] => [3,2,4,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? = 2
[3,5,4,1,2] => [3,4,1,2] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 1
[3,5,4,2,1] => [3,4,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 1
[4,1,2,3,5] => [4,1,2,3] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? = 2
[4,1,2,5,3] => [4,1,2,3] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? = 2
[4,1,3,2,5] => [4,1,3,2] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? = 2
[4,1,3,5,2] => [4,1,3,2] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? = 2
[4,1,5,2,3] => [4,1,2,3] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? = 2
[4,1,5,3,2] => [4,1,3,2] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? = 2
[4,2,1,3,5] => [4,2,1,3] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 1
[4,2,1,5,3] => [4,2,1,3] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 1
[4,2,3,1,5] => [4,2,3,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 1
[5,2,4,1,3] => [2,4,1,3] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 2
[2,3,5,1,4,6] => [2,3,5,1,4] => [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> 2
[2,3,5,1,6,4] => [2,3,5,1,4] => [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> 2
[2,3,5,6,1,4] => [2,3,5,1,4] => [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> 2
[2,3,6,5,1,4] => [2,3,5,1,4] => [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> 2
[2,4,1,5,3,6] => [2,4,1,5,3] => [3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0]
=> 2
[2,4,1,5,6,3] => [2,4,1,5,3] => [3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0]
=> 2
[2,4,1,6,5,3] => [2,4,1,5,3] => [3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0]
=> 2
[2,4,5,1,3,6] => [2,4,5,1,3] => [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
=> 2
[2,4,5,1,6,3] => [2,4,5,1,3] => [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
=> 2
[2,4,5,6,1,3] => [2,4,5,1,3] => [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
=> 2
[2,4,6,1,5,3] => [2,4,1,5,3] => [3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0]
=> 2
[2,4,6,5,1,3] => [2,4,5,1,3] => [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
=> 2
[2,5,1,3,4,6] => [2,5,1,3,4] => [3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0]
=> 2
[2,5,1,3,6,4] => [2,5,1,3,4] => [3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0]
=> 2
[2,5,1,4,3,6] => [2,5,1,4,3] => [3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0]
=> 2
[2,5,1,4,6,3] => [2,5,1,4,3] => [3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0]
=> 2
[2,5,1,6,3,4] => [2,5,1,3,4] => [3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0]
=> 2
[2,5,1,6,4,3] => [2,5,1,4,3] => [3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0]
=> 2
[2,5,3,1,4,6] => [2,5,3,1,4] => [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
=> 2
[2,5,3,1,6,4] => [2,5,3,1,4] => [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
=> 2
[2,5,3,6,1,4] => [2,5,3,1,4] => [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
=> 2
[2,5,4,1,3,6] => [2,5,4,1,3] => [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
=> 2
[2,5,4,1,6,3] => [2,5,4,1,3] => [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
=> 2
[2,5,4,6,1,3] => [2,5,4,1,3] => [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
=> 2
[2,5,6,1,3,4] => [2,5,1,3,4] => [3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0]
=> 2
[2,5,6,1,4,3] => [2,5,1,4,3] => [3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0]
=> 2
[2,5,6,3,1,4] => [2,5,3,1,4] => [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
=> 2
[2,5,6,4,1,3] => [2,5,4,1,3] => [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
=> 2
[2,6,3,5,1,4] => [2,3,5,1,4] => [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> 2
[2,6,4,1,5,3] => [2,4,1,5,3] => [3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0]
=> 2
[2,6,4,5,1,3] => [2,4,5,1,3] => [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
=> 2
[2,6,5,1,3,4] => [2,5,1,3,4] => [3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0]
=> 2
[2,6,5,1,4,3] => [2,5,1,4,3] => [3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0]
=> 2
[2,6,5,3,1,4] => [2,5,3,1,4] => [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
=> 2
[2,6,5,4,1,3] => [2,5,4,1,3] => [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
=> 2
[3,1,5,2,4,6] => [3,1,5,2,4] => [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> 2
[3,1,5,2,6,4] => [3,1,5,2,4] => [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> 2
[3,1,5,6,2,4] => [3,1,5,2,4] => [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> 2
[3,1,6,5,2,4] => [3,1,5,2,4] => [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> 2
[3,2,5,1,4,6] => [3,2,5,1,4] => [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> 2
[3,2,5,1,6,4] => [3,2,5,1,4] => [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> 2
[3,2,5,6,1,4] => [3,2,5,1,4] => [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> 2
[3,2,6,5,1,4] => [3,2,5,1,4] => [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> 2
[3,5,1,2,4,6] => [3,5,1,2,4] => [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
=> 2
[3,5,1,2,6,4] => [3,5,1,2,4] => [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
=> 2
Description
The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$.
Matching statistic: St001880
Mp00159: Permutations Demazure product with inversePermutations
Mp00062: Permutations Lehmer-code to major-code bijectionPermutations
Mp00065: Permutations permutation posetPosets
St001880: Posets ⟶ ℤResult quality: 0% values known / values provided: 0%distinct values known / distinct values provided: 50%
Values
[1,2] => [1,2] => [1,2] => ([(0,1)],2)
=> ? = 0 - 1
[2,1] => [2,1] => [2,1] => ([],2)
=> ? = 0 - 1
[2,1,3] => [2,1,3] => [2,1,3] => ([(0,2),(1,2)],3)
=> ? = 1 - 1
[2,3,1] => [3,2,1] => [3,2,1] => ([],3)
=> ? = 1 - 1
[3,2,1] => [3,2,1] => [3,2,1] => ([],3)
=> ? = 1 - 1
[2,3,1,4] => [3,2,1,4] => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> ? = 1 - 1
[2,3,4,1] => [4,2,3,1] => [2,4,3,1] => ([(1,2),(1,3)],4)
=> ? = 1 - 1
[2,4,3,1] => [4,3,2,1] => [4,3,2,1] => ([],4)
=> ? = 1 - 1
[3,1,2,4] => [3,2,1,4] => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> ? = 1 - 1
[3,1,4,2] => [4,2,3,1] => [2,4,3,1] => ([(1,2),(1,3)],4)
=> ? = 1 - 1
[3,2,1,4] => [3,2,1,4] => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> ? = 1 - 1
[3,2,4,1] => [4,2,3,1] => [2,4,3,1] => ([(1,2),(1,3)],4)
=> ? = 1 - 1
[3,4,1,2] => [4,3,2,1] => [4,3,2,1] => ([],4)
=> ? = 1 - 1
[3,4,2,1] => [4,3,2,1] => [4,3,2,1] => ([],4)
=> ? = 1 - 1
[4,2,3,1] => [4,3,2,1] => [4,3,2,1] => ([],4)
=> ? = 1 - 1
[4,3,1,2] => [4,3,2,1] => [4,3,2,1] => ([],4)
=> ? = 1 - 1
[4,3,2,1] => [4,3,2,1] => [4,3,2,1] => ([],4)
=> ? = 1 - 1
[2,3,4,1,5] => [4,2,3,1,5] => [2,4,3,1,5] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[2,3,4,5,1] => [5,2,3,4,1] => [2,3,5,4,1] => ([(1,4),(4,2),(4,3)],5)
=> ? = 2 - 1
[2,3,5,4,1] => [5,2,4,3,1] => [5,2,4,3,1] => ([(2,3),(2,4)],5)
=> ? = 2 - 1
[2,4,1,3,5] => [3,4,1,2,5] => [3,1,4,2,5] => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[2,4,1,5,3] => [3,5,1,4,2] => [1,5,3,4,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> ? = 2 - 1
[2,4,3,1,5] => [4,3,2,1,5] => [4,3,2,1,5] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? = 1 - 1
[2,4,3,5,1] => [5,3,2,4,1] => [3,2,5,4,1] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ? = 1 - 1
[2,4,5,1,3] => [4,5,3,1,2] => [4,1,5,3,2] => ([(0,4),(1,2),(1,3),(1,4)],5)
=> ? = 2 - 1
[2,4,5,3,1] => [5,4,3,2,1] => [5,4,3,2,1] => ([],5)
=> ? = 1 - 1
[2,5,3,4,1] => [5,4,3,2,1] => [5,4,3,2,1] => ([],5)
=> ? = 2 - 1
[2,5,4,1,3] => [4,5,3,1,2] => [4,1,5,3,2] => ([(0,4),(1,2),(1,3),(1,4)],5)
=> ? = 2 - 1
[2,5,4,3,1] => [5,4,3,2,1] => [5,4,3,2,1] => ([],5)
=> ? = 1 - 1
[3,1,4,2,5] => [4,2,3,1,5] => [2,4,3,1,5] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[3,1,4,5,2] => [5,2,3,4,1] => [2,3,5,4,1] => ([(1,4),(4,2),(4,3)],5)
=> ? = 2 - 1
[3,1,5,4,2] => [5,2,4,3,1] => [5,2,4,3,1] => ([(2,3),(2,4)],5)
=> ? = 2 - 1
[3,2,4,1,5] => [4,2,3,1,5] => [2,4,3,1,5] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[3,2,4,5,1] => [5,2,3,4,1] => [2,3,5,4,1] => ([(1,4),(4,2),(4,3)],5)
=> ? = 2 - 1
[3,2,5,4,1] => [5,2,4,3,1] => [5,2,4,3,1] => ([(2,3),(2,4)],5)
=> ? = 2 - 1
[3,4,1,2,5] => [4,3,2,1,5] => [4,3,2,1,5] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? = 1 - 1
[3,4,1,5,2] => [5,3,2,4,1] => [3,2,5,4,1] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ? = 1 - 1
[3,4,2,1,5] => [4,3,2,1,5] => [4,3,2,1,5] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? = 1 - 1
[3,4,2,5,1] => [5,3,2,4,1] => [3,2,5,4,1] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ? = 1 - 1
[3,4,5,1,2] => [5,4,3,2,1] => [5,4,3,2,1] => ([],5)
=> ? = 1 - 1
[3,4,5,2,1] => [5,4,3,2,1] => [5,4,3,2,1] => ([],5)
=> ? = 1 - 1
[3,5,1,4,2] => [5,4,3,2,1] => [5,4,3,2,1] => ([],5)
=> ? = 2 - 1
[3,5,2,4,1] => [5,4,3,2,1] => [5,4,3,2,1] => ([],5)
=> ? = 2 - 1
[3,5,4,1,2] => [5,4,3,2,1] => [5,4,3,2,1] => ([],5)
=> ? = 1 - 1
[3,5,4,2,1] => [5,4,3,2,1] => [5,4,3,2,1] => ([],5)
=> ? = 1 - 1
[4,1,2,3,5] => [4,2,3,1,5] => [2,4,3,1,5] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[4,1,2,5,3] => [5,2,3,4,1] => [2,3,5,4,1] => ([(1,4),(4,2),(4,3)],5)
=> ? = 2 - 1
[4,1,3,2,5] => [4,2,3,1,5] => [2,4,3,1,5] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[4,1,3,5,2] => [5,2,3,4,1] => [2,3,5,4,1] => ([(1,4),(4,2),(4,3)],5)
=> ? = 2 - 1
[4,1,5,2,3] => [5,2,4,3,1] => [5,2,4,3,1] => ([(2,3),(2,4)],5)
=> ? = 2 - 1
[2,4,1,5,3,6] => [3,5,1,4,2,6] => [1,5,3,4,2,6] => ([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6)
=> 1 = 2 - 1
[2,5,1,3,4,6] => [3,5,1,4,2,6] => [1,5,3,4,2,6] => ([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6)
=> 1 = 2 - 1
[2,5,1,4,3,6] => [3,5,1,4,2,6] => [1,5,3,4,2,6] => ([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6)
=> 1 = 2 - 1
[2,4,1,5,6,3,7] => [3,6,1,4,5,2,7] => [1,4,6,3,5,2,7] => ([(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,1),(4,5),(5,6)],7)
=> 1 = 2 - 1
[2,4,1,6,3,5,7] => [3,5,1,6,2,4,7] => [1,5,6,3,2,4,7] => ([(0,2),(0,3),(0,4),(1,6),(2,5),(3,5),(4,1),(5,6)],7)
=> 2 = 3 - 1
[2,4,5,1,6,3,7] => [4,6,3,1,5,2,7] => [1,6,4,3,5,2,7] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> 1 = 2 - 1
[2,4,6,1,5,3,7] => [4,6,5,1,3,2,7] => [1,6,4,5,3,2,7] => ([(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1)],7)
=> 1 = 2 - 1
[2,5,1,3,6,4,7] => [3,6,1,4,5,2,7] => [1,4,6,3,5,2,7] => ([(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,1),(4,5),(5,6)],7)
=> 1 = 2 - 1
[2,5,1,4,6,3,7] => [3,6,1,4,5,2,7] => [1,4,6,3,5,2,7] => ([(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,1),(4,5),(5,6)],7)
=> 1 = 2 - 1
[2,5,3,1,6,4,7] => [4,6,3,1,5,2,7] => [1,6,4,3,5,2,7] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> 1 = 2 - 1
[2,5,4,1,6,3,7] => [4,6,3,1,5,2,7] => [1,6,4,3,5,2,7] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> 1 = 2 - 1
[2,5,6,1,3,4,7] => [4,6,5,1,3,2,7] => [1,6,4,5,3,2,7] => ([(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1)],7)
=> 1 = 2 - 1
[2,5,6,1,4,3,7] => [4,6,5,1,3,2,7] => [1,6,4,5,3,2,7] => ([(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1)],7)
=> 1 = 2 - 1
[2,6,1,3,4,5,7] => [3,6,1,4,5,2,7] => [1,4,6,3,5,2,7] => ([(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,1),(4,5),(5,6)],7)
=> 1 = 2 - 1
[2,6,1,3,5,4,7] => [3,6,1,4,5,2,7] => [1,4,6,3,5,2,7] => ([(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,1),(4,5),(5,6)],7)
=> 1 = 2 - 1
[2,6,3,1,4,5,7] => [4,6,3,1,5,2,7] => [1,6,4,3,5,2,7] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> 1 = 2 - 1
[2,6,3,1,5,4,7] => [4,6,3,1,5,2,7] => [1,6,4,3,5,2,7] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> 1 = 2 - 1
[2,6,4,1,3,5,7] => [4,6,5,1,3,2,7] => [1,6,4,5,3,2,7] => ([(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1)],7)
=> 1 = 2 - 1
[2,6,4,1,5,3,7] => [4,6,5,1,3,2,7] => [1,6,4,5,3,2,7] => ([(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1)],7)
=> 1 = 2 - 1
[2,6,5,1,3,4,7] => [4,6,5,1,3,2,7] => [1,6,4,5,3,2,7] => ([(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1)],7)
=> 1 = 2 - 1
[2,6,5,1,4,3,7] => [4,6,5,1,3,2,7] => [1,6,4,5,3,2,7] => ([(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1)],7)
=> 1 = 2 - 1
[3,5,1,2,6,4,7] => [4,6,3,1,5,2,7] => [1,6,4,3,5,2,7] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> 1 = 2 - 1
[3,5,2,1,6,4,7] => [4,6,3,1,5,2,7] => [1,6,4,3,5,2,7] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> 1 = 2 - 1
[3,6,1,2,4,5,7] => [4,6,3,1,5,2,7] => [1,6,4,3,5,2,7] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> 1 = 2 - 1
[3,6,1,2,5,4,7] => [4,6,3,1,5,2,7] => [1,6,4,3,5,2,7] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> 1 = 2 - 1
[3,6,2,1,4,5,7] => [4,6,3,1,5,2,7] => [1,6,4,3,5,2,7] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> 1 = 2 - 1
[3,6,2,1,5,4,7] => [4,6,3,1,5,2,7] => [1,6,4,3,5,2,7] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> 1 = 2 - 1
Description
The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice.
Matching statistic: St001890
Mp00252: Permutations restrictionPermutations
Mp00209: Permutations pattern posetPosets
St001890: Posets ⟶ ℤResult quality: 0% values known / values provided: 0%distinct values known / distinct values provided: 25%
Values
[1,2] => [1] => ([],1)
=> ? = 0
[2,1] => [1] => ([],1)
=> ? = 0
[2,1,3] => [2,1] => ([(0,1)],2)
=> 1
[2,3,1] => [2,1] => ([(0,1)],2)
=> 1
[3,2,1] => [2,1] => ([(0,1)],2)
=> 1
[2,3,1,4] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[2,3,4,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[2,4,3,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[3,1,2,4] => [3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[3,1,4,2] => [3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[3,2,1,4] => [3,2,1] => ([(0,2),(2,1)],3)
=> 1
[3,2,4,1] => [3,2,1] => ([(0,2),(2,1)],3)
=> 1
[3,4,1,2] => [3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[3,4,2,1] => [3,2,1] => ([(0,2),(2,1)],3)
=> 1
[4,2,3,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[4,3,1,2] => [3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[4,3,2,1] => [3,2,1] => ([(0,2),(2,1)],3)
=> 1
[2,3,4,1,5] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 2
[2,3,4,5,1] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 2
[2,3,5,4,1] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 2
[2,4,1,3,5] => [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 2
[2,4,1,5,3] => [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 2
[2,4,3,1,5] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 1
[2,4,3,5,1] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 1
[2,4,5,1,3] => [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 2
[2,4,5,3,1] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 1
[2,5,3,4,1] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 2
[2,5,4,1,3] => [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 2
[2,5,4,3,1] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 1
[3,1,4,2,5] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 2
[3,1,4,5,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 2
[3,1,5,4,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 2
[3,2,4,1,5] => [3,2,4,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 2
[3,2,4,5,1] => [3,2,4,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 2
[3,2,5,4,1] => [3,2,4,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 2
[3,4,1,2,5] => [3,4,1,2] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ? = 1
[3,4,1,5,2] => [3,4,1,2] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ? = 1
[3,4,2,1,5] => [3,4,2,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 1
[3,4,2,5,1] => [3,4,2,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 1
[3,4,5,1,2] => [3,4,1,2] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ? = 1
[3,4,5,2,1] => [3,4,2,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 1
[3,5,1,4,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 2
[3,5,2,4,1] => [3,2,4,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 2
[3,5,4,1,2] => [3,4,1,2] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ? = 1
[3,5,4,2,1] => [3,4,2,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 1
[4,1,2,3,5] => [4,1,2,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 2
[4,1,2,5,3] => [4,1,2,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 2
[4,1,3,2,5] => [4,1,3,2] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 2
[4,1,3,5,2] => [4,1,3,2] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 2
[4,1,5,2,3] => [4,1,2,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 2
[4,1,5,3,2] => [4,1,3,2] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 2
[4,2,1,3,5] => [4,2,1,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 1
[4,2,1,5,3] => [4,2,1,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 1
[4,2,3,1,5] => [4,2,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 1
[4,2,3,5,1] => [4,2,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 1
[4,2,5,1,3] => [4,2,1,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 1
[4,2,5,3,1] => [4,2,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 1
[4,3,1,2,5] => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 1
[4,3,1,5,2] => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 1
[4,3,2,1,5] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> 1
[4,3,2,5,1] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> 1
[4,3,5,1,2] => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 1
[4,3,5,2,1] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> 1
[4,5,1,2,3] => [4,1,2,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 2
[4,5,1,3,2] => [4,1,3,2] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 2
[4,5,2,1,3] => [4,2,1,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 1
[4,5,2,3,1] => [4,2,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 1
[4,5,3,1,2] => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 1
[4,5,3,2,1] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> 1
[5,4,3,2,1] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> 1
[5,4,3,2,1,6] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[5,4,3,2,6,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[5,4,3,6,2,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[5,4,6,3,2,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[5,6,4,3,2,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[6,5,4,3,2,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
Description
The maximum magnitude of the Möbius function of a poset. The '''Möbius function''' of a poset is the multiplicative inverse of the zeta function in the incidence algebra. The Möbius value $\mu(x, y)$ is equal to the signed sum of chains from $x$ to $y$, where odd-length chains are counted with a minus sign, so this statistic is bounded above by the total number of chains in the poset.
Mp00209: Permutations pattern posetPosets
Mp00205: Posets maximal antichainsLattices
St001625: Lattices ⟶ ℤResult quality: 0% values known / values provided: 0%distinct values known / distinct values provided: 50%
Values
[1,2] => ([(0,1)],2)
=> ([(0,1)],2)
=> -1 = 0 - 1
[2,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> -1 = 0 - 1
[2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[3,2,1] => ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[2,3,1,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[3,1,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[3,2,4,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[3,4,1,2] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[3,4,2,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[4,2,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[2,3,4,1,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(0,6),(2,8),(3,8),(4,7),(5,1),(6,4),(7,2),(7,3),(8,5)],9)
=> ? = 2 - 1
[2,3,4,5,1] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ? = 2 - 1
[2,3,5,4,1] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ([(0,8),(1,10),(2,10),(3,10),(4,9),(5,9),(7,2),(7,3),(8,1),(8,7),(9,6),(10,4),(10,5)],11)
=> ? = 2 - 1
[2,4,1,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ([(0,8),(2,12),(3,10),(3,11),(4,9),(4,11),(5,9),(5,10),(6,1),(7,6),(8,2),(8,3),(8,4),(8,5),(9,13),(10,13),(11,13),(12,7),(13,12)],14)
=> ? = 2 - 1
[2,4,1,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,6),(2,9),(2,11),(3,6),(3,9),(3,10),(4,7),(4,9),(4,10),(4,11),(5,7),(5,9),(5,10),(5,11),(6,13),(7,12),(7,13),(9,12),(9,13),(10,12),(10,13),(11,12),(11,13),(12,8),(13,8)],14)
=> ([(0,8),(2,10),(3,10),(4,9),(5,1),(6,5),(7,2),(7,3),(7,9),(8,4),(8,7),(9,10),(10,6)],11)
=> ? = 2 - 1
[2,4,3,1,5] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ([(0,9),(2,14),(3,12),(4,12),(5,11),(5,13),(6,10),(6,13),(7,10),(7,11),(8,2),(8,13),(9,5),(9,6),(9,7),(9,8),(10,14),(11,14),(12,1),(13,14),(14,3),(14,4)],15)
=> ? = 1 - 1
[2,4,3,5,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ([(0,9),(2,14),(3,12),(4,12),(5,11),(5,13),(6,10),(6,13),(7,10),(7,11),(8,2),(8,13),(9,5),(9,6),(9,7),(9,8),(10,14),(11,14),(12,1),(13,14),(14,3),(14,4)],15)
=> ? = 1 - 1
[2,4,5,1,3] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,7),(2,10),(3,6),(3,10),(4,6),(4,8),(4,10),(5,1),(5,7),(5,8),(5,10),(6,12),(7,11),(7,12),(8,11),(8,12),(10,11),(10,12),(11,9),(12,9)],13)
=> ([(0,8),(2,9),(3,10),(4,9),(4,10),(5,1),(6,5),(7,3),(7,4),(8,2),(8,7),(9,11),(10,11),(11,6)],12)
=> ? = 2 - 1
[2,4,5,3,1] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> ([(0,7),(1,10),(2,10),(4,9),(5,8),(6,8),(6,9),(7,4),(7,5),(7,6),(8,11),(9,11),(10,3),(11,1),(11,2)],12)
=> ? = 1 - 1
[2,5,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ([(0,9),(2,14),(3,12),(4,12),(5,11),(5,13),(6,10),(6,13),(7,10),(7,11),(8,2),(8,13),(9,5),(9,6),(9,7),(9,8),(10,14),(11,14),(12,1),(13,14),(14,3),(14,4)],15)
=> ? = 2 - 1
[2,5,4,1,3] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,9),(2,10),(3,6),(3,9),(4,7),(4,9),(4,10),(5,1),(5,6),(5,7),(5,10),(6,11),(6,12),(7,11),(7,12),(9,12),(10,11),(10,12),(11,8),(12,8)],13)
=> ([(0,9),(2,11),(3,11),(4,10),(5,1),(6,5),(7,2),(7,10),(8,4),(8,7),(9,3),(9,8),(10,11),(11,6)],12)
=> ? = 2 - 1
[2,5,4,3,1] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(0,6),(2,8),(3,8),(4,7),(5,1),(6,4),(7,2),(7,3),(8,5)],9)
=> ? = 1 - 1
[3,1,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ([(0,8),(2,12),(3,10),(3,11),(4,9),(4,11),(5,9),(5,10),(6,1),(7,6),(8,2),(8,3),(8,4),(8,5),(9,13),(10,13),(11,13),(12,7),(13,12)],14)
=> ? = 2 - 1
[3,1,4,5,2] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,9),(2,10),(3,6),(3,9),(4,7),(4,9),(4,10),(5,1),(5,6),(5,7),(5,10),(6,11),(6,12),(7,11),(7,12),(9,12),(10,11),(10,12),(11,8),(12,8)],13)
=> ([(0,9),(2,11),(3,11),(4,10),(5,1),(6,5),(7,2),(7,10),(8,4),(8,7),(9,3),(9,8),(10,11),(11,6)],12)
=> ? = 2 - 1
[3,1,5,4,2] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,7),(2,10),(3,6),(3,10),(4,6),(4,8),(4,10),(5,1),(5,7),(5,8),(5,10),(6,12),(7,11),(7,12),(8,11),(8,12),(10,11),(10,12),(11,9),(12,9)],13)
=> ([(0,8),(2,9),(3,10),(4,9),(4,10),(5,1),(6,5),(7,3),(7,4),(8,2),(8,7),(9,11),(10,11),(11,6)],12)
=> ? = 2 - 1
[3,2,4,1,5] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> ([(0,7),(1,10),(2,10),(4,9),(5,8),(6,8),(6,9),(7,4),(7,5),(7,6),(8,11),(9,11),(10,3),(11,1),(11,2)],12)
=> ? = 2 - 1
[3,2,4,5,1] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ([(0,8),(1,10),(2,10),(3,10),(4,9),(5,9),(7,2),(7,3),(8,1),(8,7),(9,6),(10,4),(10,5)],11)
=> ? = 2 - 1
[3,2,5,4,1] => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(5,9),(5,10),(6,9),(6,10),(7,10),(8,9),(8,10),(9,4),(10,4)],11)
=> ([(0,7),(2,8),(3,8),(4,8),(5,1),(6,5),(7,2),(7,3),(7,4),(8,6)],9)
=> ? = 2 - 1
[3,4,1,2,5] => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(5,9),(5,10),(6,9),(6,10),(7,10),(8,9),(8,10),(9,4),(10,4)],11)
=> ([(0,7),(2,8),(3,8),(4,8),(5,1),(6,5),(7,2),(7,3),(7,4),(8,6)],9)
=> ? = 1 - 1
[3,4,1,5,2] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,7),(2,10),(3,6),(3,10),(4,6),(4,8),(4,10),(5,1),(5,7),(5,8),(5,10),(6,12),(7,11),(7,12),(8,11),(8,12),(10,11),(10,12),(11,9),(12,9)],13)
=> ([(0,8),(2,9),(3,10),(4,9),(4,10),(5,1),(6,5),(7,3),(7,4),(8,2),(8,7),(9,11),(10,11),(11,6)],12)
=> ? = 1 - 1
[3,4,2,1,5] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ([(0,8),(1,10),(2,10),(3,10),(4,9),(5,9),(7,2),(7,3),(8,1),(8,7),(9,6),(10,4),(10,5)],11)
=> ? = 1 - 1
[3,4,2,5,1] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> ([(0,7),(1,10),(2,10),(4,9),(5,8),(6,8),(6,9),(7,4),(7,5),(7,6),(8,11),(9,11),(10,3),(11,1),(11,2)],12)
=> ? = 1 - 1
[3,4,5,1,2] => ([(0,3),(0,4),(1,8),(2,7),(2,8),(3,1),(3,5),(4,2),(4,5),(5,7),(5,8),(7,6),(8,6)],9)
=> ([(0,6),(2,7),(3,7),(4,1),(5,4),(6,2),(6,3),(7,5)],8)
=> ? = 1 - 1
[3,4,5,2,1] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ([(0,6),(1,8),(2,8),(3,7),(4,7),(6,1),(6,2),(7,5),(8,3),(8,4)],9)
=> ? = 1 - 1
[3,5,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,6),(2,9),(2,11),(3,6),(3,9),(3,10),(4,7),(4,9),(4,10),(4,11),(5,7),(5,9),(5,10),(5,11),(6,13),(7,12),(7,13),(9,12),(9,13),(10,12),(10,13),(11,12),(11,13),(12,8),(13,8)],14)
=> ([(0,8),(2,10),(3,10),(4,9),(5,1),(6,5),(7,2),(7,3),(7,9),(8,4),(8,7),(9,10),(10,6)],11)
=> ? = 2 - 1
[3,5,2,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ([(0,8),(2,12),(3,10),(3,11),(4,9),(4,11),(5,9),(5,10),(6,1),(7,6),(8,2),(8,3),(8,4),(8,5),(9,13),(10,13),(11,13),(12,7),(13,12)],14)
=> ? = 2 - 1
[3,5,4,1,2] => ([(0,2),(0,3),(0,4),(1,9),(1,10),(2,6),(2,7),(3,5),(3,6),(4,1),(4,5),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ([(0,8),(2,9),(3,9),(4,9),(5,1),(6,5),(7,3),(7,4),(8,2),(8,7),(9,6)],10)
=> ? = 1 - 1
[3,5,4,2,1] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(0,6),(2,8),(3,8),(4,7),(5,1),(6,4),(7,2),(7,3),(8,5)],9)
=> ? = 1 - 1
[4,1,2,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(0,6),(2,8),(3,8),(4,7),(5,1),(6,4),(7,2),(7,3),(8,5)],9)
=> ? = 2 - 1
[4,1,2,5,3] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,9),(2,10),(3,6),(3,9),(4,7),(4,9),(4,10),(5,1),(5,6),(5,7),(5,10),(6,11),(6,12),(7,11),(7,12),(9,12),(10,11),(10,12),(11,8),(12,8)],13)
=> ([(0,9),(2,11),(3,11),(4,10),(5,1),(6,5),(7,2),(7,10),(8,4),(8,7),(9,3),(9,8),(10,11),(11,6)],12)
=> ? = 2 - 1
[4,1,3,2,5] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ([(0,9),(2,14),(3,12),(4,12),(5,11),(5,13),(6,10),(6,13),(7,10),(7,11),(8,2),(8,13),(9,5),(9,6),(9,7),(9,8),(10,14),(11,14),(12,1),(13,14),(14,3),(14,4)],15)
=> ? = 2 - 1
[4,1,3,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,10),(1,11),(1,12),(2,7),(2,11),(2,12),(3,7),(3,9),(3,10),(4,6),(4,10),(4,12),(5,6),(5,9),(5,11),(6,14),(7,13),(9,13),(9,14),(10,13),(10,14),(11,13),(11,14),(12,13),(12,14),(13,8),(14,8)],15)
=> ([(0,10),(2,12),(3,12),(4,14),(4,16),(5,14),(5,15),(6,13),(6,15),(7,13),(7,16),(8,11),(8,15),(8,16),(9,11),(9,13),(9,14),(10,4),(10,5),(10,6),(10,7),(10,8),(10,9),(11,17),(12,1),(13,17),(14,17),(15,17),(16,17),(17,2),(17,3)],18)
=> ? = 2 - 1
[4,1,5,2,3] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,7),(2,10),(3,6),(3,10),(4,6),(4,8),(4,10),(5,1),(5,7),(5,8),(5,10),(6,12),(7,11),(7,12),(8,11),(8,12),(10,11),(10,12),(11,9),(12,9)],13)
=> ([(0,8),(2,9),(3,10),(4,9),(4,10),(5,1),(6,5),(7,3),(7,4),(8,2),(8,7),(9,11),(10,11),(11,6)],12)
=> ? = 2 - 1
[4,1,5,3,2] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,9),(2,10),(3,6),(3,9),(4,7),(4,9),(4,10),(5,1),(5,6),(5,7),(5,10),(6,11),(6,12),(7,11),(7,12),(9,12),(10,11),(10,12),(11,8),(12,8)],13)
=> ([(0,9),(2,11),(3,11),(4,10),(5,1),(6,5),(7,2),(7,10),(8,4),(8,7),(9,3),(9,8),(10,11),(11,6)],12)
=> ? = 2 - 1
[4,2,1,3,5] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> ([(0,7),(1,10),(2,10),(4,9),(5,8),(6,8),(6,9),(7,4),(7,5),(7,6),(8,11),(9,11),(10,3),(11,1),(11,2)],12)
=> ? = 1 - 1
[4,2,1,5,3] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,7),(2,10),(3,6),(3,10),(4,6),(4,8),(4,10),(5,1),(5,7),(5,8),(5,10),(6,12),(7,11),(7,12),(8,11),(8,12),(10,11),(10,12),(11,9),(12,9)],13)
=> ([(0,8),(2,9),(3,10),(4,9),(4,10),(5,1),(6,5),(7,3),(7,4),(8,2),(8,7),(9,11),(10,11),(11,6)],12)
=> ? = 1 - 1
[4,2,3,1,5] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ([(0,9),(2,14),(3,12),(4,12),(5,11),(5,13),(6,10),(6,13),(7,10),(7,11),(8,2),(8,13),(9,5),(9,6),(9,7),(9,8),(10,14),(11,14),(12,1),(13,14),(14,3),(14,4)],15)
=> ? = 1 - 1
[4,2,3,5,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ([(0,9),(2,14),(3,12),(4,12),(5,11),(5,13),(6,10),(6,13),(7,10),(7,11),(8,2),(8,13),(9,5),(9,6),(9,7),(9,8),(10,14),(11,14),(12,1),(13,14),(14,3),(14,4)],15)
=> ? = 1 - 1
[4,2,5,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,6),(2,9),(2,11),(3,6),(3,9),(3,10),(4,7),(4,9),(4,10),(4,11),(5,7),(5,9),(5,10),(5,11),(6,13),(7,12),(7,13),(9,12),(9,13),(10,12),(10,13),(11,12),(11,13),(12,8),(13,8)],14)
=> ([(0,8),(2,10),(3,10),(4,9),(5,1),(6,5),(7,2),(7,3),(7,9),(8,4),(8,7),(9,10),(10,6)],11)
=> ? = 1 - 1
[4,2,5,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ([(0,8),(2,12),(3,10),(3,11),(4,9),(4,11),(5,9),(5,10),(6,1),(7,6),(8,2),(8,3),(8,4),(8,5),(9,13),(10,13),(11,13),(12,7),(13,12)],14)
=> ? = 1 - 1
[4,3,1,2,5] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ([(0,8),(1,10),(2,10),(3,10),(4,9),(5,9),(7,2),(7,3),(8,1),(8,7),(9,6),(10,4),(10,5)],11)
=> ? = 1 - 1
[4,3,1,5,2] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,9),(2,10),(3,6),(3,9),(4,7),(4,9),(4,10),(5,1),(5,6),(5,7),(5,10),(6,11),(6,12),(7,11),(7,12),(9,12),(10,11),(10,12),(11,8),(12,8)],13)
=> ([(0,9),(2,11),(3,11),(4,10),(5,1),(6,5),(7,2),(7,10),(8,4),(8,7),(9,3),(9,8),(10,11),(11,6)],12)
=> ? = 1 - 1
[4,3,2,1,5] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ? = 1 - 1
[4,3,2,5,1] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(0,6),(2,8),(3,8),(4,7),(5,1),(6,4),(7,2),(7,3),(8,5)],9)
=> ? = 1 - 1
[4,3,5,1,2] => ([(0,2),(0,3),(0,4),(1,9),(1,10),(2,6),(2,7),(3,5),(3,6),(4,1),(4,5),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ([(0,8),(2,9),(3,9),(4,9),(5,1),(6,5),(7,3),(7,4),(8,2),(8,7),(9,6)],10)
=> ? = 1 - 1
[4,3,5,2,1] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(0,6),(2,8),(3,8),(4,7),(5,1),(6,4),(7,2),(7,3),(8,5)],9)
=> ? = 1 - 1
[4,5,1,2,3] => ([(0,3),(0,4),(1,8),(2,7),(2,8),(3,1),(3,5),(4,2),(4,5),(5,7),(5,8),(7,6),(8,6)],9)
=> ([(0,6),(2,7),(3,7),(4,1),(5,4),(6,2),(6,3),(7,5)],8)
=> ? = 2 - 1
[4,5,1,3,2] => ([(0,2),(0,3),(0,4),(1,9),(1,10),(2,6),(2,7),(3,5),(3,6),(4,1),(4,5),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ([(0,8),(2,9),(3,9),(4,9),(5,1),(6,5),(7,3),(7,4),(8,2),(8,7),(9,6)],10)
=> ? = 2 - 1
[4,5,2,1,3] => ([(0,2),(0,3),(0,4),(1,9),(1,10),(2,6),(2,7),(3,5),(3,6),(4,1),(4,5),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ([(0,8),(2,9),(3,9),(4,9),(5,1),(6,5),(7,3),(7,4),(8,2),(8,7),(9,6)],10)
=> ? = 1 - 1
[4,5,2,3,1] => ([(0,1),(0,2),(0,3),(1,7),(1,8),(2,5),(2,8),(3,5),(3,7),(3,8),(5,9),(6,4),(7,6),(7,9),(8,6),(8,9),(9,4)],10)
=> ([(0,6),(2,7),(3,7),(4,1),(5,4),(6,2),(6,3),(7,5)],8)
=> ? = 1 - 1
[5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[7,6,5,4,3,2,1] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 0 = 1 - 1
Description
The Möbius invariant of a lattice. The '''Möbius invariant''' of a lattice $L$ is the value of the Möbius function applied to least and greatest element, that is $\mu(L)=\mu_L(\hat{0},\hat{1})$, where $\hat{0}$ is the least element of $L$ and $\hat{1}$ is the greatest element of $L$. For the definition of the Möbius function, see [[St000914]].
Matching statistic: St001111
Mp00067: Permutations Foata bijectionPermutations
Mp00209: Permutations pattern posetPosets
Mp00074: Posets to graphGraphs
St001111: Graphs ⟶ ℤResult quality: 0% values known / values provided: 0%distinct values known / distinct values provided: 50%
Values
[1,2] => [1,2] => ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 0 + 1
[2,1] => [2,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 0 + 1
[2,1,3] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2 = 1 + 1
[2,3,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2 = 1 + 1
[3,2,1] => [3,2,1] => ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[2,3,1,4] => [2,3,1,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([(0,5),(0,6),(1,4),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> 2 = 1 + 1
[2,3,4,1] => [2,3,4,1] => ([(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)
=> 2 = 1 + 1
[2,4,3,1] => [4,2,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([(0,5),(0,6),(1,4),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> 2 = 1 + 1
[3,1,2,4] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([(0,5),(0,6),(1,4),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> 2 = 1 + 1
[3,1,4,2] => [3,4,1,2] => ([(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)
=> 2 = 1 + 1
[3,2,1,4] => [3,2,1,4] => ([(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)
=> 2 = 1 + 1
[3,2,4,1] => [3,2,4,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([(0,5),(0,6),(1,4),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> 2 = 1 + 1
[3,4,1,2] => [1,3,4,2] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([(0,5),(0,6),(1,4),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> 2 = 1 + 1
[3,4,2,1] => [3,4,2,1] => ([(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)
=> 2 = 1 + 1
[4,2,3,1] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([(0,5),(0,6),(1,4),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> 2 = 1 + 1
[4,3,1,2] => [1,4,3,2] => ([(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)
=> 2 = 1 + 1
[4,3,2,1] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[2,3,4,1,5] => [2,3,4,1,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(0,5),(0,9),(1,4),(1,8),(2,4),(2,7),(2,8),(3,5),(3,6),(3,9),(4,6),(5,7),(6,7),(6,8),(7,9),(8,9)],10)
=> ? = 2 + 1
[2,3,4,5,1] => [2,3,4,5,1] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 2 + 1
[2,3,5,4,1] => [5,2,3,4,1] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ([(0,9),(0,10),(1,8),(1,10),(2,4),(2,5),(2,7),(3,4),(3,5),(3,6),(4,8),(5,9),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10)],11)
=> ? = 2 + 1
[2,4,1,3,5] => [2,1,4,3,5] => ([(0,1),(0,2),(0,3),(1,7),(1,8),(2,5),(2,8),(3,5),(3,7),(3,8),(5,9),(6,4),(7,6),(7,9),(8,6),(8,9),(9,4)],10)
=> ([(0,5),(0,8),(1,2),(1,3),(1,9),(2,4),(2,7),(3,7),(3,8),(4,6),(4,9),(5,6),(5,9),(6,7),(6,8),(7,9),(8,9)],10)
=> ? = 2 + 1
[2,4,1,5,3] => [4,2,5,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,6),(2,9),(2,11),(3,6),(3,9),(3,10),(4,7),(4,9),(4,10),(4,11),(5,7),(5,9),(5,10),(5,11),(6,13),(7,12),(7,13),(9,12),(9,13),(10,12),(10,13),(11,12),(11,13),(12,8),(13,8)],14)
=> ([(0,9),(0,13),(1,8),(1,11),(1,12),(2,4),(2,5),(2,13),(3,6),(3,7),(3,9),(3,13),(4,8),(4,10),(4,12),(5,8),(5,10),(5,11),(6,8),(6,10),(6,11),(6,12),(7,8),(7,10),(7,11),(7,12),(9,10),(9,11),(9,12),(10,13),(11,13),(12,13)],14)
=> ? = 2 + 1
[2,4,3,1,5] => [4,2,3,1,5] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ([(0,11),(0,12),(1,4),(1,5),(1,11),(2,3),(2,9),(2,12),(3,8),(3,10),(4,7),(4,8),(4,10),(5,6),(5,8),(5,10),(6,9),(6,11),(6,12),(7,9),(7,11),(7,12),(8,9),(10,11),(10,12)],13)
=> ? = 1 + 1
[2,4,3,5,1] => [4,2,3,5,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ([(0,11),(0,12),(1,4),(1,5),(1,11),(2,3),(2,9),(2,12),(3,8),(3,10),(4,7),(4,8),(4,10),(5,6),(5,8),(5,10),(6,9),(6,11),(6,12),(7,9),(7,11),(7,12),(8,9),(10,11),(10,12)],13)
=> ? = 1 + 1
[2,4,5,1,3] => [2,1,4,5,3] => ([(0,2),(0,3),(0,4),(1,9),(1,10),(2,6),(2,7),(3,5),(3,6),(4,1),(4,5),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ([(0,9),(0,10),(1,2),(1,3),(1,6),(2,5),(2,7),(3,7),(3,10),(4,7),(4,9),(4,10),(5,6),(5,8),(6,9),(6,10),(7,8),(8,9),(8,10)],11)
=> ? = 2 + 1
[2,4,5,3,1] => [4,2,5,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ([(0,11),(0,12),(1,9),(1,10),(1,13),(2,7),(2,8),(2,11),(2,12),(3,6),(3,8),(3,11),(3,12),(4,5),(4,8),(4,11),(4,12),(5,9),(5,10),(5,13),(6,9),(6,10),(6,13),(7,9),(7,10),(7,13),(8,9),(8,13),(10,12),(11,13),(12,13)],14)
=> ? = 1 + 1
[2,5,3,4,1] => [2,5,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ([(0,11),(0,12),(1,4),(1,5),(1,11),(2,3),(2,9),(2,12),(3,8),(3,10),(4,7),(4,8),(4,10),(5,6),(5,8),(5,10),(6,9),(6,11),(6,12),(7,9),(7,11),(7,12),(8,9),(10,11),(10,12)],13)
=> ? = 2 + 1
[2,5,4,1,3] => [2,5,1,4,3] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,7),(2,10),(3,6),(3,10),(4,6),(4,8),(4,10),(5,1),(5,7),(5,8),(5,10),(6,12),(7,11),(7,12),(8,11),(8,12),(10,11),(10,12),(11,9),(12,9)],13)
=> ([(0,10),(0,12),(1,9),(1,10),(1,12),(2,3),(2,8),(2,12),(3,6),(3,11),(4,5),(4,6),(4,11),(5,9),(5,10),(5,12),(6,8),(6,9),(7,8),(7,9),(7,10),(7,12),(8,11),(9,11),(10,11),(11,12)],13)
=> ? = 2 + 1
[2,5,4,3,1] => [5,4,2,3,1] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(0,5),(0,9),(1,4),(1,8),(2,4),(2,7),(2,8),(3,5),(3,6),(3,9),(4,6),(5,7),(6,7),(6,8),(7,9),(8,9)],10)
=> ? = 1 + 1
[3,1,4,2,5] => [3,4,1,2,5] => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(5,9),(5,10),(6,9),(6,10),(7,10),(8,9),(8,10),(9,4),(10,4)],11)
=> ([(0,9),(0,10),(1,5),(1,6),(1,10),(2,3),(2,5),(2,6),(3,7),(3,8),(4,5),(4,6),(4,9),(4,10),(5,8),(6,7),(7,9),(7,10),(8,9),(8,10)],11)
=> ? = 2 + 1
[3,1,4,5,2] => [3,4,1,5,2] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,7),(2,10),(3,6),(3,10),(4,6),(4,8),(4,10),(5,1),(5,7),(5,8),(5,10),(6,12),(7,11),(7,12),(8,11),(8,12),(10,11),(10,12),(11,9),(12,9)],13)
=> ([(0,10),(0,12),(1,9),(1,10),(1,12),(2,3),(2,8),(2,12),(3,6),(3,11),(4,5),(4,6),(4,11),(5,9),(5,10),(5,12),(6,8),(6,9),(7,8),(7,9),(7,10),(7,12),(8,11),(9,11),(10,11),(11,12)],13)
=> ? = 2 + 1
[3,1,5,4,2] => [5,3,4,1,2] => ([(0,1),(0,2),(0,3),(1,7),(1,8),(2,5),(2,8),(3,5),(3,7),(3,8),(5,9),(6,4),(7,6),(7,9),(8,6),(8,9),(9,4)],10)
=> ([(0,5),(0,8),(1,2),(1,3),(1,9),(2,4),(2,7),(3,7),(3,8),(4,6),(4,9),(5,6),(5,9),(6,7),(6,8),(7,9),(8,9)],10)
=> ? = 2 + 1
[3,2,4,1,5] => [3,2,4,1,5] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> ([(0,9),(0,10),(1,4),(1,7),(1,10),(2,3),(2,6),(2,9),(3,8),(3,11),(4,8),(4,11),(5,6),(5,7),(5,9),(5,10),(6,8),(6,11),(7,8),(7,11),(9,11),(10,11)],12)
=> ? = 2 + 1
[3,2,4,5,1] => [3,2,4,5,1] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ([(0,9),(0,10),(1,8),(1,10),(2,4),(2,5),(2,7),(3,4),(3,5),(3,6),(4,8),(5,9),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10)],11)
=> ? = 2 + 1
[3,2,5,4,1] => [5,3,2,4,1] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ([(0,9),(0,11),(1,8),(1,10),(2,6),(2,7),(2,8),(2,10),(3,5),(3,7),(3,8),(3,10),(4,5),(4,6),(4,8),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(10,11)],12)
=> ? = 2 + 1
[3,4,1,2,5] => [1,3,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ([(0,9),(0,11),(1,8),(1,10),(2,6),(2,7),(2,8),(2,10),(3,5),(3,7),(3,8),(3,10),(4,5),(4,6),(4,8),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(10,11)],12)
=> ? = 1 + 1
[3,4,1,5,2] => [3,4,5,1,2] => ([(0,3),(0,4),(1,8),(2,7),(2,8),(3,1),(3,5),(4,2),(4,5),(5,7),(5,8),(7,6),(8,6)],9)
=> ([(0,5),(0,8),(1,4),(1,6),(2,6),(2,8),(3,4),(3,5),(3,8),(4,7),(5,7),(6,7),(7,8)],9)
=> ? = 1 + 1
[3,4,2,1,5] => [3,4,2,1,5] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ([(0,9),(0,10),(1,8),(1,10),(2,4),(2,5),(2,7),(3,4),(3,5),(3,6),(4,8),(5,9),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10)],11)
=> ? = 1 + 1
[3,4,2,5,1] => [3,4,2,5,1] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> ([(0,9),(0,10),(1,4),(1,7),(1,10),(2,3),(2,6),(2,9),(3,8),(3,11),(4,8),(4,11),(5,6),(5,7),(5,9),(5,10),(6,8),(6,11),(7,8),(7,11),(9,11),(10,11)],12)
=> ? = 1 + 1
[3,4,5,1,2] => [1,3,4,5,2] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(0,5),(0,9),(1,4),(1,8),(2,4),(2,7),(2,8),(3,5),(3,6),(3,9),(4,6),(5,7),(6,7),(6,8),(7,9),(8,9)],10)
=> ? = 1 + 1
[3,4,5,2,1] => [3,4,5,2,1] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ([(0,6),(0,7),(1,4),(1,5),(2,5),(2,7),(3,4),(3,6),(4,8),(5,8),(6,8),(7,8)],9)
=> ? = 1 + 1
[3,5,1,4,2] => [3,5,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,6),(2,9),(2,11),(3,6),(3,9),(3,10),(4,7),(4,9),(4,10),(4,11),(5,7),(5,9),(5,10),(5,11),(6,13),(7,12),(7,13),(9,12),(9,13),(10,12),(10,13),(11,12),(11,13),(12,8),(13,8)],14)
=> ([(0,9),(0,13),(1,8),(1,11),(1,12),(2,4),(2,5),(2,13),(3,6),(3,7),(3,9),(3,13),(4,8),(4,10),(4,12),(5,8),(5,10),(5,11),(6,8),(6,10),(6,11),(6,12),(7,8),(7,10),(7,11),(7,12),(9,10),(9,11),(9,12),(10,13),(11,13),(12,13)],14)
=> ? = 2 + 1
[3,5,2,4,1] => [3,2,5,4,1] => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(5,9),(5,10),(6,9),(6,10),(7,10),(8,9),(8,10),(9,4),(10,4)],11)
=> ([(0,9),(0,10),(1,5),(1,6),(1,10),(2,3),(2,5),(2,6),(3,7),(3,8),(4,5),(4,6),(4,9),(4,10),(5,8),(6,7),(7,9),(7,10),(8,9),(8,10)],11)
=> ? = 2 + 1
[3,5,4,1,2] => [1,5,3,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ([(0,11),(0,12),(1,4),(1,5),(1,11),(2,3),(2,9),(2,12),(3,8),(3,10),(4,7),(4,8),(4,10),(5,6),(5,8),(5,10),(6,9),(6,11),(6,12),(7,9),(7,11),(7,12),(8,9),(10,11),(10,12)],13)
=> ? = 1 + 1
[3,5,4,2,1] => [5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(0,5),(0,9),(1,4),(1,8),(2,4),(2,7),(2,8),(3,5),(3,6),(3,9),(4,6),(5,7),(6,7),(6,8),(7,9),(8,9)],10)
=> ? = 1 + 1
[4,1,2,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(0,5),(0,9),(1,4),(1,8),(2,4),(2,7),(2,8),(3,5),(3,6),(3,9),(4,6),(5,7),(6,7),(6,8),(7,9),(8,9)],10)
=> ? = 2 + 1
[4,1,2,5,3] => [4,1,5,2,3] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,7),(2,10),(3,6),(3,10),(4,6),(4,8),(4,10),(5,1),(5,7),(5,8),(5,10),(6,12),(7,11),(7,12),(8,11),(8,12),(10,11),(10,12),(11,9),(12,9)],13)
=> ([(0,10),(0,12),(1,9),(1,10),(1,12),(2,3),(2,8),(2,12),(3,6),(3,11),(4,5),(4,6),(4,11),(5,9),(5,10),(5,12),(6,8),(6,9),(7,8),(7,9),(7,10),(7,12),(8,11),(9,11),(10,11),(11,12)],13)
=> ? = 2 + 1
[4,1,3,2,5] => [4,1,3,2,5] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ([(0,11),(0,12),(1,4),(1,5),(1,11),(2,3),(2,9),(2,12),(3,8),(3,10),(4,7),(4,8),(4,10),(5,6),(5,8),(5,10),(6,9),(6,11),(6,12),(7,9),(7,11),(7,12),(8,9),(10,11),(10,12)],13)
=> ? = 2 + 1
[4,1,3,5,2] => [4,1,3,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,10),(1,11),(1,12),(2,7),(2,11),(2,12),(3,7),(3,9),(3,10),(4,6),(4,10),(4,12),(5,6),(5,9),(5,11),(6,14),(7,13),(9,13),(9,14),(10,13),(10,14),(11,13),(11,14),(12,13),(12,14),(13,8),(14,8)],15)
=> ([(0,13),(0,14),(1,5),(1,6),(1,14),(2,3),(2,4),(2,13),(3,10),(3,11),(3,12),(4,8),(4,9),(4,12),(5,9),(5,11),(5,12),(6,8),(6,10),(6,12),(7,8),(7,9),(7,10),(7,11),(7,12),(8,13),(8,14),(9,13),(9,14),(10,13),(10,14),(11,13),(11,14)],15)
=> ? = 2 + 1
[4,1,5,2,3] => [1,4,5,2,3] => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(5,9),(5,10),(6,9),(6,10),(7,10),(8,9),(8,10),(9,4),(10,4)],11)
=> ([(0,9),(0,10),(1,5),(1,6),(1,10),(2,3),(2,5),(2,6),(3,7),(3,8),(4,5),(4,6),(4,9),(4,10),(5,8),(6,7),(7,9),(7,10),(8,9),(8,10)],11)
=> ? = 2 + 1
[4,1,5,3,2] => [4,5,3,1,2] => ([(0,1),(0,2),(0,3),(1,7),(1,8),(2,5),(2,8),(3,5),(3,7),(5,9),(6,4),(7,6),(7,9),(8,6),(8,9),(9,4)],10)
=> ([(0,6),(0,9),(1,3),(1,4),(1,5),(2,3),(2,4),(2,9),(3,8),(4,7),(5,7),(5,8),(6,7),(6,8),(7,9),(8,9)],10)
=> ? = 2 + 1
[4,2,1,3,5] => [2,4,1,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ([(0,11),(0,12),(1,9),(1,10),(1,13),(2,7),(2,8),(2,11),(2,12),(3,6),(3,8),(3,11),(3,12),(4,5),(4,8),(4,11),(4,12),(5,9),(5,10),(5,13),(6,9),(6,10),(6,13),(7,9),(7,10),(7,13),(8,9),(8,13),(10,12),(11,13),(12,13)],14)
=> ? = 1 + 1
[4,2,1,5,3] => [4,5,2,1,3] => ([(0,2),(0,3),(0,4),(1,9),(1,10),(2,6),(2,7),(3,5),(3,6),(4,1),(4,5),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ([(0,9),(0,10),(1,2),(1,3),(1,6),(2,5),(2,7),(3,7),(3,10),(4,7),(4,9),(4,10),(5,6),(5,8),(6,9),(6,10),(7,8),(8,9),(8,10)],11)
=> ? = 1 + 1
[4,2,3,1,5] => [2,4,3,1,5] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ([(0,11),(0,12),(1,4),(1,5),(1,11),(2,3),(2,9),(2,12),(3,8),(3,10),(4,7),(4,8),(4,10),(5,6),(5,8),(5,10),(6,9),(6,11),(6,12),(7,9),(7,11),(7,12),(8,9),(10,11),(10,12)],13)
=> ? = 1 + 1
[4,2,3,5,1] => [2,4,3,5,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ([(0,11),(0,12),(1,4),(1,5),(1,11),(2,3),(2,9),(2,12),(3,8),(3,10),(4,7),(4,8),(4,10),(5,6),(5,8),(5,10),(6,9),(6,11),(6,12),(7,9),(7,11),(7,12),(8,9),(10,11),(10,12)],13)
=> ? = 1 + 1
[4,2,5,1,3] => [2,4,1,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,6),(2,9),(2,11),(3,6),(3,9),(3,10),(4,7),(4,9),(4,10),(4,11),(5,7),(5,9),(5,10),(5,11),(6,13),(7,12),(7,13),(9,12),(9,13),(10,12),(10,13),(11,12),(11,13),(12,8),(13,8)],14)
=> ([(0,9),(0,13),(1,8),(1,11),(1,12),(2,4),(2,5),(2,13),(3,6),(3,7),(3,9),(3,13),(4,8),(4,10),(4,12),(5,8),(5,10),(5,11),(6,8),(6,10),(6,11),(6,12),(7,8),(7,10),(7,11),(7,12),(9,10),(9,11),(9,12),(10,13),(11,13),(12,13)],14)
=> ? = 1 + 1
[4,2,5,3,1] => [4,5,2,3,1] => ([(0,1),(0,2),(0,3),(1,7),(1,8),(2,5),(2,8),(3,5),(3,7),(3,8),(5,9),(6,4),(7,6),(7,9),(8,6),(8,9),(9,4)],10)
=> ([(0,5),(0,8),(1,2),(1,3),(1,9),(2,4),(2,7),(3,7),(3,8),(4,6),(4,9),(5,6),(5,9),(6,7),(6,8),(7,9),(8,9)],10)
=> ? = 1 + 1
[4,3,1,2,5] => [1,4,3,2,5] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ([(0,9),(0,10),(1,8),(1,10),(2,4),(2,5),(2,7),(3,4),(3,5),(3,6),(4,8),(5,9),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10)],11)
=> ? = 1 + 1
[4,3,1,5,2] => [4,3,5,1,2] => ([(0,2),(0,3),(0,4),(1,9),(1,10),(2,6),(2,7),(3,5),(3,6),(4,1),(4,5),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ([(0,9),(0,10),(1,2),(1,3),(1,6),(2,5),(2,7),(3,7),(3,10),(4,7),(4,9),(4,10),(5,6),(5,8),(6,9),(6,10),(7,8),(8,9),(8,10)],11)
=> ? = 1 + 1
[4,3,2,1,5] => [4,3,2,1,5] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 1 + 1
[4,3,2,5,1] => [4,3,2,5,1] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(0,5),(0,9),(1,4),(1,8),(2,4),(2,7),(2,8),(3,5),(3,6),(3,9),(4,6),(5,7),(6,7),(6,8),(7,9),(8,9)],10)
=> ? = 1 + 1
[4,3,5,1,2] => [1,4,3,5,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ([(0,11),(0,12),(1,4),(1,5),(1,11),(2,3),(2,9),(2,12),(3,8),(3,10),(4,7),(4,8),(4,10),(5,6),(5,8),(5,10),(6,9),(6,11),(6,12),(7,9),(7,11),(7,12),(8,9),(10,11),(10,12)],13)
=> ? = 1 + 1
[4,3,5,2,1] => [4,3,5,2,1] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(0,5),(0,9),(1,4),(1,8),(2,4),(2,7),(2,8),(3,5),(3,6),(3,9),(4,6),(5,7),(6,7),(6,8),(7,9),(8,9)],10)
=> ? = 1 + 1
[4,5,1,2,3] => [1,2,4,5,3] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(0,5),(0,9),(1,4),(1,8),(2,4),(2,7),(2,8),(3,5),(3,6),(3,9),(4,6),(5,7),(6,7),(6,8),(7,9),(8,9)],10)
=> ? = 2 + 1
[4,5,1,3,2] => [4,1,5,3,2] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,9),(2,10),(3,6),(3,9),(4,7),(4,9),(4,10),(5,1),(5,6),(5,7),(5,10),(6,11),(6,12),(7,11),(7,12),(9,12),(10,11),(10,12),(11,8),(12,8)],13)
=> ([(0,11),(0,12),(1,6),(1,7),(1,8),(2,7),(2,8),(2,10),(3,9),(3,11),(3,12),(4,5),(4,7),(4,8),(4,10),(5,9),(5,11),(5,12),(6,9),(6,11),(6,12),(7,12),(8,9),(9,10),(10,11),(10,12)],13)
=> ? = 2 + 1
[4,5,2,1,3] => [2,4,5,1,3] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,7),(2,10),(3,6),(3,10),(4,6),(4,8),(4,10),(5,1),(5,7),(5,8),(5,10),(6,12),(7,11),(7,12),(8,11),(8,12),(10,11),(10,12),(11,9),(12,9)],13)
=> ([(0,10),(0,12),(1,9),(1,10),(1,12),(2,3),(2,8),(2,12),(3,6),(3,11),(4,5),(4,6),(4,11),(5,9),(5,10),(5,12),(6,8),(6,9),(7,8),(7,9),(7,10),(7,12),(8,11),(9,11),(10,11),(11,12)],13)
=> ? = 1 + 1
[4,5,2,3,1] => [2,4,5,3,1] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> ([(0,9),(0,10),(1,4),(1,7),(1,10),(2,3),(2,6),(2,9),(3,8),(3,11),(4,8),(4,11),(5,6),(5,7),(5,9),(5,10),(6,8),(6,11),(7,8),(7,11),(9,11),(10,11)],12)
=> ? = 1 + 1
[5,4,3,2,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 1 + 1
[6,5,4,3,2,1] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 2 = 1 + 1
[7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 2 = 1 + 1
Description
The weak 2-dynamic chromatic number of a graph. A $k$-weak-dynamic coloring of a graph $G$ is a (non-proper) coloring of $G$ in such a way that each vertex $v$ sees at least $\min\{d(v), k\}$ colors in its neighborhood. The $k$-weak-dynamic number of a graph is the smallest number of colors needed to find an $k$-dynamic coloring. This statistic records the $2$-weak-dynamic number of a graph.
Matching statistic: St001716
Mp00067: Permutations Foata bijectionPermutations
Mp00209: Permutations pattern posetPosets
Mp00074: Posets to graphGraphs
St001716: Graphs ⟶ ℤResult quality: 0% values known / values provided: 0%distinct values known / distinct values provided: 50%
Values
[1,2] => [1,2] => ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 0 + 1
[2,1] => [2,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 0 + 1
[2,1,3] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2 = 1 + 1
[2,3,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2 = 1 + 1
[3,2,1] => [3,2,1] => ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[2,3,1,4] => [2,3,1,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([(0,5),(0,6),(1,4),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> 2 = 1 + 1
[2,3,4,1] => [2,3,4,1] => ([(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)
=> 2 = 1 + 1
[2,4,3,1] => [4,2,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([(0,5),(0,6),(1,4),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> 2 = 1 + 1
[3,1,2,4] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([(0,5),(0,6),(1,4),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> 2 = 1 + 1
[3,1,4,2] => [3,4,1,2] => ([(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)
=> 2 = 1 + 1
[3,2,1,4] => [3,2,1,4] => ([(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)
=> 2 = 1 + 1
[3,2,4,1] => [3,2,4,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([(0,5),(0,6),(1,4),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> 2 = 1 + 1
[3,4,1,2] => [1,3,4,2] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([(0,5),(0,6),(1,4),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> 2 = 1 + 1
[3,4,2,1] => [3,4,2,1] => ([(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)
=> 2 = 1 + 1
[4,2,3,1] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([(0,5),(0,6),(1,4),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> 2 = 1 + 1
[4,3,1,2] => [1,4,3,2] => ([(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)
=> 2 = 1 + 1
[4,3,2,1] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[2,3,4,1,5] => [2,3,4,1,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(0,5),(0,9),(1,4),(1,8),(2,4),(2,7),(2,8),(3,5),(3,6),(3,9),(4,6),(5,7),(6,7),(6,8),(7,9),(8,9)],10)
=> ? = 2 + 1
[2,3,4,5,1] => [2,3,4,5,1] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 2 + 1
[2,3,5,4,1] => [5,2,3,4,1] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ([(0,9),(0,10),(1,8),(1,10),(2,4),(2,5),(2,7),(3,4),(3,5),(3,6),(4,8),(5,9),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10)],11)
=> ? = 2 + 1
[2,4,1,3,5] => [2,1,4,3,5] => ([(0,1),(0,2),(0,3),(1,7),(1,8),(2,5),(2,8),(3,5),(3,7),(3,8),(5,9),(6,4),(7,6),(7,9),(8,6),(8,9),(9,4)],10)
=> ([(0,5),(0,8),(1,2),(1,3),(1,9),(2,4),(2,7),(3,7),(3,8),(4,6),(4,9),(5,6),(5,9),(6,7),(6,8),(7,9),(8,9)],10)
=> ? = 2 + 1
[2,4,1,5,3] => [4,2,5,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,6),(2,9),(2,11),(3,6),(3,9),(3,10),(4,7),(4,9),(4,10),(4,11),(5,7),(5,9),(5,10),(5,11),(6,13),(7,12),(7,13),(9,12),(9,13),(10,12),(10,13),(11,12),(11,13),(12,8),(13,8)],14)
=> ([(0,9),(0,13),(1,8),(1,11),(1,12),(2,4),(2,5),(2,13),(3,6),(3,7),(3,9),(3,13),(4,8),(4,10),(4,12),(5,8),(5,10),(5,11),(6,8),(6,10),(6,11),(6,12),(7,8),(7,10),(7,11),(7,12),(9,10),(9,11),(9,12),(10,13),(11,13),(12,13)],14)
=> ? = 2 + 1
[2,4,3,1,5] => [4,2,3,1,5] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ([(0,11),(0,12),(1,4),(1,5),(1,11),(2,3),(2,9),(2,12),(3,8),(3,10),(4,7),(4,8),(4,10),(5,6),(5,8),(5,10),(6,9),(6,11),(6,12),(7,9),(7,11),(7,12),(8,9),(10,11),(10,12)],13)
=> ? = 1 + 1
[2,4,3,5,1] => [4,2,3,5,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ([(0,11),(0,12),(1,4),(1,5),(1,11),(2,3),(2,9),(2,12),(3,8),(3,10),(4,7),(4,8),(4,10),(5,6),(5,8),(5,10),(6,9),(6,11),(6,12),(7,9),(7,11),(7,12),(8,9),(10,11),(10,12)],13)
=> ? = 1 + 1
[2,4,5,1,3] => [2,1,4,5,3] => ([(0,2),(0,3),(0,4),(1,9),(1,10),(2,6),(2,7),(3,5),(3,6),(4,1),(4,5),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ([(0,9),(0,10),(1,2),(1,3),(1,6),(2,5),(2,7),(3,7),(3,10),(4,7),(4,9),(4,10),(5,6),(5,8),(6,9),(6,10),(7,8),(8,9),(8,10)],11)
=> ? = 2 + 1
[2,4,5,3,1] => [4,2,5,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ([(0,11),(0,12),(1,9),(1,10),(1,13),(2,7),(2,8),(2,11),(2,12),(3,6),(3,8),(3,11),(3,12),(4,5),(4,8),(4,11),(4,12),(5,9),(5,10),(5,13),(6,9),(6,10),(6,13),(7,9),(7,10),(7,13),(8,9),(8,13),(10,12),(11,13),(12,13)],14)
=> ? = 1 + 1
[2,5,3,4,1] => [2,5,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ([(0,11),(0,12),(1,4),(1,5),(1,11),(2,3),(2,9),(2,12),(3,8),(3,10),(4,7),(4,8),(4,10),(5,6),(5,8),(5,10),(6,9),(6,11),(6,12),(7,9),(7,11),(7,12),(8,9),(10,11),(10,12)],13)
=> ? = 2 + 1
[2,5,4,1,3] => [2,5,1,4,3] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,7),(2,10),(3,6),(3,10),(4,6),(4,8),(4,10),(5,1),(5,7),(5,8),(5,10),(6,12),(7,11),(7,12),(8,11),(8,12),(10,11),(10,12),(11,9),(12,9)],13)
=> ([(0,10),(0,12),(1,9),(1,10),(1,12),(2,3),(2,8),(2,12),(3,6),(3,11),(4,5),(4,6),(4,11),(5,9),(5,10),(5,12),(6,8),(6,9),(7,8),(7,9),(7,10),(7,12),(8,11),(9,11),(10,11),(11,12)],13)
=> ? = 2 + 1
[2,5,4,3,1] => [5,4,2,3,1] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(0,5),(0,9),(1,4),(1,8),(2,4),(2,7),(2,8),(3,5),(3,6),(3,9),(4,6),(5,7),(6,7),(6,8),(7,9),(8,9)],10)
=> ? = 1 + 1
[3,1,4,2,5] => [3,4,1,2,5] => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(5,9),(5,10),(6,9),(6,10),(7,10),(8,9),(8,10),(9,4),(10,4)],11)
=> ([(0,9),(0,10),(1,5),(1,6),(1,10),(2,3),(2,5),(2,6),(3,7),(3,8),(4,5),(4,6),(4,9),(4,10),(5,8),(6,7),(7,9),(7,10),(8,9),(8,10)],11)
=> ? = 2 + 1
[3,1,4,5,2] => [3,4,1,5,2] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,7),(2,10),(3,6),(3,10),(4,6),(4,8),(4,10),(5,1),(5,7),(5,8),(5,10),(6,12),(7,11),(7,12),(8,11),(8,12),(10,11),(10,12),(11,9),(12,9)],13)
=> ([(0,10),(0,12),(1,9),(1,10),(1,12),(2,3),(2,8),(2,12),(3,6),(3,11),(4,5),(4,6),(4,11),(5,9),(5,10),(5,12),(6,8),(6,9),(7,8),(7,9),(7,10),(7,12),(8,11),(9,11),(10,11),(11,12)],13)
=> ? = 2 + 1
[3,1,5,4,2] => [5,3,4,1,2] => ([(0,1),(0,2),(0,3),(1,7),(1,8),(2,5),(2,8),(3,5),(3,7),(3,8),(5,9),(6,4),(7,6),(7,9),(8,6),(8,9),(9,4)],10)
=> ([(0,5),(0,8),(1,2),(1,3),(1,9),(2,4),(2,7),(3,7),(3,8),(4,6),(4,9),(5,6),(5,9),(6,7),(6,8),(7,9),(8,9)],10)
=> ? = 2 + 1
[3,2,4,1,5] => [3,2,4,1,5] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> ([(0,9),(0,10),(1,4),(1,7),(1,10),(2,3),(2,6),(2,9),(3,8),(3,11),(4,8),(4,11),(5,6),(5,7),(5,9),(5,10),(6,8),(6,11),(7,8),(7,11),(9,11),(10,11)],12)
=> ? = 2 + 1
[3,2,4,5,1] => [3,2,4,5,1] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ([(0,9),(0,10),(1,8),(1,10),(2,4),(2,5),(2,7),(3,4),(3,5),(3,6),(4,8),(5,9),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10)],11)
=> ? = 2 + 1
[3,2,5,4,1] => [5,3,2,4,1] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ([(0,9),(0,11),(1,8),(1,10),(2,6),(2,7),(2,8),(2,10),(3,5),(3,7),(3,8),(3,10),(4,5),(4,6),(4,8),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(10,11)],12)
=> ? = 2 + 1
[3,4,1,2,5] => [1,3,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ([(0,9),(0,11),(1,8),(1,10),(2,6),(2,7),(2,8),(2,10),(3,5),(3,7),(3,8),(3,10),(4,5),(4,6),(4,8),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(10,11)],12)
=> ? = 1 + 1
[3,4,1,5,2] => [3,4,5,1,2] => ([(0,3),(0,4),(1,8),(2,7),(2,8),(3,1),(3,5),(4,2),(4,5),(5,7),(5,8),(7,6),(8,6)],9)
=> ([(0,5),(0,8),(1,4),(1,6),(2,6),(2,8),(3,4),(3,5),(3,8),(4,7),(5,7),(6,7),(7,8)],9)
=> ? = 1 + 1
[3,4,2,1,5] => [3,4,2,1,5] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ([(0,9),(0,10),(1,8),(1,10),(2,4),(2,5),(2,7),(3,4),(3,5),(3,6),(4,8),(5,9),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10)],11)
=> ? = 1 + 1
[3,4,2,5,1] => [3,4,2,5,1] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> ([(0,9),(0,10),(1,4),(1,7),(1,10),(2,3),(2,6),(2,9),(3,8),(3,11),(4,8),(4,11),(5,6),(5,7),(5,9),(5,10),(6,8),(6,11),(7,8),(7,11),(9,11),(10,11)],12)
=> ? = 1 + 1
[3,4,5,1,2] => [1,3,4,5,2] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(0,5),(0,9),(1,4),(1,8),(2,4),(2,7),(2,8),(3,5),(3,6),(3,9),(4,6),(5,7),(6,7),(6,8),(7,9),(8,9)],10)
=> ? = 1 + 1
[3,4,5,2,1] => [3,4,5,2,1] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ([(0,6),(0,7),(1,4),(1,5),(2,5),(2,7),(3,4),(3,6),(4,8),(5,8),(6,8),(7,8)],9)
=> ? = 1 + 1
[3,5,1,4,2] => [3,5,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,6),(2,9),(2,11),(3,6),(3,9),(3,10),(4,7),(4,9),(4,10),(4,11),(5,7),(5,9),(5,10),(5,11),(6,13),(7,12),(7,13),(9,12),(9,13),(10,12),(10,13),(11,12),(11,13),(12,8),(13,8)],14)
=> ([(0,9),(0,13),(1,8),(1,11),(1,12),(2,4),(2,5),(2,13),(3,6),(3,7),(3,9),(3,13),(4,8),(4,10),(4,12),(5,8),(5,10),(5,11),(6,8),(6,10),(6,11),(6,12),(7,8),(7,10),(7,11),(7,12),(9,10),(9,11),(9,12),(10,13),(11,13),(12,13)],14)
=> ? = 2 + 1
[3,5,2,4,1] => [3,2,5,4,1] => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(5,9),(5,10),(6,9),(6,10),(7,10),(8,9),(8,10),(9,4),(10,4)],11)
=> ([(0,9),(0,10),(1,5),(1,6),(1,10),(2,3),(2,5),(2,6),(3,7),(3,8),(4,5),(4,6),(4,9),(4,10),(5,8),(6,7),(7,9),(7,10),(8,9),(8,10)],11)
=> ? = 2 + 1
[3,5,4,1,2] => [1,5,3,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ([(0,11),(0,12),(1,4),(1,5),(1,11),(2,3),(2,9),(2,12),(3,8),(3,10),(4,7),(4,8),(4,10),(5,6),(5,8),(5,10),(6,9),(6,11),(6,12),(7,9),(7,11),(7,12),(8,9),(10,11),(10,12)],13)
=> ? = 1 + 1
[3,5,4,2,1] => [5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(0,5),(0,9),(1,4),(1,8),(2,4),(2,7),(2,8),(3,5),(3,6),(3,9),(4,6),(5,7),(6,7),(6,8),(7,9),(8,9)],10)
=> ? = 1 + 1
[4,1,2,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(0,5),(0,9),(1,4),(1,8),(2,4),(2,7),(2,8),(3,5),(3,6),(3,9),(4,6),(5,7),(6,7),(6,8),(7,9),(8,9)],10)
=> ? = 2 + 1
[4,1,2,5,3] => [4,1,5,2,3] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,7),(2,10),(3,6),(3,10),(4,6),(4,8),(4,10),(5,1),(5,7),(5,8),(5,10),(6,12),(7,11),(7,12),(8,11),(8,12),(10,11),(10,12),(11,9),(12,9)],13)
=> ([(0,10),(0,12),(1,9),(1,10),(1,12),(2,3),(2,8),(2,12),(3,6),(3,11),(4,5),(4,6),(4,11),(5,9),(5,10),(5,12),(6,8),(6,9),(7,8),(7,9),(7,10),(7,12),(8,11),(9,11),(10,11),(11,12)],13)
=> ? = 2 + 1
[4,1,3,2,5] => [4,1,3,2,5] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ([(0,11),(0,12),(1,4),(1,5),(1,11),(2,3),(2,9),(2,12),(3,8),(3,10),(4,7),(4,8),(4,10),(5,6),(5,8),(5,10),(6,9),(6,11),(6,12),(7,9),(7,11),(7,12),(8,9),(10,11),(10,12)],13)
=> ? = 2 + 1
[4,1,3,5,2] => [4,1,3,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,10),(1,11),(1,12),(2,7),(2,11),(2,12),(3,7),(3,9),(3,10),(4,6),(4,10),(4,12),(5,6),(5,9),(5,11),(6,14),(7,13),(9,13),(9,14),(10,13),(10,14),(11,13),(11,14),(12,13),(12,14),(13,8),(14,8)],15)
=> ([(0,13),(0,14),(1,5),(1,6),(1,14),(2,3),(2,4),(2,13),(3,10),(3,11),(3,12),(4,8),(4,9),(4,12),(5,9),(5,11),(5,12),(6,8),(6,10),(6,12),(7,8),(7,9),(7,10),(7,11),(7,12),(8,13),(8,14),(9,13),(9,14),(10,13),(10,14),(11,13),(11,14)],15)
=> ? = 2 + 1
[4,1,5,2,3] => [1,4,5,2,3] => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(5,9),(5,10),(6,9),(6,10),(7,10),(8,9),(8,10),(9,4),(10,4)],11)
=> ([(0,9),(0,10),(1,5),(1,6),(1,10),(2,3),(2,5),(2,6),(3,7),(3,8),(4,5),(4,6),(4,9),(4,10),(5,8),(6,7),(7,9),(7,10),(8,9),(8,10)],11)
=> ? = 2 + 1
[4,1,5,3,2] => [4,5,3,1,2] => ([(0,1),(0,2),(0,3),(1,7),(1,8),(2,5),(2,8),(3,5),(3,7),(5,9),(6,4),(7,6),(7,9),(8,6),(8,9),(9,4)],10)
=> ([(0,6),(0,9),(1,3),(1,4),(1,5),(2,3),(2,4),(2,9),(3,8),(4,7),(5,7),(5,8),(6,7),(6,8),(7,9),(8,9)],10)
=> ? = 2 + 1
[4,2,1,3,5] => [2,4,1,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ([(0,11),(0,12),(1,9),(1,10),(1,13),(2,7),(2,8),(2,11),(2,12),(3,6),(3,8),(3,11),(3,12),(4,5),(4,8),(4,11),(4,12),(5,9),(5,10),(5,13),(6,9),(6,10),(6,13),(7,9),(7,10),(7,13),(8,9),(8,13),(10,12),(11,13),(12,13)],14)
=> ? = 1 + 1
[4,2,1,5,3] => [4,5,2,1,3] => ([(0,2),(0,3),(0,4),(1,9),(1,10),(2,6),(2,7),(3,5),(3,6),(4,1),(4,5),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ([(0,9),(0,10),(1,2),(1,3),(1,6),(2,5),(2,7),(3,7),(3,10),(4,7),(4,9),(4,10),(5,6),(5,8),(6,9),(6,10),(7,8),(8,9),(8,10)],11)
=> ? = 1 + 1
[4,2,3,1,5] => [2,4,3,1,5] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ([(0,11),(0,12),(1,4),(1,5),(1,11),(2,3),(2,9),(2,12),(3,8),(3,10),(4,7),(4,8),(4,10),(5,6),(5,8),(5,10),(6,9),(6,11),(6,12),(7,9),(7,11),(7,12),(8,9),(10,11),(10,12)],13)
=> ? = 1 + 1
[4,2,3,5,1] => [2,4,3,5,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ([(0,11),(0,12),(1,4),(1,5),(1,11),(2,3),(2,9),(2,12),(3,8),(3,10),(4,7),(4,8),(4,10),(5,6),(5,8),(5,10),(6,9),(6,11),(6,12),(7,9),(7,11),(7,12),(8,9),(10,11),(10,12)],13)
=> ? = 1 + 1
[4,2,5,1,3] => [2,4,1,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,6),(2,9),(2,11),(3,6),(3,9),(3,10),(4,7),(4,9),(4,10),(4,11),(5,7),(5,9),(5,10),(5,11),(6,13),(7,12),(7,13),(9,12),(9,13),(10,12),(10,13),(11,12),(11,13),(12,8),(13,8)],14)
=> ([(0,9),(0,13),(1,8),(1,11),(1,12),(2,4),(2,5),(2,13),(3,6),(3,7),(3,9),(3,13),(4,8),(4,10),(4,12),(5,8),(5,10),(5,11),(6,8),(6,10),(6,11),(6,12),(7,8),(7,10),(7,11),(7,12),(9,10),(9,11),(9,12),(10,13),(11,13),(12,13)],14)
=> ? = 1 + 1
[4,2,5,3,1] => [4,5,2,3,1] => ([(0,1),(0,2),(0,3),(1,7),(1,8),(2,5),(2,8),(3,5),(3,7),(3,8),(5,9),(6,4),(7,6),(7,9),(8,6),(8,9),(9,4)],10)
=> ([(0,5),(0,8),(1,2),(1,3),(1,9),(2,4),(2,7),(3,7),(3,8),(4,6),(4,9),(5,6),(5,9),(6,7),(6,8),(7,9),(8,9)],10)
=> ? = 1 + 1
[4,3,1,2,5] => [1,4,3,2,5] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ([(0,9),(0,10),(1,8),(1,10),(2,4),(2,5),(2,7),(3,4),(3,5),(3,6),(4,8),(5,9),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10)],11)
=> ? = 1 + 1
[4,3,1,5,2] => [4,3,5,1,2] => ([(0,2),(0,3),(0,4),(1,9),(1,10),(2,6),(2,7),(3,5),(3,6),(4,1),(4,5),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ([(0,9),(0,10),(1,2),(1,3),(1,6),(2,5),(2,7),(3,7),(3,10),(4,7),(4,9),(4,10),(5,6),(5,8),(6,9),(6,10),(7,8),(8,9),(8,10)],11)
=> ? = 1 + 1
[4,3,2,1,5] => [4,3,2,1,5] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 1 + 1
[4,3,2,5,1] => [4,3,2,5,1] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(0,5),(0,9),(1,4),(1,8),(2,4),(2,7),(2,8),(3,5),(3,6),(3,9),(4,6),(5,7),(6,7),(6,8),(7,9),(8,9)],10)
=> ? = 1 + 1
[4,3,5,1,2] => [1,4,3,5,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ([(0,11),(0,12),(1,4),(1,5),(1,11),(2,3),(2,9),(2,12),(3,8),(3,10),(4,7),(4,8),(4,10),(5,6),(5,8),(5,10),(6,9),(6,11),(6,12),(7,9),(7,11),(7,12),(8,9),(10,11),(10,12)],13)
=> ? = 1 + 1
[4,3,5,2,1] => [4,3,5,2,1] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(0,5),(0,9),(1,4),(1,8),(2,4),(2,7),(2,8),(3,5),(3,6),(3,9),(4,6),(5,7),(6,7),(6,8),(7,9),(8,9)],10)
=> ? = 1 + 1
[4,5,1,2,3] => [1,2,4,5,3] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(0,5),(0,9),(1,4),(1,8),(2,4),(2,7),(2,8),(3,5),(3,6),(3,9),(4,6),(5,7),(6,7),(6,8),(7,9),(8,9)],10)
=> ? = 2 + 1
[4,5,1,3,2] => [4,1,5,3,2] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,9),(2,10),(3,6),(3,9),(4,7),(4,9),(4,10),(5,1),(5,6),(5,7),(5,10),(6,11),(6,12),(7,11),(7,12),(9,12),(10,11),(10,12),(11,8),(12,8)],13)
=> ([(0,11),(0,12),(1,6),(1,7),(1,8),(2,7),(2,8),(2,10),(3,9),(3,11),(3,12),(4,5),(4,7),(4,8),(4,10),(5,9),(5,11),(5,12),(6,9),(6,11),(6,12),(7,12),(8,9),(9,10),(10,11),(10,12)],13)
=> ? = 2 + 1
[4,5,2,1,3] => [2,4,5,1,3] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,7),(2,10),(3,6),(3,10),(4,6),(4,8),(4,10),(5,1),(5,7),(5,8),(5,10),(6,12),(7,11),(7,12),(8,11),(8,12),(10,11),(10,12),(11,9),(12,9)],13)
=> ([(0,10),(0,12),(1,9),(1,10),(1,12),(2,3),(2,8),(2,12),(3,6),(3,11),(4,5),(4,6),(4,11),(5,9),(5,10),(5,12),(6,8),(6,9),(7,8),(7,9),(7,10),(7,12),(8,11),(9,11),(10,11),(11,12)],13)
=> ? = 1 + 1
[4,5,2,3,1] => [2,4,5,3,1] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> ([(0,9),(0,10),(1,4),(1,7),(1,10),(2,3),(2,6),(2,9),(3,8),(3,11),(4,8),(4,11),(5,6),(5,7),(5,9),(5,10),(6,8),(6,11),(7,8),(7,11),(9,11),(10,11)],12)
=> ? = 1 + 1
[5,4,3,2,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 1 + 1
[6,5,4,3,2,1] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 2 = 1 + 1
[7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 2 = 1 + 1
Description
The 1-improper chromatic number of a graph. This is the least number of colours in a vertex-colouring, such that each vertex has at most one neighbour with the same colour.