Identifier
-
Mp00023:
Dyck paths
—to non-crossing permutation⟶
Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000260: Graphs ⟶ ℤ
Values
[1,0] => [1] => [1] => ([],1) => 0
[1,1,0,0] => [2,1] => [2,1] => ([(0,1)],2) => 1
[1,0,1,1,0,0] => [1,3,2] => [3,1,2] => ([(0,2),(1,2)],3) => 1
[1,1,1,0,0,0] => [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3) => 1
[1,0,1,0,1,1,0,0] => [1,2,4,3] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4) => 1
[1,0,1,1,0,1,0,0] => [1,3,4,2] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4) => 2
[1,0,1,1,1,0,0,0] => [1,4,3,2] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 1
[1,1,0,1,1,0,0,0] => [2,4,3,1] => [4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4) => 1
[1,1,1,0,1,0,0,0] => [4,2,3,1] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4) => 1
[1,1,1,1,0,0,0,0] => [4,3,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 1
[1,0,1,0,1,0,1,1,0,0] => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5) => 1
[1,0,1,0,1,1,0,1,0,0] => [1,2,4,5,3] => [3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5) => 2
[1,0,1,0,1,1,1,0,0,0] => [1,2,5,4,3] => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 1
[1,0,1,1,0,0,1,1,0,0] => [1,3,2,5,4] => [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5) => 2
[1,0,1,1,0,1,0,1,0,0] => [1,3,4,5,2] => [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5) => 2
[1,0,1,1,0,1,1,0,0,0] => [1,3,5,4,2] => [5,2,4,1,3] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5) => 1
[1,0,1,1,1,0,0,1,0,0] => [1,4,3,5,2] => [3,2,5,1,4] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5) => 2
[1,0,1,1,1,0,1,0,0,0] => [1,5,3,4,2] => [3,5,4,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5) => 2
[1,0,1,1,1,1,0,0,0,0] => [1,5,4,3,2] => [5,4,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 1
[1,1,0,0,1,1,1,0,0,0] => [2,1,5,4,3] => [5,1,4,2,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 1
[1,1,0,1,0,1,1,0,0,0] => [2,3,5,4,1] => [5,1,2,4,3] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5) => 1
[1,1,0,1,1,0,1,0,0,0] => [2,5,3,4,1] => [3,5,1,4,2] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5) => 2
[1,1,0,1,1,1,0,0,0,0] => [2,5,4,3,1] => [5,4,1,3,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 1
[1,1,1,0,1,0,1,0,0,0] => [5,2,3,4,1] => [2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5) => 1
[1,1,1,0,1,1,0,0,0,0] => [5,2,4,3,1] => [5,2,4,3,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 1
[1,1,1,1,0,0,1,0,0,0] => [5,3,2,4,1] => [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5) => 1
[1,1,1,1,0,1,0,0,0,0] => [5,3,4,2,1] => [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 1
[1,1,1,1,1,0,0,0,0,0] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 1
[1,0,1,0,1,0,1,0,1,1,0,0] => [1,2,3,4,6,5] => [6,1,2,3,4,5] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 1
[1,0,1,0,1,0,1,1,0,1,0,0] => [1,2,3,5,6,4] => [4,6,1,2,3,5] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 2
[1,0,1,0,1,0,1,1,1,0,0,0] => [1,2,3,6,5,4] => [6,5,1,2,3,4] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 1
[1,0,1,0,1,1,0,0,1,1,0,0] => [1,2,4,3,6,5] => [3,6,1,2,4,5] => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6) => 2
[1,0,1,0,1,1,0,1,0,1,0,0] => [1,2,4,5,6,3] => [3,4,6,1,2,5] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => 2
[1,0,1,0,1,1,0,1,1,0,0,0] => [1,2,4,6,5,3] => [6,3,5,1,2,4] => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 1
[1,0,1,0,1,1,1,0,0,1,0,0] => [1,2,5,4,6,3] => [4,3,6,1,2,5] => ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => 2
[1,0,1,0,1,1,1,0,1,0,0,0] => [1,2,6,4,5,3] => [4,6,5,1,2,3] => ([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 2
[1,0,1,0,1,1,1,1,0,0,0,0] => [1,2,6,5,4,3] => [6,5,4,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 1
[1,0,1,1,0,0,1,0,1,1,0,0] => [1,3,2,4,6,5] => [2,6,1,3,4,5] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => 2
[1,0,1,1,0,0,1,1,0,1,0,0] => [1,3,2,5,6,4] => [2,4,6,1,3,5] => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => 2
[1,0,1,1,0,0,1,1,1,0,0,0] => [1,3,2,6,5,4] => [6,2,5,1,3,4] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6) => 1
[1,0,1,1,0,1,0,0,1,1,0,0] => [1,3,4,2,6,5] => [2,3,6,1,4,5] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => 2
[1,0,1,1,0,1,0,1,0,1,0,0] => [1,3,4,5,6,2] => [2,3,4,6,1,5] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => 2
[1,0,1,1,0,1,0,1,1,0,0,0] => [1,3,4,6,5,2] => [6,2,3,5,1,4] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6) => 1
[1,0,1,1,0,1,1,0,0,1,0,0] => [1,3,5,4,6,2] => [4,2,3,6,1,5] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6) => 2
[1,0,1,1,0,1,1,0,1,0,0,0] => [1,3,6,4,5,2] => [4,6,2,5,1,3] => ([(0,1),(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 2
[1,0,1,1,0,1,1,1,0,0,0,0] => [1,3,6,5,4,2] => [6,5,2,4,1,3] => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 1
[1,0,1,1,1,0,0,0,1,1,0,0] => [1,4,3,2,6,5] => [3,2,6,1,4,5] => ([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6) => 2
[1,0,1,1,1,0,0,1,0,1,0,0] => [1,4,3,5,6,2] => [3,2,4,6,1,5] => ([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => 2
[1,0,1,1,1,0,0,1,1,0,0,0] => [1,4,3,6,5,2] => [2,6,3,5,1,4] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => 2
[1,0,1,1,1,0,1,0,0,1,0,0] => [1,5,3,4,6,2] => [3,4,2,6,1,5] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6) => 2
[1,0,1,1,1,0,1,0,1,0,0,0] => [1,6,3,4,5,2] => [3,4,6,5,1,2] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6) => 2
[1,0,1,1,1,0,1,1,0,0,0,0] => [1,6,3,5,4,2] => [6,3,5,4,1,2] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6) => 1
[1,0,1,1,1,1,0,0,0,1,0,0] => [1,5,4,3,6,2] => [4,3,2,6,1,5] => ([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 2
[1,0,1,1,1,1,0,0,1,0,0,0] => [1,6,4,3,5,2] => [4,3,6,5,1,2] => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 2
[1,0,1,1,1,1,0,1,0,0,0,0] => [1,6,4,5,3,2] => [3,6,5,4,1,2] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6) => 2
[1,0,1,1,1,1,1,0,0,0,0,0] => [1,6,5,4,3,2] => [6,5,4,3,1,2] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 1
[1,1,0,0,1,0,1,1,1,0,0,0] => [2,1,3,6,5,4] => [6,1,5,2,3,4] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 1
[1,1,0,0,1,1,0,1,1,0,0,0] => [2,1,4,6,5,3] => [6,1,3,5,2,4] => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6) => 1
[1,1,0,0,1,1,1,0,0,1,0,0] => [2,1,5,4,6,3] => [4,1,3,6,2,5] => ([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6) => 2
[1,1,0,0,1,1,1,0,1,0,0,0] => [2,1,6,4,5,3] => [4,6,1,5,2,3] => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 2
[1,1,0,0,1,1,1,1,0,0,0,0] => [2,1,6,5,4,3] => [6,5,1,4,2,3] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 1
[1,1,0,1,0,0,1,1,1,0,0,0] => [2,3,1,6,5,4] => [6,1,2,5,3,4] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 1
[1,1,0,1,0,1,0,1,1,0,0,0] => [2,3,4,6,5,1] => [6,1,2,3,5,4] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6) => 1
[1,1,0,1,0,1,1,0,1,0,0,0] => [2,3,6,4,5,1] => [4,6,1,2,5,3] => ([(0,3),(0,5),(1,3),(1,5),(2,4),(2,5),(3,4),(4,5)],6) => 2
[1,1,0,1,0,1,1,1,0,0,0,0] => [2,3,6,5,4,1] => [6,5,1,2,4,3] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 1
[1,1,0,1,1,0,0,1,1,0,0,0] => [2,4,3,6,5,1] => [2,6,1,3,5,4] => ([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => 2
[1,1,0,1,1,0,1,0,1,0,0,0] => [2,6,3,4,5,1] => [3,4,6,1,5,2] => ([(0,3),(0,5),(1,3),(1,5),(2,4),(2,5),(3,4),(4,5)],6) => 2
[1,1,0,1,1,0,1,1,0,0,0,0] => [2,6,3,5,4,1] => [6,3,5,1,4,2] => ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 1
[1,1,0,1,1,1,0,0,1,0,0,0] => [2,6,4,3,5,1] => [4,3,6,1,5,2] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(2,3),(2,5),(3,5),(4,5)],6) => 2
[1,1,0,1,1,1,0,1,0,0,0,0] => [2,6,4,5,3,1] => [3,6,5,1,4,2] => ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => 2
[1,1,0,1,1,1,1,0,0,0,0,0] => [2,6,5,4,3,1] => [6,5,4,1,3,2] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 1
[1,1,1,0,0,1,1,1,0,0,0,0] => [3,2,6,5,4,1] => [6,1,5,2,4,3] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 1
[1,1,1,0,1,0,0,1,1,0,0,0] => [4,2,3,6,5,1] => [6,2,1,3,5,4] => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => 1
[1,1,1,0,1,0,1,0,1,0,0,0] => [6,2,3,4,5,1] => [2,3,4,6,5,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6) => 1
[1,1,1,0,1,0,1,1,0,0,0,0] => [6,2,3,5,4,1] => [6,2,3,5,4,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 1
[1,1,1,0,1,1,0,0,1,0,0,0] => [6,2,4,3,5,1] => [4,2,3,6,5,1] => ([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 1
[1,1,1,0,1,1,0,1,0,0,0,0] => [6,2,4,5,3,1] => [3,6,2,5,4,1] => ([(0,3),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 1
[1,1,1,0,1,1,1,0,0,0,0,0] => [6,2,5,4,3,1] => [6,5,2,4,3,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 1
[1,1,1,1,0,0,1,0,1,0,0,0] => [6,3,2,4,5,1] => [3,2,4,6,5,1] => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => 1
[1,1,1,1,0,0,1,1,0,0,0,0] => [6,3,2,5,4,1] => [2,6,3,5,4,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 1
[1,1,1,1,0,1,0,0,1,0,0,0] => [6,3,4,2,5,1] => [2,4,3,6,5,1] => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => 1
[1,1,1,1,0,1,0,1,0,0,0,0] => [6,3,4,5,2,1] => [2,3,6,5,4,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 1
[1,1,1,1,0,1,1,0,0,0,0,0] => [6,3,5,4,2,1] => [6,2,5,4,3,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 1
[1,1,1,1,1,0,0,0,1,0,0,0] => [6,4,3,2,5,1] => [4,3,2,6,5,1] => ([(0,1),(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 1
[1,1,1,1,1,0,0,1,0,0,0,0] => [6,4,3,5,2,1] => [3,2,6,5,4,1] => ([(0,1),(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 1
[1,1,1,1,1,0,1,0,0,0,0,0] => [6,5,3,4,2,1] => [3,6,5,4,2,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 1
[1,1,1,1,1,1,0,0,0,0,0,0] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0] => [1,2,3,4,5,7,6] => [7,1,2,3,4,5,6] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0] => [1,2,3,4,6,7,5] => [5,7,1,2,3,4,6] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7) => 2
[1,0,1,0,1,0,1,0,1,1,1,0,0,0] => [1,2,3,4,7,6,5] => [7,6,1,2,3,4,5] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0] => [1,2,3,5,4,7,6] => [4,7,1,2,3,5,6] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7) => 2
[1,0,1,0,1,0,1,1,0,1,0,1,0,0] => [1,2,3,5,6,7,4] => [4,5,7,1,2,3,6] => ([(0,6),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,6),(4,6),(5,6)],7) => 2
[1,0,1,0,1,0,1,1,0,1,1,0,0,0] => [1,2,3,5,7,6,4] => [7,4,6,1,2,3,5] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7) => 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,0] => [1,2,3,6,5,7,4] => [5,4,7,1,2,3,6] => ([(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5)],7) => 2
[1,0,1,0,1,0,1,1,1,0,1,0,0,0] => [1,2,3,7,5,6,4] => [5,7,6,1,2,3,4] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(5,6)],7) => 2
[1,0,1,0,1,0,1,1,1,1,0,0,0,0] => [1,2,3,7,6,5,4] => [7,6,5,1,2,3,4] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,0] => [1,2,4,3,5,7,6] => [3,7,1,2,4,5,6] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6)],7) => 2
[1,0,1,0,1,1,0,0,1,1,0,1,0,0] => [1,2,4,3,6,7,5] => [3,5,7,1,2,4,6] => ([(0,6),(1,4),(1,5),(2,3),(2,6),(3,4),(3,5),(4,6),(5,6)],7) => 2
[1,0,1,0,1,1,0,0,1,1,1,0,0,0] => [1,2,4,3,7,6,5] => [7,3,6,1,2,4,5] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,0] => [1,2,4,5,3,7,6] => [3,4,7,1,2,5,6] => ([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7) => 2
[1,0,1,0,1,1,0,1,0,1,0,1,0,0] => [1,2,4,5,6,7,3] => [3,4,5,7,1,2,6] => ([(0,4),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7) => 2
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Description
The radius of a connected graph.
This is the minimum eccentricity of any vertex.
This is the minimum eccentricity of any vertex.
Map
Lehmer-code to major-code bijection
Description
Sends a permutation to the unique permutation such that the Lehmer code is sent to the major code.
The Lehmer code encodes the inversions of a permutation and the major code encodes its major index. In particular, the number of inversions of a permutation equals the major index of its image under this map.
* The Lehmer code of a permutation σ is given by L(σ)=l1…ln with li=#{j>i:σj<σi}. In particular, li is the number of boxes in the i-th column of the Rothe diagram. For example, the Lehmer code of σ=[4,3,1,5,2] is 32010. The Lehmer code L:Sn ˜⟶ Sn is a bijection between permutations of size n and sequences l1…ln∈Nn with li≤i.
* The major code M(σ) of a permutation σ∈Sn is a way to encode a permutation as a sequence m1m2…mn with mi≥i. To define mi, let deli(σ) be the normalized permutation obtained by removing all σj<i from the one-line notation of σ. The i-th index is then given by
mi=maj(deli(σ))−maj(deli−1(σ)).
For example, the permutation [9,3,5,7,2,1,4,6,8] has major code [5,0,1,0,1,2,0,1,0] since
maj([8,2,4,6,1,3,5,7])=5,maj([7,1,3,5,2,4,6])=5,maj([6,2,4,1,3,5])=4,
maj([5,1,3,2,4])=4,maj([4,2,1,3])=3,maj([3,1,2])=1,maj([2,1])=1.
Observe that the sum of the major code of σ equals the major index of σ.
The Lehmer code encodes the inversions of a permutation and the major code encodes its major index. In particular, the number of inversions of a permutation equals the major index of its image under this map.
* The Lehmer code of a permutation σ is given by L(σ)=l1…ln with li=#{j>i:σj<σi}. In particular, li is the number of boxes in the i-th column of the Rothe diagram. For example, the Lehmer code of σ=[4,3,1,5,2] is 32010. The Lehmer code L:Sn ˜⟶ Sn is a bijection between permutations of size n and sequences l1…ln∈Nn with li≤i.
* The major code M(σ) of a permutation σ∈Sn is a way to encode a permutation as a sequence m1m2…mn with mi≥i. To define mi, let deli(σ) be the normalized permutation obtained by removing all σj<i from the one-line notation of σ. The i-th index is then given by
mi=maj(deli(σ))−maj(deli−1(σ)).
For example, the permutation [9,3,5,7,2,1,4,6,8] has major code [5,0,1,0,1,2,0,1,0] since
maj([8,2,4,6,1,3,5,7])=5,maj([7,1,3,5,2,4,6])=5,maj([6,2,4,1,3,5])=4,
maj([5,1,3,2,4])=4,maj([4,2,1,3])=3,maj([3,1,2])=1,maj([2,1])=1.
Observe that the sum of the major code of σ equals the major index of σ.
Map
to non-crossing permutation
Description
Sends a Dyck path D with valley at positions {(i1,j1),…,(ik,jk)} to the unique non-crossing permutation π having descents {i1,…,ik} and whose inverse has descents {j1,…,jk}.
It sends the area St000012The area of a Dyck path. to the number of inversions St000018The number of inversions of a permutation. and the major index St000027The major index of a Dyck path. to n(n−1) minus the sum of the major index St000004The major index of a permutation. and the inverse major index St000305The inverse major index of a permutation..
It sends the area St000012The area of a Dyck path. to the number of inversions St000018The number of inversions of a permutation. and the major index St000027The major index of a Dyck path. to n(n−1) minus the sum of the major index St000004The major index of a permutation. and the inverse major index St000305The inverse major index of a permutation..
Map
graph of inversions
Description
The graph of inversions of a permutation.
For a permutation of {1,…,n}, this is the graph with vertices {1,…,n}, where (i,j) is an edge if and only if it is an inversion of the permutation.
For a permutation of {1,…,n}, this is the graph with vertices {1,…,n}, where (i,j) is an edge if and only if it is an inversion of the permutation.
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