Identifier
-
Mp00129:
Dyck paths
—to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶
Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000259: Graphs ⟶ ℤ
Values
[1,0] => [1] => [1] => ([],1) => 0
[1,0,1,0] => [2,1] => [2,1] => ([(0,1)],2) => 1
[1,0,1,0,1,0] => [2,3,1] => [3,1,2] => ([(0,2),(1,2)],3) => 2
[1,1,0,0,1,0] => [1,3,2] => [2,3,1] => ([(0,2),(1,2)],3) => 2
[1,0,1,0,1,0,1,0] => [2,3,4,1] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4) => 2
[1,0,1,1,0,0,1,0] => [2,1,4,3] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4) => 2
[1,1,0,0,1,0,1,0] => [1,3,4,2] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4) => 3
[1,1,0,1,0,0,1,0] => [3,1,4,2] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4) => 2
[1,1,1,0,0,0,1,0] => [1,2,4,3] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4) => 2
[1,0,1,0,1,0,1,0,1,0] => [2,3,4,5,1] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5) => 2
[1,0,1,0,1,1,0,0,1,0] => [2,3,1,5,4] => [4,2,3,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 2
[1,0,1,1,0,0,1,0,1,0] => [2,1,4,5,3] => [3,2,5,1,4] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5) => 3
[1,0,1,1,0,1,0,0,1,0] => [2,4,1,5,3] => [4,2,5,1,3] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5) => 2
[1,0,1,1,1,0,0,0,1,0] => [2,1,3,5,4] => [3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5) => 2
[1,1,0,0,1,0,1,0,1,0] => [1,3,4,5,2] => [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5) => 3
[1,1,0,0,1,1,0,0,1,0] => [1,3,2,5,4] => [3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 2
[1,1,0,1,0,0,1,0,1,0] => [3,1,4,5,2] => [3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5) => 3
[1,1,0,1,0,1,0,0,1,0] => [3,4,1,5,2] => [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5) => 2
[1,1,0,1,1,0,0,0,1,0] => [3,1,2,5,4] => [2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5) => 2
[1,1,0,1,1,0,0,1,0,0] => [3,1,5,2,4] => [3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5) => 3
[1,1,1,0,0,0,1,0,1,0] => [1,2,4,5,3] => [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5) => 3
[1,1,1,0,0,1,0,0,1,0] => [1,4,2,5,3] => [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5) => 2
[1,1,1,0,1,0,0,0,1,0] => [4,1,2,5,3] => [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5) => 3
[1,1,1,0,1,0,0,1,0,0] => [4,1,5,2,3] => [3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5) => 4
[1,1,1,1,0,0,0,0,1,0] => [1,2,3,5,4] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5) => 2
[1,0,1,0,1,0,1,0,1,0,1,0] => [2,3,4,5,6,1] => [6,1,2,3,4,5] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 2
[1,0,1,0,1,0,1,1,0,0,1,0] => [2,3,4,1,6,5] => [5,2,3,4,6,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 2
[1,0,1,0,1,1,0,0,1,0,1,0] => [2,3,1,5,6,4] => [4,2,3,6,1,5] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6) => 3
[1,0,1,0,1,1,0,1,0,0,1,0] => [2,3,5,1,6,4] => [5,2,3,6,1,4] => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 2
[1,0,1,0,1,1,1,0,0,0,1,0] => [2,3,1,4,6,5] => [4,2,3,5,6,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 2
[1,0,1,1,0,0,1,0,1,0,1,0] => [2,1,4,5,6,3] => [3,2,6,1,4,5] => ([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6) => 3
[1,0,1,1,0,0,1,1,0,0,1,0] => [2,1,4,3,6,5] => [4,3,5,2,6,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 2
[1,0,1,1,0,1,0,0,1,0,1,0] => [2,4,1,5,6,3] => [4,2,6,1,3,5] => ([(0,5),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5)],6) => 3
[1,0,1,1,0,1,0,1,0,0,1,0] => [2,4,5,1,6,3] => [5,2,6,1,3,4] => ([(0,3),(0,5),(1,3),(1,5),(2,4),(2,5),(3,4),(4,5)],6) => 2
[1,0,1,1,0,1,1,0,0,0,1,0] => [2,4,1,3,6,5] => [2,4,5,3,6,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 2
[1,0,1,1,0,1,1,0,0,1,0,0] => [2,4,1,6,3,5] => [4,2,1,5,6,3] => ([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6) => 3
[1,0,1,1,1,0,0,0,1,0,1,0] => [2,1,3,5,6,4] => [3,2,4,6,1,5] => ([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => 3
[1,0,1,1,1,0,0,1,0,0,1,0] => [2,1,5,3,6,4] => [4,3,5,6,1,2] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6) => 2
[1,0,1,1,1,0,1,0,0,0,1,0] => [2,5,1,3,6,4] => [2,4,5,6,1,3] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 3
[1,0,1,1,1,0,1,0,0,1,0,0] => [2,5,1,6,3,4] => [4,2,1,6,3,5] => ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6) => 4
[1,0,1,1,1,1,0,0,0,0,1,0] => [2,1,3,4,6,5] => [3,2,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6) => 2
[1,1,0,0,1,0,1,0,1,0,1,0] => [1,3,4,5,6,2] => [2,6,1,3,4,5] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => 3
[1,1,0,0,1,0,1,1,0,0,1,0] => [1,3,4,2,6,5] => [3,5,2,4,6,1] => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6) => 2
[1,1,0,0,1,1,0,0,1,0,1,0] => [1,3,2,5,6,4] => [3,4,2,6,1,5] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6) => 3
[1,1,0,0,1,1,0,1,0,0,1,0] => [1,3,5,2,6,4] => [3,5,2,6,1,4] => ([(0,1),(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
[1,1,0,0,1,1,1,0,0,0,1,0] => [1,3,2,4,6,5] => [3,4,2,5,6,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 2
[1,1,0,1,0,0,1,0,1,0,1,0] => [3,1,4,5,6,2] => [3,6,1,2,4,5] => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6) => 3
[1,1,0,1,0,0,1,1,0,0,1,0] => [3,1,4,2,6,5] => [4,5,2,3,6,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6) => 2
[1,1,0,1,0,0,1,1,0,1,0,0] => [3,1,4,6,2,5] => [3,1,4,5,6,2] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => 3
[1,1,0,1,0,1,0,0,1,0,1,0] => [3,4,1,5,6,2] => [4,6,1,2,3,5] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 3
[1,1,0,1,0,1,0,1,0,0,1,0] => [3,4,5,1,6,2] => [5,6,1,2,3,4] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 2
[1,1,0,1,0,1,1,0,0,0,1,0] => [3,4,1,2,6,5] => [2,5,3,4,6,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 2
[1,1,0,1,0,1,1,0,0,1,0,0] => [3,4,1,6,2,5] => [4,1,3,5,6,2] => ([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => 3
[1,1,0,1,1,0,0,0,1,0,1,0] => [3,1,2,5,6,4] => [2,4,3,6,1,5] => ([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => 3
[1,1,0,1,1,0,0,1,0,0,1,0] => [3,1,5,2,6,4] => [4,5,2,6,1,3] => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 2
[1,1,0,1,1,0,0,1,0,1,0,0] => [3,1,5,6,2,4] => [3,1,4,6,2,5] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => 4
[1,1,0,1,1,0,1,0,0,0,1,0] => [3,5,1,2,6,4] => [2,5,3,6,1,4] => ([(0,5),(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => 3
[1,1,0,1,1,0,1,0,0,1,0,0] => [3,5,1,6,2,4] => [4,1,3,6,2,5] => ([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6) => 4
[1,1,0,1,1,1,0,0,0,0,1,0] => [3,1,2,4,6,5] => [2,4,3,5,6,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6) => 2
[1,1,1,0,0,0,1,0,1,0,1,0] => [1,2,4,5,6,3] => [2,3,6,1,4,5] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => 3
[1,1,1,0,0,0,1,1,0,0,1,0] => [1,2,4,3,6,5] => [3,4,5,2,6,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 2
[1,1,1,0,0,1,0,0,1,0,1,0] => [1,4,2,5,6,3] => [3,4,6,1,2,5] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => 3
[1,1,1,0,0,1,0,1,0,0,1,0] => [1,4,5,2,6,3] => [3,5,6,1,2,4] => ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6) => 3
[1,1,1,0,0,1,1,0,0,0,1,0] => [1,4,2,3,6,5] => [3,2,5,4,6,1] => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => 2
[1,1,1,0,0,1,1,0,0,1,0,0] => [1,4,2,6,3,5] => [3,4,1,5,6,2] => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6) => 3
[1,1,1,0,1,0,0,0,1,0,1,0] => [4,1,2,5,6,3] => [2,4,6,1,3,5] => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => 3
[1,1,1,0,1,0,0,1,0,0,1,0] => [4,1,5,2,6,3] => [4,5,6,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6) => 2
[1,1,1,0,1,0,0,1,0,1,0,0] => [4,1,5,6,2,3] => [3,1,6,2,4,5] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => 4
[1,1,1,0,1,0,1,0,0,0,1,0] => [4,5,1,2,6,3] => [2,5,6,1,3,4] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => 3
[1,1,1,0,1,0,1,0,0,1,0,0] => [4,5,1,6,2,3] => [4,1,6,2,3,5] => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6) => 4
[1,1,1,0,1,1,0,0,0,0,1,0] => [4,1,2,3,6,5] => [2,3,5,4,6,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6) => 2
[1,1,1,0,1,1,0,0,0,1,0,0] => [4,1,2,6,3,5] => [2,4,1,5,6,3] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => 4
[1,1,1,1,0,0,0,0,1,0,1,0] => [1,2,3,5,6,4] => [2,3,4,6,1,5] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => 3
[1,1,1,1,0,0,0,1,0,0,1,0] => [1,2,5,3,6,4] => [3,4,5,6,1,2] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 2
[1,1,1,1,0,0,1,0,0,0,1,0] => [1,5,2,3,6,4] => [3,2,5,6,1,4] => ([(0,3),(0,4),(1,2),(1,5),(2,5),(3,5),(4,5)],6) => 3
[1,1,1,1,0,0,1,0,0,1,0,0] => [1,5,2,6,3,4] => [3,4,1,6,2,5] => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6) => 4
[1,1,1,1,0,1,0,0,0,0,1,0] => [5,1,2,3,6,4] => [2,3,5,6,1,4] => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6) => 3
[1,1,1,1,0,1,0,0,0,1,0,0] => [5,1,2,6,3,4] => [2,4,1,6,3,5] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 5
[1,1,1,1,1,0,0,0,0,0,1,0] => [1,2,3,4,6,5] => [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 2
[1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [2,3,4,5,6,7,1] => [7,1,2,3,4,5,6] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 2
[1,0,1,0,1,0,1,0,1,1,0,0,1,0] => [2,3,4,5,1,7,6] => [6,2,3,4,5,7,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 2
[1,0,1,0,1,0,1,1,0,0,1,0,1,0] => [2,3,4,1,6,7,5] => [5,2,3,4,7,1,6] => ([(0,4),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,6),(5,6)],7) => 3
[1,0,1,0,1,0,1,1,0,1,0,0,1,0] => [2,3,4,6,1,7,5] => [6,2,3,4,7,1,5] => ([(0,1),(0,6),(1,5),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 2
[1,0,1,0,1,0,1,1,1,0,0,0,1,0] => [2,3,4,1,5,7,6] => [5,2,3,4,6,7,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 2
[1,0,1,0,1,1,0,0,1,0,1,0,1,0] => [2,3,1,5,6,7,4] => [4,2,3,7,1,5,6] => ([(0,4),(1,4),(2,5),(2,6),(3,5),(3,6),(4,6),(5,6)],7) => 3
[1,0,1,0,1,1,0,0,1,1,0,0,1,0] => [2,3,1,5,4,7,6] => [5,3,4,6,2,7,1] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 2
[1,0,1,0,1,1,0,1,0,0,1,0,1,0] => [2,3,5,1,6,7,4] => [5,2,3,7,1,4,6] => ([(0,4),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(4,6),(5,6)],7) => 3
[1,0,1,0,1,1,0,1,0,1,0,0,1,0] => [2,3,5,6,1,7,4] => [6,2,3,7,1,4,5] => ([(0,5),(0,6),(1,5),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(5,6)],7) => 2
[1,0,1,0,1,1,0,1,1,0,0,0,1,0] => [2,3,5,1,4,7,6] => [2,4,5,6,3,7,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 2
[1,0,1,0,1,1,0,1,1,0,0,1,0,0] => [2,3,5,1,7,4,6] => [5,2,3,1,6,7,4] => ([(0,4),(1,4),(2,5),(2,6),(3,5),(3,6),(4,6),(5,6)],7) => 3
[1,0,1,0,1,1,1,0,0,0,1,0,1,0] => [2,3,1,4,6,7,5] => [4,2,3,5,7,1,6] => ([(0,4),(1,6),(2,5),(2,6),(3,5),(3,6),(4,6),(5,6)],7) => 3
[1,0,1,0,1,1,1,0,0,1,0,0,1,0] => [2,3,1,6,4,7,5] => [5,3,4,6,7,1,2] => ([(0,5),(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7) => 2
[1,0,1,0,1,1,1,0,1,0,0,0,1,0] => [2,3,6,1,4,7,5] => [2,4,5,6,7,1,3] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7) => 3
[1,0,1,0,1,1,1,0,1,0,0,1,0,0] => [2,3,6,1,7,4,5] => [5,2,3,1,7,4,6] => ([(0,4),(1,4),(1,6),(2,5),(2,6),(3,5),(3,6),(5,6)],7) => 4
[1,0,1,0,1,1,1,1,0,0,0,0,1,0] => [2,3,1,4,5,7,6] => [4,2,3,5,6,7,1] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 2
[1,0,1,1,0,0,1,0,1,0,1,0,1,0] => [2,1,4,5,6,7,3] => [3,2,7,1,4,5,6] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(4,5),(5,6)],7) => 3
[1,0,1,1,0,0,1,0,1,1,0,0,1,0] => [2,1,4,5,3,7,6] => [4,3,6,2,5,7,1] => ([(0,6),(1,4),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 2
[1,0,1,1,0,0,1,1,0,0,1,0,1,0] => [2,1,4,3,6,7,5] => [4,3,5,2,7,1,6] => ([(0,2),(1,5),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[1,0,1,1,0,0,1,1,0,1,0,0,1,0] => [2,1,4,6,3,7,5] => [4,3,6,2,7,1,5] => ([(0,1),(0,4),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[1,0,1,1,0,0,1,1,1,0,0,0,1,0] => [2,1,4,3,5,7,6] => [4,3,5,2,6,7,1] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 2
[1,0,1,1,0,1,0,0,1,0,1,0,1,0] => [2,4,1,5,6,7,3] => [4,2,7,1,3,5,6] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(4,5),(4,6)],7) => 3
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Description
The diameter of a connected graph.
This is the greatest distance between any pair of vertices.
This is the greatest distance between any pair of vertices.
Map
to 321-avoiding permutation (Billey-Jockusch-Stanley)
Description
The Billey-Jockusch-Stanley bijection to 321-avoiding permutations.
Map
graph of inversions
Description
The graph of inversions of a permutation.
For a permutation of $\{1,\dots,n\}$, this is the graph with vertices $\{1,\dots,n\}$, where $(i,j)$ is an edge if and only if it is an inversion of the permutation.
For a permutation of $\{1,\dots,n\}$, this is the graph with vertices $\{1,\dots,n\}$, where $(i,j)$ is an edge if and only if it is an inversion of the permutation.
Map
major-index to inversion-number bijection
Description
Return the permutation whose Lehmer code equals the major code of the preimage.
This map sends the major index to the number of inversions.
This map sends the major index to the number of inversions.
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