Identifier
-
Mp00050:
Ordered trees
—to binary tree: right brother = right child⟶
Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000777: Graphs ⟶ ℤ
Values
[[]] => [.,.] => [1] => ([],1) => 1
[[],[]] => [.,[.,.]] => [2,1] => ([(0,1)],2) => 2
[[],[],[]] => [.,[.,[.,.]]] => [3,2,1] => ([(0,1),(0,2),(1,2)],3) => 2
[[],[[]]] => [.,[[.,.],.]] => [2,3,1] => ([(0,2),(1,2)],3) => 3
[[],[],[],[]] => [.,[.,[.,[.,.]]]] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 2
[[],[],[[]]] => [.,[.,[[.,.],.]]] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 3
[[],[[]],[]] => [.,[[.,.],[.,.]]] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4) => 4
[[],[[],[]]] => [.,[[.,[.,.]],.]] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4) => 4
[[],[[[]]]] => [.,[[[.,.],.],.]] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4) => 3
[[],[],[],[],[]] => [.,[.,[.,[.,[.,.]]]]] => [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) => 2
[[],[],[],[[]]] => [.,[.,[.,[[.,.],.]]]] => [4,5,3,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 3
[[],[],[[]],[]] => [.,[.,[[.,.],[.,.]]]] => [3,5,4,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 4
[[],[],[[],[]]] => [.,[.,[[.,[.,.]],.]]] => [4,3,5,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 4
[[],[],[[[]]]] => [.,[.,[[[.,.],.],.]]] => [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 3
[[],[[]],[],[]] => [.,[[.,.],[.,[.,.]]]] => [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 4
[[],[[]],[[]]] => [.,[[.,.],[[.,.],.]]] => [2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 5
[[],[[],[]],[]] => [.,[[.,[.,.]],[.,.]]] => [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5) => 4
[[],[[[]]],[]] => [.,[[[.,.],.],[.,.]]] => [2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5) => 4
[[],[[],[],[]]] => [.,[[.,[.,[.,.]]],.]] => [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 4
[[],[[],[[]]]] => [.,[[.,[[.,.],.]],.]] => [3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 5
[[],[[[]],[]]] => [.,[[[.,.],[.,.]],.]] => [2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5) => 4
[[],[[[],[]]]] => [.,[[[.,[.,.]],.],.]] => [3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5) => 4
[[],[[[[]]]]] => [.,[[[[.,.],.],.],.]] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5) => 3
[[],[],[],[],[],[]] => [.,[.,[.,[.,[.,[.,.]]]]]] => [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) => 2
[[],[],[],[],[[]]] => [.,[.,[.,[.,[[.,.],.]]]]] => [5,6,4,3,2,1] => ([(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) => 3
[[],[],[],[[]],[]] => [.,[.,[.,[[.,.],[.,.]]]]] => [4,6,5,3,2,1] => ([(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) => 4
[[],[],[],[[],[]]] => [.,[.,[.,[[.,[.,.]],.]]]] => [5,4,6,3,2,1] => ([(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) => 4
[[],[],[],[[[]]]] => [.,[.,[.,[[[.,.],.],.]]]] => [4,5,6,3,2,1] => ([(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) => 3
[[],[],[[]],[],[]] => [.,[.,[[.,.],[.,[.,.]]]]] => [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) => 4
[[],[],[[]],[[]]] => [.,[.,[[.,.],[[.,.],.]]]] => [3,5,6,4,2,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 5
[[],[],[[],[]],[]] => [.,[.,[[.,[.,.]],[.,.]]]] => [4,3,6,5,2,1] => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4
[[],[],[[[]]],[]] => [.,[.,[[[.,.],.],[.,.]]]] => [3,4,6,5,2,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4
[[],[],[[],[],[]]] => [.,[.,[[.,[.,[.,.]]],.]]] => [5,4,3,6,2,1] => ([(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) => 4
[[],[],[[],[[]]]] => [.,[.,[[.,[[.,.],.]],.]]] => [4,5,3,6,2,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 5
[[],[],[[[]],[]]] => [.,[.,[[[.,.],[.,.]],.]]] => [3,5,4,6,2,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4
[[],[],[[[],[]]]] => [.,[.,[[[.,[.,.]],.],.]]] => [4,3,5,6,2,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4
[[],[],[[[[]]]]] => [.,[.,[[[[.,.],.],.],.]]] => [3,4,5,6,2,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
[[],[[]],[],[],[]] => [.,[[.,.],[.,[.,[.,.]]]]] => [2,6,5,4,3,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4
[[],[[]],[],[[]]] => [.,[[.,.],[.,[[.,.],.]]]] => [2,5,6,4,3,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 5
[[],[[]],[[]],[]] => [.,[[.,.],[[.,.],[.,.]]]] => [2,4,6,5,3,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 6
[[],[[]],[[],[]]] => [.,[[.,.],[[.,[.,.]],.]]] => [2,5,4,6,3,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 6
[[],[[]],[[[]]]] => [.,[[.,.],[[[.,.],.],.]]] => [2,4,5,6,3,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 5
[[],[[],[]],[],[]] => [.,[[.,[.,.]],[.,[.,.]]]] => [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) => 5
[[],[[[]]],[],[]] => [.,[[[.,.],.],[.,[.,.]]]] => [2,3,6,5,4,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4
[[],[[],[]],[[]]] => [.,[[.,[.,.]],[[.,.],.]]] => [3,2,5,6,4,1] => ([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 6
[[],[[[]]],[[]]] => [.,[[[.,.],.],[[.,.],.]]] => [2,3,5,6,4,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 5
[[],[[],[],[]],[]] => [.,[[.,[.,[.,.]]],[.,.]]] => [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) => 5
[[],[[],[[]]],[]] => [.,[[.,[[.,.],.]],[.,.]]] => [3,4,2,6,5,1] => ([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 6
[[],[[[]],[]],[]] => [.,[[[.,.],[.,.]],[.,.]]] => [2,4,3,6,5,1] => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => 4
[[],[[[],[]]],[]] => [.,[[[.,[.,.]],.],[.,.]]] => [3,2,4,6,5,1] => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => 4
[[],[[[[]]]],[]] => [.,[[[[.,.],.],.],[.,.]]] => [2,3,4,6,5,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6) => 4
[[],[[],[],[],[]]] => [.,[[.,[.,[.,[.,.]]]],.]] => [5,4,3,2,6,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4
[[],[[],[],[[]]]] => [.,[[.,[.,[[.,.],.]]],.]] => [4,5,3,2,6,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 5
[[],[[],[[]],[]]] => [.,[[.,[[.,.],[.,.]]],.]] => [3,5,4,2,6,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 6
[[],[[],[[],[]]]] => [.,[[.,[[.,[.,.]],.]],.]] => [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) => 6
[[],[[],[[[]]]]] => [.,[[.,[[[.,.],.],.]],.]] => [3,4,5,2,6,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 5
[[],[[[]],[],[]]] => [.,[[[.,.],[.,[.,.]]],.]] => [2,5,4,3,6,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4
[[],[[[]],[[]]]] => [.,[[[.,.],[[.,.],.]],.]] => [2,4,5,3,6,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 5
[[],[[[],[]],[]]] => [.,[[[.,[.,.]],[.,.]],.]] => [3,2,5,4,6,1] => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => 4
[[],[[[[]]],[]]] => [.,[[[[.,.],.],[.,.]],.]] => [2,3,5,4,6,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6) => 4
[[],[[[],[],[]]]] => [.,[[[.,[.,[.,.]]],.],.]] => [4,3,2,5,6,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4
[[],[[[],[[]]]]] => [.,[[[.,[[.,.],.]],.],.]] => [3,4,2,5,6,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 5
[[],[[[[]],[]]]] => [.,[[[[.,.],[.,.]],.],.]] => [2,4,3,5,6,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6) => 4
[[],[[[[],[]]]]] => [.,[[[[.,[.,.]],.],.],.]] => [3,2,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6) => 4
[[],[[[[[]]]]]] => [.,[[[[[.,.],.],.],.],.]] => [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 3
[[],[],[],[],[],[],[]] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 2
[[],[],[],[],[],[[]]] => [.,[.,[.,[.,[.,[[.,.],.]]]]]] => [6,7,5,4,3,2,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[[],[],[],[],[[]],[]] => [.,[.,[.,[.,[[.,.],[.,.]]]]]] => [5,7,6,4,3,2,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 4
[[],[],[],[],[[],[]]] => [.,[.,[.,[.,[[.,[.,.]],.]]]]] => [6,5,7,4,3,2,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 4
[[],[],[],[],[[[]]]] => [.,[.,[.,[.,[[[.,.],.],.]]]]] => [5,6,7,4,3,2,1] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[[],[],[],[[]],[],[]] => [.,[.,[.,[[.,.],[.,[.,.]]]]]] => [4,7,6,5,3,2,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 4
[[],[],[],[[]],[[]]] => [.,[.,[.,[[.,.],[[.,.],.]]]]] => [4,6,7,5,3,2,1] => ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 5
[[],[],[],[[],[]],[]] => [.,[.,[.,[[.,[.,.]],[.,.]]]]] => [5,4,7,6,3,2,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(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) => 4
[[],[],[],[[[]]],[]] => [.,[.,[.,[[[.,.],.],[.,.]]]]] => [4,5,7,6,3,2,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 4
[[],[],[],[[],[],[]]] => [.,[.,[.,[[.,[.,[.,.]]],.]]]] => [6,5,4,7,3,2,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 4
[[],[],[],[[],[[]]]] => [.,[.,[.,[[.,[[.,.],.]],.]]]] => [5,6,4,7,3,2,1] => ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 5
[[],[],[],[[[]],[]]] => [.,[.,[.,[[[.,.],[.,.]],.]]]] => [4,6,5,7,3,2,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 4
[[],[],[],[[[],[]]]] => [.,[.,[.,[[[.,[.,.]],.],.]]]] => [5,4,6,7,3,2,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 4
[[],[],[],[[[[]]]]] => [.,[.,[.,[[[[.,.],.],.],.]]]] => [4,5,6,7,3,2,1] => ([(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) => 3
[[],[],[[]],[],[],[]] => [.,[.,[[.,.],[.,[.,[.,.]]]]]] => [3,7,6,5,4,2,1] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 4
[[],[],[[]],[],[[]]] => [.,[.,[[.,.],[.,[[.,.],.]]]]] => [3,6,7,5,4,2,1] => ([(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 5
[[],[],[[]],[[]],[]] => [.,[.,[[.,.],[[.,.],[.,.]]]]] => [3,5,7,6,4,2,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 6
[[],[],[[]],[[],[]]] => [.,[.,[[.,.],[[.,[.,.]],.]]]] => [3,6,5,7,4,2,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 6
[[],[],[[]],[[[]]]] => [.,[.,[[.,.],[[[.,.],.],.]]]] => [3,5,6,7,4,2,1] => ([(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) => 5
[[],[],[[],[]],[],[]] => [.,[.,[[.,[.,.]],[.,[.,.]]]]] => [4,3,7,6,5,2,1] => ([(0,1),(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 5
[[],[],[[[]]],[],[]] => [.,[.,[[[.,.],.],[.,[.,.]]]]] => [3,4,7,6,5,2,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 4
[[],[],[[],[]],[[]]] => [.,[.,[[.,[.,.]],[[.,.],.]]]] => [4,3,6,7,5,2,1] => ([(0,1),(0,5),(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) => 6
[[],[],[[[]]],[[]]] => [.,[.,[[[.,.],.],[[.,.],.]]]] => [3,4,6,7,5,2,1] => ([(0,5),(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) => 5
[[],[],[[],[],[]],[]] => [.,[.,[[.,[.,[.,.]]],[.,.]]]] => [5,4,3,7,6,2,1] => ([(0,1),(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 5
[[],[],[[],[[]]],[]] => [.,[.,[[.,[[.,.],.]],[.,.]]]] => [4,5,3,7,6,2,1] => ([(0,1),(0,5),(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) => 6
[[],[],[[[]],[]],[]] => [.,[.,[[[.,.],[.,.]],[.,.]]]] => [3,5,4,7,6,2,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 4
[[],[],[[[],[]]],[]] => [.,[.,[[[.,[.,.]],.],[.,.]]]] => [4,3,5,7,6,2,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 4
[[],[],[[[[]]]],[]] => [.,[.,[[[[.,.],.],.],[.,.]]]] => [3,4,5,7,6,2,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 4
[[],[],[[],[],[],[]]] => [.,[.,[[.,[.,[.,[.,.]]]],.]]] => [6,5,4,3,7,2,1] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 4
[[],[],[[],[],[[]]]] => [.,[.,[[.,[.,[[.,.],.]]],.]]] => [5,6,4,3,7,2,1] => ([(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 5
[[],[],[[],[[]],[]]] => [.,[.,[[.,[[.,.],[.,.]]],.]]] => [4,6,5,3,7,2,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 6
[[],[],[[],[[],[]]]] => [.,[.,[[.,[[.,[.,.]],.]],.]]] => [5,4,6,3,7,2,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 6
[[],[],[[],[[[]]]]] => [.,[.,[[.,[[[.,.],.],.]],.]]] => [4,5,6,3,7,2,1] => ([(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) => 5
[[],[],[[[]],[],[]]] => [.,[.,[[[.,.],[.,[.,.]]],.]]] => [3,6,5,4,7,2,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 4
[[],[],[[[]],[[]]]] => [.,[.,[[[.,.],[[.,.],.]],.]]] => [3,5,6,4,7,2,1] => ([(0,5),(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) => 5
[[],[],[[[],[]],[]]] => [.,[.,[[[.,[.,.]],[.,.]],.]]] => [4,3,6,5,7,2,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 4
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Description
The number of distinct eigenvalues of the distance Laplacian of a connected graph.
Map
to binary tree: right brother = right child
Description
Return a binary tree of size $n-1$ (where $n$ is the size of an ordered tree $t$) obtained from $t$ by the following recursive rule:
- if $x$ is the right brother of $y$ in $t$, then $x$ becomes the right child of $y$;
- if $x$ is the first child of $y$ in $t$, then $x$ becomes the left child of $y$,
and removing the root of $t$.
- if $x$ is the right brother of $y$ in $t$, then $x$ becomes the right child of $y$;
- if $x$ is the first child of $y$ in $t$, then $x$ becomes the left child of $y$,
and removing the root of $t$.
Map
to 312-avoiding permutation
Description
Return a 312-avoiding permutation corresponding to a binary tree.
The linear extensions of a binary tree form an interval of the weak order called the Sylvester class of the tree. This permutation is the minimal element of this Sylvester class.
The linear extensions of a binary tree form an interval of the weak order called the Sylvester class of the tree. This permutation is the minimal element of this Sylvester class.
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.
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