Identifier
-
Mp00029:
Dyck paths
—to binary tree: left tree, up step, right tree, down step⟶
Binary trees
Mp00009: Binary trees —left rotate⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St001880: Posets ⟶ ℤ
Values
[1,0,1,0,1,0] => [[[.,.],.],.] => [.,[[.,.],.]] => ([(0,2),(2,1)],3) => 3
[1,0,1,1,0,0] => [[.,.],[.,.]] => [[[.,.],.],.] => ([(0,2),(2,1)],3) => 3
[1,1,0,0,1,0] => [[.,[.,.]],.] => [.,[.,[.,.]]] => ([(0,2),(2,1)],3) => 3
[1,1,0,1,0,0] => [.,[[.,.],.]] => [[.,[.,.]],.] => ([(0,2),(2,1)],3) => 3
[1,0,1,0,1,0,1,0] => [[[[.,.],.],.],.] => [.,[[[.,.],.],.]] => ([(0,3),(2,1),(3,2)],4) => 4
[1,0,1,0,1,1,0,0] => [[[.,.],.],[.,.]] => [[[[.,.],.],.],.] => ([(0,3),(2,1),(3,2)],4) => 4
[1,1,0,0,1,0,1,0] => [[[.,[.,.]],.],.] => [.,[[.,[.,.]],.]] => ([(0,3),(2,1),(3,2)],4) => 4
[1,1,0,0,1,1,0,0] => [[.,[.,.]],[.,.]] => [[[.,[.,.]],.],.] => ([(0,3),(2,1),(3,2)],4) => 4
[1,1,0,1,0,0,1,0] => [[.,[[.,.],.]],.] => [.,[.,[[.,.],.]]] => ([(0,3),(2,1),(3,2)],4) => 4
[1,1,0,1,0,1,0,0] => [.,[[[.,.],.],.]] => [[.,[[.,.],.]],.] => ([(0,3),(2,1),(3,2)],4) => 4
[1,1,1,0,0,0,1,0] => [[.,[.,[.,.]]],.] => [.,[.,[.,[.,.]]]] => ([(0,3),(2,1),(3,2)],4) => 4
[1,1,1,0,0,1,0,0] => [.,[[.,[.,.]],.]] => [[.,[.,[.,.]]],.] => ([(0,3),(2,1),(3,2)],4) => 4
[1,0,1,0,1,0,1,0,1,0] => [[[[[.,.],.],.],.],.] => [.,[[[[.,.],.],.],.]] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[1,0,1,0,1,0,1,1,0,0] => [[[[.,.],.],.],[.,.]] => [[[[[.,.],.],.],.],.] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[1,1,0,0,1,0,1,0,1,0] => [[[[.,[.,.]],.],.],.] => [.,[[[.,[.,.]],.],.]] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[1,1,0,0,1,0,1,1,0,0] => [[[.,[.,.]],.],[.,.]] => [[[[.,[.,.]],.],.],.] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[1,1,0,1,0,0,1,0,1,0] => [[[.,[[.,.],.]],.],.] => [.,[[.,[[.,.],.]],.]] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[1,1,0,1,0,0,1,1,0,0] => [[.,[[.,.],.]],[.,.]] => [[[.,[[.,.],.]],.],.] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[1,1,0,1,0,1,0,0,1,0] => [[.,[[[.,.],.],.]],.] => [.,[.,[[[.,.],.],.]]] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[1,1,0,1,0,1,0,1,0,0] => [.,[[[[.,.],.],.],.]] => [[.,[[[.,.],.],.]],.] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[1,1,1,0,0,0,1,0,1,0] => [[[.,[.,[.,.]]],.],.] => [.,[[.,[.,[.,.]]],.]] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[1,1,1,0,0,0,1,1,0,0] => [[.,[.,[.,.]]],[.,.]] => [[[.,[.,[.,.]]],.],.] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[1,1,1,0,0,1,0,0,1,0] => [[.,[[.,[.,.]],.]],.] => [.,[.,[[.,[.,.]],.]]] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[1,1,1,0,0,1,0,1,0,0] => [.,[[[.,[.,.]],.],.]] => [[.,[[.,[.,.]],.]],.] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[1,1,1,0,1,0,0,0,1,0] => [[.,[.,[[.,.],.]]],.] => [.,[.,[.,[[.,.],.]]]] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[1,1,1,0,1,0,0,1,0,0] => [.,[[.,[[.,.],.]],.]] => [[.,[.,[[.,.],.]]],.] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[1,1,1,1,0,0,0,0,1,0] => [[.,[.,[.,[.,.]]]],.] => [.,[.,[.,[.,[.,.]]]]] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[1,1,1,1,0,0,0,1,0,0] => [.,[[.,[.,[.,.]]],.]] => [[.,[.,[.,[.,.]]]],.] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[1,0,1,0,1,0,1,0,1,0,1,0] => [[[[[[.,.],.],.],.],.],.] => [.,[[[[[.,.],.],.],.],.]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[1,0,1,0,1,0,1,0,1,1,0,0] => [[[[[.,.],.],.],.],[.,.]] => [[[[[[.,.],.],.],.],.],.] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[1,1,0,0,1,0,1,0,1,0,1,0] => [[[[[.,[.,.]],.],.],.],.] => [.,[[[[.,[.,.]],.],.],.]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[1,1,0,0,1,0,1,0,1,1,0,0] => [[[[.,[.,.]],.],.],[.,.]] => [[[[[.,[.,.]],.],.],.],.] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[1,1,0,1,0,0,1,0,1,0,1,0] => [[[[.,[[.,.],.]],.],.],.] => [.,[[[.,[[.,.],.]],.],.]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[1,1,0,1,0,0,1,0,1,1,0,0] => [[[.,[[.,.],.]],.],[.,.]] => [[[[.,[[.,.],.]],.],.],.] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[1,1,0,1,0,1,0,0,1,0,1,0] => [[[.,[[[.,.],.],.]],.],.] => [.,[[.,[[[.,.],.],.]],.]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[1,1,0,1,0,1,0,0,1,1,0,0] => [[.,[[[.,.],.],.]],[.,.]] => [[[.,[[[.,.],.],.]],.],.] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[1,1,0,1,0,1,0,1,0,0,1,0] => [[.,[[[[.,.],.],.],.]],.] => [.,[.,[[[[.,.],.],.],.]]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[1,1,0,1,0,1,0,1,0,1,0,0] => [.,[[[[[.,.],.],.],.],.]] => [[.,[[[[.,.],.],.],.]],.] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[1,1,1,0,0,0,1,0,1,0,1,0] => [[[[.,[.,[.,.]]],.],.],.] => [.,[[[.,[.,[.,.]]],.],.]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[1,1,1,0,0,0,1,0,1,1,0,0] => [[[.,[.,[.,.]]],.],[.,.]] => [[[[.,[.,[.,.]]],.],.],.] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[1,1,1,0,0,1,0,0,1,0,1,0] => [[[.,[[.,[.,.]],.]],.],.] => [.,[[.,[[.,[.,.]],.]],.]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[1,1,1,0,0,1,0,0,1,1,0,0] => [[.,[[.,[.,.]],.]],[.,.]] => [[[.,[[.,[.,.]],.]],.],.] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[1,1,1,0,0,1,0,1,0,0,1,0] => [[.,[[[.,[.,.]],.],.]],.] => [.,[.,[[[.,[.,.]],.],.]]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[1,1,1,0,0,1,0,1,0,1,0,0] => [.,[[[[.,[.,.]],.],.],.]] => [[.,[[[.,[.,.]],.],.]],.] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[1,1,1,0,1,0,0,0,1,0,1,0] => [[[.,[.,[[.,.],.]]],.],.] => [.,[[.,[.,[[.,.],.]]],.]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[1,1,1,0,1,0,0,0,1,1,0,0] => [[.,[.,[[.,.],.]]],[.,.]] => [[[.,[.,[[.,.],.]]],.],.] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[1,1,1,0,1,0,0,1,0,0,1,0] => [[.,[[.,[[.,.],.]],.]],.] => [.,[.,[[.,[[.,.],.]],.]]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[1,1,1,0,1,0,0,1,0,1,0,0] => [.,[[[.,[[.,.],.]],.],.]] => [[.,[[.,[[.,.],.]],.]],.] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[1,1,1,0,1,0,1,0,0,0,1,0] => [[.,[.,[[[.,.],.],.]]],.] => [.,[.,[.,[[[.,.],.],.]]]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[1,1,1,0,1,0,1,0,0,1,0,0] => [.,[[.,[[[.,.],.],.]],.]] => [[.,[.,[[[.,.],.],.]]],.] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[1,1,1,1,0,0,0,0,1,0,1,0] => [[[.,[.,[.,[.,.]]]],.],.] => [.,[[.,[.,[.,[.,.]]]],.]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[1,1,1,1,0,0,0,0,1,1,0,0] => [[.,[.,[.,[.,.]]]],[.,.]] => [[[.,[.,[.,[.,.]]]],.],.] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[1,1,1,1,0,0,0,1,0,0,1,0] => [[.,[[.,[.,[.,.]]],.]],.] => [.,[.,[[.,[.,[.,.]]],.]]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[1,1,1,1,0,0,0,1,0,1,0,0] => [.,[[[.,[.,[.,.]]],.],.]] => [[.,[[.,[.,[.,.]]],.]],.] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[1,1,1,1,0,0,1,0,0,0,1,0] => [[.,[.,[[.,[.,.]],.]]],.] => [.,[.,[.,[[.,[.,.]],.]]]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[1,1,1,1,0,0,1,0,0,1,0,0] => [.,[[.,[[.,[.,.]],.]],.]] => [[.,[.,[[.,[.,.]],.]]],.] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[1,1,1,1,0,1,0,0,0,0,1,0] => [[.,[.,[.,[[.,.],.]]]],.] => [.,[.,[.,[.,[[.,.],.]]]]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[1,1,1,1,0,1,0,0,0,1,0,0] => [.,[[.,[.,[[.,.],.]]],.]] => [[.,[.,[.,[[.,.],.]]]],.] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[1,1,1,1,1,0,0,0,0,0,1,0] => [[.,[.,[.,[.,[.,.]]]]],.] => [.,[.,[.,[.,[.,[.,.]]]]]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[1,1,1,1,1,0,0,0,0,1,0,0] => [.,[[.,[.,[.,[.,.]]]],.]] => [[.,[.,[.,[.,[.,.]]]]],.] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [[[[[[[.,.],.],.],.],.],.],.] => [.,[[[[[[.,.],.],.],.],.],.]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
[1,0,1,0,1,0,1,0,1,0,1,1,0,0] => [[[[[[.,.],.],.],.],.],[.,.]] => [[[[[[[.,.],.],.],.],.],.],.] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
[1,1,0,0,1,0,1,0,1,0,1,0,1,0] => [[[[[[.,[.,.]],.],.],.],.],.] => [.,[[[[[.,[.,.]],.],.],.],.]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
[1,1,0,0,1,0,1,0,1,0,1,1,0,0] => [[[[[.,[.,.]],.],.],.],[.,.]] => [[[[[[.,[.,.]],.],.],.],.],.] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
[1,1,0,1,0,0,1,0,1,0,1,0,1,0] => [[[[[.,[[.,.],.]],.],.],.],.] => [.,[[[[.,[[.,.],.]],.],.],.]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
[1,1,0,1,0,0,1,0,1,0,1,1,0,0] => [[[[.,[[.,.],.]],.],.],[.,.]] => [[[[[.,[[.,.],.]],.],.],.],.] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
[1,1,0,1,0,1,0,0,1,0,1,0,1,0] => [[[[.,[[[.,.],.],.]],.],.],.] => [.,[[[.,[[[.,.],.],.]],.],.]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
[1,1,0,1,0,1,0,0,1,0,1,1,0,0] => [[[.,[[[.,.],.],.]],.],[.,.]] => [[[[.,[[[.,.],.],.]],.],.],.] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
[1,1,0,1,0,1,0,1,0,0,1,0,1,0] => [[[.,[[[[.,.],.],.],.]],.],.] => [.,[[.,[[[[.,.],.],.],.]],.]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
[1,1,0,1,0,1,0,1,0,0,1,1,0,0] => [[.,[[[[.,.],.],.],.]],[.,.]] => [[[.,[[[[.,.],.],.],.]],.],.] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
[1,1,0,1,0,1,0,1,0,1,0,0,1,0] => [[.,[[[[[.,.],.],.],.],.]],.] => [.,[.,[[[[[.,.],.],.],.],.]]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
[1,1,0,1,0,1,0,1,0,1,0,1,0,0] => [.,[[[[[[.,.],.],.],.],.],.]] => [[.,[[[[[.,.],.],.],.],.]],.] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
[1,1,1,0,0,0,1,0,1,0,1,0,1,0] => [[[[[.,[.,[.,.]]],.],.],.],.] => [.,[[[[.,[.,[.,.]]],.],.],.]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
[1,1,1,0,0,0,1,0,1,0,1,1,0,0] => [[[[.,[.,[.,.]]],.],.],[.,.]] => [[[[[.,[.,[.,.]]],.],.],.],.] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
[1,1,1,0,0,1,0,0,1,0,1,0,1,0] => [[[[.,[[.,[.,.]],.]],.],.],.] => [.,[[[.,[[.,[.,.]],.]],.],.]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
[1,1,1,0,0,1,0,0,1,0,1,1,0,0] => [[[.,[[.,[.,.]],.]],.],[.,.]] => [[[[.,[[.,[.,.]],.]],.],.],.] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
[1,1,1,0,0,1,0,1,0,0,1,0,1,0] => [[[.,[[[.,[.,.]],.],.]],.],.] => [.,[[.,[[[.,[.,.]],.],.]],.]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
[1,1,1,0,0,1,0,1,0,0,1,1,0,0] => [[.,[[[.,[.,.]],.],.]],[.,.]] => [[[.,[[[.,[.,.]],.],.]],.],.] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
[1,1,1,0,0,1,0,1,0,1,0,0,1,0] => [[.,[[[[.,[.,.]],.],.],.]],.] => [.,[.,[[[[.,[.,.]],.],.],.]]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
[1,1,1,0,0,1,0,1,0,1,0,1,0,0] => [.,[[[[[.,[.,.]],.],.],.],.]] => [[.,[[[[.,[.,.]],.],.],.]],.] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
[1,1,1,0,1,0,0,0,1,0,1,0,1,0] => [[[[.,[.,[[.,.],.]]],.],.],.] => [.,[[[.,[.,[[.,.],.]]],.],.]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
[1,1,1,0,1,0,0,0,1,0,1,1,0,0] => [[[.,[.,[[.,.],.]]],.],[.,.]] => [[[[.,[.,[[.,.],.]]],.],.],.] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
[1,1,1,0,1,0,0,1,0,0,1,0,1,0] => [[[.,[[.,[[.,.],.]],.]],.],.] => [.,[[.,[[.,[[.,.],.]],.]],.]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
[1,1,1,0,1,0,0,1,0,0,1,1,0,0] => [[.,[[.,[[.,.],.]],.]],[.,.]] => [[[.,[[.,[[.,.],.]],.]],.],.] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
[1,1,1,0,1,0,0,1,0,1,0,0,1,0] => [[.,[[[.,[[.,.],.]],.],.]],.] => [.,[.,[[[.,[[.,.],.]],.],.]]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
[1,1,1,0,1,0,0,1,0,1,0,1,0,0] => [.,[[[[.,[[.,.],.]],.],.],.]] => [[.,[[[.,[[.,.],.]],.],.]],.] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
[1,1,1,0,1,0,1,0,0,0,1,0,1,0] => [[[.,[.,[[[.,.],.],.]]],.],.] => [.,[[.,[.,[[[.,.],.],.]]],.]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
[1,1,1,0,1,0,1,0,0,0,1,1,0,0] => [[.,[.,[[[.,.],.],.]]],[.,.]] => [[[.,[.,[[[.,.],.],.]]],.],.] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
[1,1,1,0,1,0,1,0,0,1,0,0,1,0] => [[.,[[.,[[[.,.],.],.]],.]],.] => [.,[.,[[.,[[[.,.],.],.]],.]]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
[1,1,1,0,1,0,1,0,0,1,0,1,0,0] => [.,[[[.,[[[.,.],.],.]],.],.]] => [[.,[[.,[[[.,.],.],.]],.]],.] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
[1,1,1,0,1,0,1,0,1,0,0,0,1,0] => [[.,[.,[[[[.,.],.],.],.]]],.] => [.,[.,[.,[[[[.,.],.],.],.]]]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
[1,1,1,0,1,0,1,0,1,0,0,1,0,0] => [.,[[.,[[[[.,.],.],.],.]],.]] => [[.,[.,[[[[.,.],.],.],.]]],.] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
[1,1,1,1,0,0,0,0,1,0,1,0,1,0] => [[[[.,[.,[.,[.,.]]]],.],.],.] => [.,[[[.,[.,[.,[.,.]]]],.],.]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
[1,1,1,1,0,0,0,0,1,0,1,1,0,0] => [[[.,[.,[.,[.,.]]]],.],[.,.]] => [[[[.,[.,[.,[.,.]]]],.],.],.] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
[1,1,1,1,0,0,0,1,0,0,1,0,1,0] => [[[.,[[.,[.,[.,.]]],.]],.],.] => [.,[[.,[[.,[.,[.,.]]],.]],.]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
[1,1,1,1,0,0,0,1,0,0,1,1,0,0] => [[.,[[.,[.,[.,.]]],.]],[.,.]] => [[[.,[[.,[.,[.,.]]],.]],.],.] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
[1,1,1,1,0,0,0,1,0,1,0,0,1,0] => [[.,[[[.,[.,[.,.]]],.],.]],.] => [.,[.,[[[.,[.,[.,.]]],.],.]]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
[1,1,1,1,0,0,0,1,0,1,0,1,0,0] => [.,[[[[.,[.,[.,.]]],.],.],.]] => [[.,[[[.,[.,[.,.]]],.],.]],.] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
[1,1,1,1,0,0,1,0,0,0,1,0,1,0] => [[[.,[.,[[.,[.,.]],.]]],.],.] => [.,[[.,[.,[[.,[.,.]],.]]],.]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
[1,1,1,1,0,0,1,0,0,0,1,1,0,0] => [[.,[.,[[.,[.,.]],.]]],[.,.]] => [[[.,[.,[[.,[.,.]],.]]],.],.] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
[1,1,1,1,0,0,1,0,0,1,0,0,1,0] => [[.,[[.,[[.,[.,.]],.]],.]],.] => [.,[.,[[.,[[.,[.,.]],.]],.]]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
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Description
The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice.
Map
to poset
Description
Return the poset obtained by interpreting the tree as a Hasse diagram.
Map
left rotate
Description
Return the result of left rotation applied to a binary tree.
Left rotation on binary trees is defined as follows: Let $T$ be a binary tree such that the right child of the root of $T$ is a node. Let $A$ be the left child of the root of $T$, and let $B$ and $C$ be the left and right children of the right child of the root of $T$. (Keep in mind that nodes of trees are identified with the subtrees consisting of their descendants.) Then, the left rotation of $T$ is the binary tree in which the right child of the root is $C$, whereas the left child of the root is a node whose left and right children are $A$ and $B$.
Left rotation on binary trees is defined as follows: Let $T$ be a binary tree such that the right child of the root of $T$ is a node. Let $A$ be the left child of the root of $T$, and let $B$ and $C$ be the left and right children of the right child of the root of $T$. (Keep in mind that nodes of trees are identified with the subtrees consisting of their descendants.) Then, the left rotation of $T$ is the binary tree in which the right child of the root is $C$, whereas the left child of the root is a node whose left and right children are $A$ and $B$.
Map
to binary tree: left tree, up step, right tree, down step
Description
Return the binary tree corresponding to the Dyck path under the transformation left tree - up step - right tree - down step.
A Dyck path $D$ of semilength $n$ with $n > 1$ may be uniquely decomposed into $L 1 R 0$ for Dyck paths $L,R$ of respective semilengths $n_1,n_2$ with $n_1+n_2 = n-1$.
This map sends $D$ to the binary tree $T$ consisting of a root node with a left child according to $L$ and a right child according to $R$ and then recursively proceeds.
The base case of the unique Dyck path of semilength $1$ is sent to a single node.
This map may also be described as the unique map sending the Tamari orders on Dyck paths to the Tamari order on binary trees.
A Dyck path $D$ of semilength $n$ with $n > 1$ may be uniquely decomposed into $L 1 R 0$ for Dyck paths $L,R$ of respective semilengths $n_1,n_2$ with $n_1+n_2 = n-1$.
This map sends $D$ to the binary tree $T$ consisting of a root node with a left child according to $L$ and a right child according to $R$ and then recursively proceeds.
The base case of the unique Dyck path of semilength $1$ is sent to a single node.
This map may also be described as the unique map sending the Tamari orders on Dyck paths to the Tamari order on binary trees.
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