Identifier
-
Mp00199:
Dyck paths
—prime Dyck path⟶
Dyck paths
Mp00034: Dyck paths —to binary tree: up step, left tree, down step, right tree⟶ Binary trees
Mp00011: Binary trees —to graph⟶ Graphs
St001692: Graphs ⟶ ℤ
Values
[1,0] => [1,1,0,0] => [[.,.],.] => ([(0,1)],2) => 0
[1,0,1,0] => [1,1,0,1,0,0] => [[.,[.,.]],.] => ([(0,2),(1,2)],3) => 1
[1,1,0,0] => [1,1,1,0,0,0] => [[[.,.],.],.] => ([(0,2),(1,2)],3) => 1
[1,0,1,0,1,0] => [1,1,0,1,0,1,0,0] => [[.,[.,[.,.]]],.] => ([(0,3),(1,2),(2,3)],4) => 2
[1,0,1,1,0,0] => [1,1,0,1,1,0,0,0] => [[.,[[.,.],.]],.] => ([(0,3),(1,2),(2,3)],4) => 2
[1,1,0,0,1,0] => [1,1,1,0,0,1,0,0] => [[[.,.],[.,.]],.] => ([(0,3),(1,3),(2,3)],4) => 1
[1,1,0,1,0,0] => [1,1,1,0,1,0,0,0] => [[[.,[.,.]],.],.] => ([(0,3),(1,2),(2,3)],4) => 2
[1,1,1,0,0,0] => [1,1,1,1,0,0,0,0] => [[[[.,.],.],.],.] => ([(0,3),(1,2),(2,3)],4) => 2
[1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,0] => [[.,[.,[.,[.,.]]]],.] => ([(0,4),(1,3),(2,3),(2,4)],5) => 3
[1,0,1,0,1,1,0,0] => [1,1,0,1,0,1,1,0,0,0] => [[.,[.,[[.,.],.]]],.] => ([(0,4),(1,3),(2,3),(2,4)],5) => 3
[1,0,1,1,0,0,1,0] => [1,1,0,1,1,0,0,1,0,0] => [[.,[[.,.],[.,.]]],.] => ([(0,4),(1,4),(2,3),(3,4)],5) => 2
[1,0,1,1,0,1,0,0] => [1,1,0,1,1,0,1,0,0,0] => [[.,[[.,[.,.]],.]],.] => ([(0,4),(1,3),(2,3),(2,4)],5) => 3
[1,0,1,1,1,0,0,0] => [1,1,0,1,1,1,0,0,0,0] => [[.,[[[.,.],.],.]],.] => ([(0,4),(1,3),(2,3),(2,4)],5) => 3
[1,1,0,0,1,0,1,0] => [1,1,1,0,0,1,0,1,0,0] => [[[.,.],[.,[.,.]]],.] => ([(0,4),(1,4),(2,3),(3,4)],5) => 2
[1,1,0,0,1,1,0,0] => [1,1,1,0,0,1,1,0,0,0] => [[[.,.],[[.,.],.]],.] => ([(0,4),(1,4),(2,3),(3,4)],5) => 2
[1,1,0,1,0,0,1,0] => [1,1,1,0,1,0,0,1,0,0] => [[[.,[.,.]],[.,.]],.] => ([(0,4),(1,4),(2,3),(3,4)],5) => 2
[1,1,0,1,0,1,0,0] => [1,1,1,0,1,0,1,0,0,0] => [[[.,[.,[.,.]]],.],.] => ([(0,4),(1,3),(2,3),(2,4)],5) => 3
[1,1,0,1,1,0,0,0] => [1,1,1,0,1,1,0,0,0,0] => [[[.,[[.,.],.]],.],.] => ([(0,4),(1,3),(2,3),(2,4)],5) => 3
[1,1,1,0,0,0,1,0] => [1,1,1,1,0,0,0,1,0,0] => [[[[.,.],.],[.,.]],.] => ([(0,4),(1,4),(2,3),(3,4)],5) => 2
[1,1,1,0,0,1,0,0] => [1,1,1,1,0,0,1,0,0,0] => [[[[.,.],[.,.]],.],.] => ([(0,4),(1,4),(2,3),(3,4)],5) => 2
[1,1,1,0,1,0,0,0] => [1,1,1,1,0,1,0,0,0,0] => [[[[.,[.,.]],.],.],.] => ([(0,4),(1,3),(2,3),(2,4)],5) => 3
[1,1,1,1,0,0,0,0] => [1,1,1,1,1,0,0,0,0,0] => [[[[[.,.],.],.],.],.] => ([(0,4),(1,3),(2,3),(2,4)],5) => 3
[1,0,1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,1,0,0] => [[.,[.,[.,[.,[.,.]]]]],.] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 4
[1,0,1,0,1,0,1,1,0,0] => [1,1,0,1,0,1,0,1,1,0,0,0] => [[.,[.,[.,[[.,.],.]]]],.] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 4
[1,0,1,0,1,1,0,0,1,0] => [1,1,0,1,0,1,1,0,0,1,0,0] => [[.,[.,[[.,.],[.,.]]]],.] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => 3
[1,0,1,0,1,1,0,1,0,0] => [1,1,0,1,0,1,1,0,1,0,0,0] => [[.,[.,[[.,[.,.]],.]]],.] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 4
[1,0,1,0,1,1,1,0,0,0] => [1,1,0,1,0,1,1,1,0,0,0,0] => [[.,[.,[[[.,.],.],.]]],.] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 4
[1,0,1,1,0,0,1,0,1,0] => [1,1,0,1,1,0,0,1,0,1,0,0] => [[.,[[.,.],[.,[.,.]]]],.] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => 3
[1,0,1,1,0,0,1,1,0,0] => [1,1,0,1,1,0,0,1,1,0,0,0] => [[.,[[.,.],[[.,.],.]]],.] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => 3
[1,0,1,1,0,1,0,0,1,0] => [1,1,0,1,1,0,1,0,0,1,0,0] => [[.,[[.,[.,.]],[.,.]]],.] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => 3
[1,0,1,1,0,1,0,1,0,0] => [1,1,0,1,1,0,1,0,1,0,0,0] => [[.,[[.,[.,[.,.]]],.]],.] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 4
[1,0,1,1,0,1,1,0,0,0] => [1,1,0,1,1,0,1,1,0,0,0,0] => [[.,[[.,[[.,.],.]],.]],.] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 4
[1,0,1,1,1,0,0,0,1,0] => [1,1,0,1,1,1,0,0,0,1,0,0] => [[.,[[[.,.],.],[.,.]]],.] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => 3
[1,0,1,1,1,0,0,1,0,0] => [1,1,0,1,1,1,0,0,1,0,0,0] => [[.,[[[.,.],[.,.]],.]],.] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => 3
[1,0,1,1,1,0,1,0,0,0] => [1,1,0,1,1,1,0,1,0,0,0,0] => [[.,[[[.,[.,.]],.],.]],.] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 4
[1,0,1,1,1,1,0,0,0,0] => [1,1,0,1,1,1,1,0,0,0,0,0] => [[.,[[[[.,.],.],.],.]],.] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 4
[1,1,0,0,1,0,1,0,1,0] => [1,1,1,0,0,1,0,1,0,1,0,0] => [[[.,.],[.,[.,[.,.]]]],.] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => 3
[1,1,0,0,1,0,1,1,0,0] => [1,1,1,0,0,1,0,1,1,0,0,0] => [[[.,.],[.,[[.,.],.]]],.] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => 3
[1,1,0,0,1,1,0,0,1,0] => [1,1,1,0,0,1,1,0,0,1,0,0] => [[[.,.],[[.,.],[.,.]]],.] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => 2
[1,1,0,0,1,1,0,1,0,0] => [1,1,1,0,0,1,1,0,1,0,0,0] => [[[.,.],[[.,[.,.]],.]],.] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => 3
[1,1,0,0,1,1,1,0,0,0] => [1,1,1,0,0,1,1,1,0,0,0,0] => [[[.,.],[[[.,.],.],.]],.] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => 3
[1,1,0,1,0,0,1,0,1,0] => [1,1,1,0,1,0,0,1,0,1,0,0] => [[[.,[.,.]],[.,[.,.]]],.] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => 3
[1,1,0,1,0,0,1,1,0,0] => [1,1,1,0,1,0,0,1,1,0,0,0] => [[[.,[.,.]],[[.,.],.]],.] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => 3
[1,1,0,1,0,1,0,0,1,0] => [1,1,1,0,1,0,1,0,0,1,0,0] => [[[.,[.,[.,.]]],[.,.]],.] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => 3
[1,1,0,1,0,1,0,1,0,0] => [1,1,1,0,1,0,1,0,1,0,0,0] => [[[.,[.,[.,[.,.]]]],.],.] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 4
[1,1,0,1,0,1,1,0,0,0] => [1,1,1,0,1,0,1,1,0,0,0,0] => [[[.,[.,[[.,.],.]]],.],.] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 4
[1,1,0,1,1,0,0,0,1,0] => [1,1,1,0,1,1,0,0,0,1,0,0] => [[[.,[[.,.],.]],[.,.]],.] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => 3
[1,1,0,1,1,0,0,1,0,0] => [1,1,1,0,1,1,0,0,1,0,0,0] => [[[.,[[.,.],[.,.]]],.],.] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => 3
[1,1,0,1,1,0,1,0,0,0] => [1,1,1,0,1,1,0,1,0,0,0,0] => [[[.,[[.,[.,.]],.]],.],.] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 4
[1,1,0,1,1,1,0,0,0,0] => [1,1,1,0,1,1,1,0,0,0,0,0] => [[[.,[[[.,.],.],.]],.],.] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 4
[1,1,1,0,0,0,1,0,1,0] => [1,1,1,1,0,0,0,1,0,1,0,0] => [[[[.,.],.],[.,[.,.]]],.] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => 3
[1,1,1,0,0,0,1,1,0,0] => [1,1,1,1,0,0,0,1,1,0,0,0] => [[[[.,.],.],[[.,.],.]],.] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => 3
[1,1,1,0,0,1,0,0,1,0] => [1,1,1,1,0,0,1,0,0,1,0,0] => [[[[.,.],[.,.]],[.,.]],.] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => 2
[1,1,1,0,0,1,0,1,0,0] => [1,1,1,1,0,0,1,0,1,0,0,0] => [[[[.,.],[.,[.,.]]],.],.] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => 3
[1,1,1,0,0,1,1,0,0,0] => [1,1,1,1,0,0,1,1,0,0,0,0] => [[[[.,.],[[.,.],.]],.],.] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => 3
[1,1,1,0,1,0,0,0,1,0] => [1,1,1,1,0,1,0,0,0,1,0,0] => [[[[.,[.,.]],.],[.,.]],.] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => 3
[1,1,1,0,1,0,0,1,0,0] => [1,1,1,1,0,1,0,0,1,0,0,0] => [[[[.,[.,.]],[.,.]],.],.] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => 3
[1,1,1,0,1,0,1,0,0,0] => [1,1,1,1,0,1,0,1,0,0,0,0] => [[[[.,[.,[.,.]]],.],.],.] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 4
[1,1,1,0,1,1,0,0,0,0] => [1,1,1,1,0,1,1,0,0,0,0,0] => [[[[.,[[.,.],.]],.],.],.] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 4
[1,1,1,1,0,0,0,0,1,0] => [1,1,1,1,1,0,0,0,0,1,0,0] => [[[[[.,.],.],.],[.,.]],.] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => 3
[1,1,1,1,0,0,0,1,0,0] => [1,1,1,1,1,0,0,0,1,0,0,0] => [[[[[.,.],.],[.,.]],.],.] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => 3
[1,1,1,1,0,0,1,0,0,0] => [1,1,1,1,1,0,0,1,0,0,0,0] => [[[[[.,.],[.,.]],.],.],.] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => 3
[1,1,1,1,0,1,0,0,0,0] => [1,1,1,1,1,0,1,0,0,0,0,0] => [[[[[.,[.,.]],.],.],.],.] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 4
[1,1,1,1,1,0,0,0,0,0] => [1,1,1,1,1,1,0,0,0,0,0,0] => [[[[[[.,.],.],.],.],.],.] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 4
[1,0,1,0,1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,1,0,1,0,0] => [[.,[.,[.,[.,[.,[.,.]]]]]],.] => ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7) => 5
[1,0,1,0,1,0,1,0,1,1,0,0] => [1,1,0,1,0,1,0,1,0,1,1,0,0,0] => [[.,[.,[.,[.,[[.,.],.]]]]],.] => ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7) => 5
[1,0,1,0,1,0,1,1,0,0,1,0] => [1,1,0,1,0,1,0,1,1,0,0,1,0,0] => [[.,[.,[.,[[.,.],[.,.]]]]],.] => ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7) => 4
[1,0,1,0,1,0,1,1,0,1,0,0] => [1,1,0,1,0,1,0,1,1,0,1,0,0,0] => [[.,[.,[.,[[.,[.,.]],.]]]],.] => ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7) => 5
[1,0,1,0,1,0,1,1,1,0,0,0] => [1,1,0,1,0,1,0,1,1,1,0,0,0,0] => [[.,[.,[.,[[[.,.],.],.]]]],.] => ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7) => 5
[1,0,1,0,1,1,0,0,1,0,1,0] => [1,1,0,1,0,1,1,0,0,1,0,1,0,0] => [[.,[.,[[.,.],[.,[.,.]]]]],.] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7) => 4
[1,0,1,0,1,1,0,0,1,1,0,0] => [1,1,0,1,0,1,1,0,0,1,1,0,0,0] => [[.,[.,[[.,.],[[.,.],.]]]],.] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7) => 4
[1,0,1,0,1,1,0,1,0,0,1,0] => [1,1,0,1,0,1,1,0,1,0,0,1,0,0] => [[.,[.,[[.,[.,.]],[.,.]]]],.] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7) => 4
[1,0,1,0,1,1,0,1,0,1,0,0] => [1,1,0,1,0,1,1,0,1,0,1,0,0,0] => [[.,[.,[[.,[.,[.,.]]],.]]],.] => ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7) => 5
[1,0,1,0,1,1,0,1,1,0,0,0] => [1,1,0,1,0,1,1,0,1,1,0,0,0,0] => [[.,[.,[[.,[[.,.],.]],.]]],.] => ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7) => 5
[1,0,1,0,1,1,1,0,0,0,1,0] => [1,1,0,1,0,1,1,1,0,0,0,1,0,0] => [[.,[.,[[[.,.],.],[.,.]]]],.] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7) => 4
[1,0,1,0,1,1,1,0,0,1,0,0] => [1,1,0,1,0,1,1,1,0,0,1,0,0,0] => [[.,[.,[[[.,.],[.,.]],.]]],.] => ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7) => 4
[1,0,1,0,1,1,1,0,1,0,0,0] => [1,1,0,1,0,1,1,1,0,1,0,0,0,0] => [[.,[.,[[[.,[.,.]],.],.]]],.] => ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7) => 5
[1,0,1,0,1,1,1,1,0,0,0,0] => [1,1,0,1,0,1,1,1,1,0,0,0,0,0] => [[.,[.,[[[[.,.],.],.],.]]],.] => ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7) => 5
[1,0,1,1,0,0,1,0,1,0,1,0] => [1,1,0,1,1,0,0,1,0,1,0,1,0,0] => [[.,[[.,.],[.,[.,[.,.]]]]],.] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7) => 4
[1,0,1,1,0,0,1,0,1,1,0,0] => [1,1,0,1,1,0,0,1,0,1,1,0,0,0] => [[.,[[.,.],[.,[[.,.],.]]]],.] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7) => 4
[1,0,1,1,0,0,1,1,0,0,1,0] => [1,1,0,1,1,0,0,1,1,0,0,1,0,0] => [[.,[[.,.],[[.,.],[.,.]]]],.] => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7) => 3
[1,0,1,1,0,0,1,1,0,1,0,0] => [1,1,0,1,1,0,0,1,1,0,1,0,0,0] => [[.,[[.,.],[[.,[.,.]],.]]],.] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7) => 4
[1,0,1,1,0,0,1,1,1,0,0,0] => [1,1,0,1,1,0,0,1,1,1,0,0,0,0] => [[.,[[.,.],[[[.,.],.],.]]],.] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7) => 4
[1,0,1,1,0,1,0,0,1,0,1,0] => [1,1,0,1,1,0,1,0,0,1,0,1,0,0] => [[.,[[.,[.,.]],[.,[.,.]]]],.] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 4
[1,0,1,1,0,1,0,0,1,1,0,0] => [1,1,0,1,1,0,1,0,0,1,1,0,0,0] => [[.,[[.,[.,.]],[[.,.],.]]],.] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 4
[1,0,1,1,0,1,0,1,0,0,1,0] => [1,1,0,1,1,0,1,0,1,0,0,1,0,0] => [[.,[[.,[.,[.,.]]],[.,.]]],.] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7) => 4
[1,0,1,1,0,1,0,1,0,1,0,0] => [1,1,0,1,1,0,1,0,1,0,1,0,0,0] => [[.,[[.,[.,[.,[.,.]]]],.]],.] => ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7) => 5
[1,0,1,1,0,1,0,1,1,0,0,0] => [1,1,0,1,1,0,1,0,1,1,0,0,0,0] => [[.,[[.,[.,[[.,.],.]]],.]],.] => ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7) => 5
[1,0,1,1,0,1,1,0,0,0,1,0] => [1,1,0,1,1,0,1,1,0,0,0,1,0,0] => [[.,[[.,[[.,.],.]],[.,.]]],.] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7) => 4
[1,0,1,1,0,1,1,0,0,1,0,0] => [1,1,0,1,1,0,1,1,0,0,1,0,0,0] => [[.,[[.,[[.,.],[.,.]]],.]],.] => ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7) => 4
[1,0,1,1,0,1,1,0,1,0,0,0] => [1,1,0,1,1,0,1,1,0,1,0,0,0,0] => [[.,[[.,[[.,[.,.]],.]],.]],.] => ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7) => 5
[1,0,1,1,0,1,1,1,0,0,0,0] => [1,1,0,1,1,0,1,1,1,0,0,0,0,0] => [[.,[[.,[[[.,.],.],.]],.]],.] => ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7) => 5
[1,0,1,1,1,0,0,0,1,0,1,0] => [1,1,0,1,1,1,0,0,0,1,0,1,0,0] => [[.,[[[.,.],.],[.,[.,.]]]],.] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 4
[1,0,1,1,1,0,0,0,1,1,0,0] => [1,1,0,1,1,1,0,0,0,1,1,0,0,0] => [[.,[[[.,.],.],[[.,.],.]]],.] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 4
[1,0,1,1,1,0,0,1,0,0,1,0] => [1,1,0,1,1,1,0,0,1,0,0,1,0,0] => [[.,[[[.,.],[.,.]],[.,.]]],.] => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7) => 3
[1,0,1,1,1,0,0,1,0,1,0,0] => [1,1,0,1,1,1,0,0,1,0,1,0,0,0] => [[.,[[[.,.],[.,[.,.]]],.]],.] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7) => 4
[1,0,1,1,1,0,0,1,1,0,0,0] => [1,1,0,1,1,1,0,0,1,1,0,0,0,0] => [[.,[[[.,.],[[.,.],.]],.]],.] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7) => 4
[1,0,1,1,1,0,1,0,0,0,1,0] => [1,1,0,1,1,1,0,1,0,0,0,1,0,0] => [[.,[[[.,[.,.]],.],[.,.]]],.] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7) => 4
[1,0,1,1,1,0,1,0,0,1,0,0] => [1,1,0,1,1,1,0,1,0,0,1,0,0,0] => [[.,[[[.,[.,.]],[.,.]],.]],.] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7) => 4
[1,0,1,1,1,0,1,0,1,0,0,0] => [1,1,0,1,1,1,0,1,0,1,0,0,0,0] => [[.,[[[.,[.,[.,.]]],.],.]],.] => ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7) => 5
[1,0,1,1,1,0,1,1,0,0,0,0] => [1,1,0,1,1,1,0,1,1,0,0,0,0,0] => [[.,[[[.,[[.,.],.]],.],.]],.] => ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7) => 5
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Description
The number of vertices with higher degree than the average degree in a graph.
Map
to binary tree: up step, left tree, down step, right tree
Description
Return the binary tree corresponding to the Dyck path under the transformation up step - left tree - down step - right tree.
A Dyck path D of semilength n with n>1 may be uniquely decomposed into 1L0R for Dyck paths L,R of respective semilengths n1,n2 with n1+n2=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.
A Dyck path D of semilength n with n>1 may be uniquely decomposed into 1L0R for Dyck paths L,R of respective semilengths n1,n2 with n1+n2=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.
Map
prime Dyck path
Description
Return the Dyck path obtained by adding an initial up and a final down step.
Map
to graph
Description
Return the undirected graph obtained from the tree nodes and edges, with leaves being ignored.
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