edit this statistic or download as text // json
Identifier
Values
[.,.] => 0
[.,[.,.]] => 0
[[.,.],.] => 0
[.,[.,[.,.]]] => 0
[.,[[.,.],.]] => 0
[[.,.],[.,.]] => 0
[[.,[.,.]],.] => 0
[[[.,.],.],.] => 0
[.,[.,[.,[.,.]]]] => 0
[.,[.,[[.,.],.]]] => 0
[.,[[.,.],[.,.]]] => 1
[.,[[.,[.,.]],.]] => 0
[.,[[[.,.],.],.]] => 0
[[.,.],[.,[.,.]]] => 0
[[.,.],[[.,.],.]] => 0
[[.,[.,.]],[.,.]] => 0
[[[.,.],.],[.,.]] => 0
[[.,[.,[.,.]]],.] => 0
[[.,[[.,.],.]],.] => 0
[[[.,.],[.,.]],.] => 1
[[[.,[.,.]],.],.] => 0
[[[[.,.],.],.],.] => 0
[.,[.,[.,[.,[.,.]]]]] => 0
[.,[.,[.,[[.,.],.]]]] => 0
[.,[.,[[.,.],[.,.]]]] => 1
[.,[.,[[.,[.,.]],.]]] => 0
[.,[.,[[[.,.],.],.]]] => 0
[.,[[.,.],[.,[.,.]]]] => 1
[.,[[.,.],[[.,.],.]]] => 1
[.,[[.,[.,.]],[.,.]]] => 1
[.,[[[.,.],.],[.,.]]] => 1
[.,[[.,[.,[.,.]]],.]] => 0
[.,[[.,[[.,.],.]],.]] => 0
[.,[[[.,.],[.,.]],.]] => 1
[.,[[[.,[.,.]],.],.]] => 0
[.,[[[[.,.],.],.],.]] => 0
[[.,.],[.,[.,[.,.]]]] => 0
[[.,.],[.,[[.,.],.]]] => 0
[[.,.],[[.,.],[.,.]]] => 1
[[.,.],[[.,[.,.]],.]] => 0
[[.,.],[[[.,.],.],.]] => 0
[[.,[.,.]],[.,[.,.]]] => 0
[[.,[.,.]],[[.,.],.]] => 0
[[[.,.],.],[.,[.,.]]] => 0
[[[.,.],.],[[.,.],.]] => 0
[[.,[.,[.,.]]],[.,.]] => 0
[[.,[[.,.],.]],[.,.]] => 0
[[[.,.],[.,.]],[.,.]] => 1
[[[.,[.,.]],.],[.,.]] => 0
[[[[.,.],.],.],[.,.]] => 0
[[.,[.,[.,[.,.]]]],.] => 0
[[.,[.,[[.,.],.]]],.] => 0
[[.,[[.,.],[.,.]]],.] => 1
[[.,[[.,[.,.]],.]],.] => 0
[[.,[[[.,.],.],.]],.] => 0
[[[.,.],[.,[.,.]]],.] => 1
[[[.,.],[[.,.],.]],.] => 1
[[[.,[.,.]],[.,.]],.] => 1
[[[[.,.],.],[.,.]],.] => 1
[[[.,[.,[.,.]]],.],.] => 0
[[[.,[[.,.],.]],.],.] => 0
[[[[.,.],[.,.]],.],.] => 1
[[[[.,[.,.]],.],.],.] => 0
[[[[[.,.],.],.],.],.] => 0
[.,[.,[.,[.,[.,[.,.]]]]]] => 0
[.,[.,[.,[.,[[.,.],.]]]]] => 0
[.,[.,[.,[[.,.],[.,.]]]]] => 1
[.,[.,[.,[[.,[.,.]],.]]]] => 0
[.,[.,[.,[[[.,.],.],.]]]] => 0
[.,[.,[[.,.],[.,[.,.]]]]] => 1
[.,[.,[[.,.],[[.,.],.]]]] => 1
[.,[.,[[.,[.,.]],[.,.]]]] => 1
[.,[.,[[[.,.],.],[.,.]]]] => 1
[.,[.,[[.,[.,[.,.]]],.]]] => 0
[.,[.,[[.,[[.,.],.]],.]]] => 0
[.,[.,[[[.,.],[.,.]],.]]] => 1
[.,[.,[[[.,[.,.]],.],.]]] => 0
[.,[.,[[[[.,.],.],.],.]]] => 0
[.,[[.,.],[.,[.,[.,.]]]]] => 1
[.,[[.,.],[.,[[.,.],.]]]] => 1
[.,[[.,.],[[.,.],[.,.]]]] => 2
[.,[[.,.],[[.,[.,.]],.]]] => 1
[.,[[.,.],[[[.,.],.],.]]] => 1
[.,[[.,[.,.]],[.,[.,.]]]] => 1
[.,[[.,[.,.]],[[.,.],.]]] => 1
[.,[[[.,.],.],[.,[.,.]]]] => 1
[.,[[[.,.],.],[[.,.],.]]] => 1
[.,[[.,[.,[.,.]]],[.,.]]] => 1
[.,[[.,[[.,.],.]],[.,.]]] => 1
[.,[[[.,.],[.,.]],[.,.]]] => 2
[.,[[[.,[.,.]],.],[.,.]]] => 1
[.,[[[[.,.],.],.],[.,.]]] => 1
[.,[[.,[.,[.,[.,.]]]],.]] => 0
[.,[[.,[.,[[.,.],.]]],.]] => 0
[.,[[.,[[.,.],[.,.]]],.]] => 1
[.,[[.,[[.,[.,.]],.]],.]] => 0
[.,[[.,[[[.,.],.],.]],.]] => 0
[.,[[[.,.],[.,[.,.]]],.]] => 1
[.,[[[.,.],[[.,.],.]],.]] => 1
[.,[[[.,[.,.]],[.,.]],.]] => 1
[.,[[[[.,.],.],[.,.]],.]] => 1
>>> Load all 196 entries. <<<
[.,[[[.,[.,[.,.]]],.],.]] => 0
[.,[[[.,[[.,.],.]],.],.]] => 0
[.,[[[[.,.],[.,.]],.],.]] => 1
[.,[[[[.,[.,.]],.],.],.]] => 0
[.,[[[[[.,.],.],.],.],.]] => 0
[[.,.],[.,[.,[.,[.,.]]]]] => 0
[[.,.],[.,[.,[[.,.],.]]]] => 0
[[.,.],[.,[[.,.],[.,.]]]] => 1
[[.,.],[.,[[.,[.,.]],.]]] => 0
[[.,.],[.,[[[.,.],.],.]]] => 0
[[.,.],[[.,.],[.,[.,.]]]] => 1
[[.,.],[[.,.],[[.,.],.]]] => 1
[[.,.],[[.,[.,.]],[.,.]]] => 1
[[.,.],[[[.,.],.],[.,.]]] => 1
[[.,.],[[.,[.,[.,.]]],.]] => 0
[[.,.],[[.,[[.,.],.]],.]] => 0
[[.,.],[[[.,.],[.,.]],.]] => 1
[[.,.],[[[.,[.,.]],.],.]] => 0
[[.,.],[[[[.,.],.],.],.]] => 0
[[.,[.,.]],[.,[.,[.,.]]]] => 0
[[.,[.,.]],[.,[[.,.],.]]] => 0
[[.,[.,.]],[[.,.],[.,.]]] => 1
[[.,[.,.]],[[.,[.,.]],.]] => 0
[[.,[.,.]],[[[.,.],.],.]] => 0
[[[.,.],.],[.,[.,[.,.]]]] => 0
[[[.,.],.],[.,[[.,.],.]]] => 0
[[[.,.],.],[[.,.],[.,.]]] => 1
[[[.,.],.],[[.,[.,.]],.]] => 0
[[[.,.],.],[[[.,.],.],.]] => 0
[[.,[.,[.,.]]],[.,[.,.]]] => 0
[[.,[.,[.,.]]],[[.,.],.]] => 0
[[.,[[.,.],.]],[.,[.,.]]] => 0
[[.,[[.,.],.]],[[.,.],.]] => 0
[[[.,.],[.,.]],[.,[.,.]]] => 1
[[[.,.],[.,.]],[[.,.],.]] => 1
[[[.,[.,.]],.],[.,[.,.]]] => 0
[[[.,[.,.]],.],[[.,.],.]] => 0
[[[[.,.],.],.],[.,[.,.]]] => 0
[[[[.,.],.],.],[[.,.],.]] => 0
[[.,[.,[.,[.,.]]]],[.,.]] => 0
[[.,[.,[[.,.],.]]],[.,.]] => 0
[[.,[[.,.],[.,.]]],[.,.]] => 1
[[.,[[.,[.,.]],.]],[.,.]] => 0
[[.,[[[.,.],.],.]],[.,.]] => 0
[[[.,.],[.,[.,.]]],[.,.]] => 1
[[[.,.],[[.,.],.]],[.,.]] => 1
[[[.,[.,.]],[.,.]],[.,.]] => 1
[[[[.,.],.],[.,.]],[.,.]] => 1
[[[.,[.,[.,.]]],.],[.,.]] => 0
[[[.,[[.,.],.]],.],[.,.]] => 0
[[[[.,.],[.,.]],.],[.,.]] => 1
[[[[.,[.,.]],.],.],[.,.]] => 0
[[[[[.,.],.],.],.],[.,.]] => 0
[[.,[.,[.,[.,[.,.]]]]],.] => 0
[[.,[.,[.,[[.,.],.]]]],.] => 0
[[.,[.,[[.,.],[.,.]]]],.] => 1
[[.,[.,[[.,[.,.]],.]]],.] => 0
[[.,[.,[[[.,.],.],.]]],.] => 0
[[.,[[.,.],[.,[.,.]]]],.] => 1
[[.,[[.,.],[[.,.],.]]],.] => 1
[[.,[[.,[.,.]],[.,.]]],.] => 1
[[.,[[[.,.],.],[.,.]]],.] => 1
[[.,[[.,[.,[.,.]]],.]],.] => 0
[[.,[[.,[[.,.],.]],.]],.] => 0
[[.,[[[.,.],[.,.]],.]],.] => 1
[[.,[[[.,[.,.]],.],.]],.] => 0
[[.,[[[[.,.],.],.],.]],.] => 0
[[[.,.],[.,[.,[.,.]]]],.] => 1
[[[.,.],[.,[[.,.],.]]],.] => 1
[[[.,.],[[.,.],[.,.]]],.] => 2
[[[.,.],[[.,[.,.]],.]],.] => 1
[[[.,.],[[[.,.],.],.]],.] => 1
[[[.,[.,.]],[.,[.,.]]],.] => 1
[[[.,[.,.]],[[.,.],.]],.] => 1
[[[[.,.],.],[.,[.,.]]],.] => 1
[[[[.,.],.],[[.,.],.]],.] => 1
[[[.,[.,[.,.]]],[.,.]],.] => 1
[[[.,[[.,.],.]],[.,.]],.] => 1
[[[[.,.],[.,.]],[.,.]],.] => 2
[[[[.,[.,.]],.],[.,.]],.] => 1
[[[[[.,.],.],.],[.,.]],.] => 1
[[[.,[.,[.,[.,.]]]],.],.] => 0
[[[.,[.,[[.,.],.]]],.],.] => 0
[[[.,[[.,.],[.,.]]],.],.] => 1
[[[.,[[.,[.,.]],.]],.],.] => 0
[[[.,[[[.,.],.],.]],.],.] => 0
[[[[.,.],[.,[.,.]]],.],.] => 1
[[[[.,.],[[.,.],.]],.],.] => 1
[[[[.,[.,.]],[.,.]],.],.] => 1
[[[[[.,.],.],[.,.]],.],.] => 1
[[[[.,[.,[.,.]]],.],.],.] => 0
[[[[.,[[.,.],.]],.],.],.] => 0
[[[[[.,.],[.,.]],.],.],.] => 1
[[[[[.,[.,.]],.],.],.],.] => 0
[[[[[[.,.],.],.],.],.],.] => 0
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Description
The number of nodes of degree 3 of a binary tree.
Equivalently, the number of internal triangles in the associated triangulation of an $(n+2)$-gon.
References
[1] Dochtermann, A. Face rings of cycles, associahedra, and standard Young tableaux arXiv:1503.06243
Code
def statistic(T):
    return sum(statistic_aux(C) for C in T)

def statistic_aux(T):
    if T == BinaryTree() or (T[0] == BinaryTree() and T[1] == BinaryTree()):
        return 0
    if T[0] == BinaryTree():
        return statistic_aux(T[1])
    if T[1] == BinaryTree():
        return statistic_aux(T[0])
    return 1 + statistic_aux(T[0]) + statistic_aux(T[1])
Created
May 27, 2015 at 15:00 by Anton Dochtermann
Updated
Aug 05, 2016 at 20:11 by Martin Rubey