edit this statistic or download as text // json
Identifier
Values
=>
Cc0010;cc-rep
[.,.]=>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 [.,[[[.,[.,[.,.]]],.],.]]=>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