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Identifier
Values
[.,[.,.]] => 1
[[.,.],.] => 2
[.,[.,[.,.]]] => 1
[.,[[.,.],.]] => 1
[[.,.],[.,.]] => 2
[[.,[.,.]],.] => 2
[[[.,.],.],.] => 3
[.,[.,[.,[.,.]]]] => 1
[.,[.,[[.,.],.]]] => 1
[.,[[.,.],[.,.]]] => 1
[.,[[.,[.,.]],.]] => 1
[.,[[[.,.],.],.]] => 1
[[.,.],[.,[.,.]]] => 2
[[.,.],[[.,.],.]] => 2
[[.,[.,.]],[.,.]] => 2
[[[.,.],.],[.,.]] => 3
[[.,[.,[.,.]]],.] => 2
[[.,[[.,.],.]],.] => 2
[[[.,.],[.,.]],.] => 3
[[[.,[.,.]],.],.] => 3
[[[[.,.],.],.],.] => 4
[.,[.,[.,[.,[.,.]]]]] => 1
[.,[.,[.,[[.,.],.]]]] => 1
[.,[.,[[.,.],[.,.]]]] => 1
[.,[.,[[.,[.,.]],.]]] => 1
[.,[.,[[[.,.],.],.]]] => 1
[.,[[.,.],[.,[.,.]]]] => 1
[.,[[.,.],[[.,.],.]]] => 1
[.,[[.,[.,.]],[.,.]]] => 1
[.,[[[.,.],.],[.,.]]] => 1
[.,[[.,[.,[.,.]]],.]] => 1
[.,[[.,[[.,.],.]],.]] => 1
[.,[[[.,.],[.,.]],.]] => 1
[.,[[[.,[.,.]],.],.]] => 1
[.,[[[[.,.],.],.],.]] => 1
[[.,.],[.,[.,[.,.]]]] => 2
[[.,.],[.,[[.,.],.]]] => 2
[[.,.],[[.,.],[.,.]]] => 2
[[.,.],[[.,[.,.]],.]] => 2
[[.,.],[[[.,.],.],.]] => 2
[[.,[.,.]],[.,[.,.]]] => 2
[[.,[.,.]],[[.,.],.]] => 2
[[[.,.],.],[.,[.,.]]] => 3
[[[.,.],.],[[.,.],.]] => 3
[[.,[.,[.,.]]],[.,.]] => 2
[[.,[[.,.],.]],[.,.]] => 2
[[[.,.],[.,.]],[.,.]] => 3
[[[.,[.,.]],.],[.,.]] => 3
[[[[.,.],.],.],[.,.]] => 4
[[.,[.,[.,[.,.]]]],.] => 2
[[.,[.,[[.,.],.]]],.] => 2
[[.,[[.,.],[.,.]]],.] => 2
[[.,[[.,[.,.]],.]],.] => 2
[[.,[[[.,.],.],.]],.] => 2
[[[.,.],[.,[.,.]]],.] => 3
[[[.,.],[[.,.],.]],.] => 3
[[[.,[.,.]],[.,.]],.] => 3
[[[[.,.],.],[.,.]],.] => 4
[[[.,[.,[.,.]]],.],.] => 3
[[[.,[[.,.],.]],.],.] => 3
[[[[.,.],[.,.]],.],.] => 4
[[[[.,[.,.]],.],.],.] => 4
[[[[[.,.],.],.],.],.] => 5
[.,[.,[.,[.,[.,[.,.]]]]]] => 1
[.,[.,[.,[.,[[.,.],.]]]]] => 1
[.,[.,[.,[[.,.],[.,.]]]]] => 1
[.,[.,[.,[[.,[.,.]],.]]]] => 1
[.,[.,[.,[[[.,.],.],.]]]] => 1
[.,[.,[[.,.],[.,[.,.]]]]] => 1
[.,[.,[[.,.],[[.,.],.]]]] => 1
[.,[.,[[.,[.,.]],[.,.]]]] => 1
[.,[.,[[[.,.],.],[.,.]]]] => 1
[.,[.,[[.,[.,[.,.]]],.]]] => 1
[.,[.,[[.,[[.,.],.]],.]]] => 1
[.,[.,[[[.,.],[.,.]],.]]] => 1
[.,[.,[[[.,[.,.]],.],.]]] => 1
[.,[.,[[[[.,.],.],.],.]]] => 1
[.,[[.,.],[.,[.,[.,.]]]]] => 1
[.,[[.,.],[.,[[.,.],.]]]] => 1
[.,[[.,.],[[.,.],[.,.]]]] => 1
[.,[[.,.],[[.,[.,.]],.]]] => 1
[.,[[.,.],[[[.,.],.],.]]] => 1
[.,[[.,[.,.]],[.,[.,.]]]] => 1
[.,[[.,[.,.]],[[.,.],.]]] => 1
[.,[[[.,.],.],[.,[.,.]]]] => 1
[.,[[[.,.],.],[[.,.],.]]] => 1
[.,[[.,[.,[.,.]]],[.,.]]] => 1
[.,[[.,[[.,.],.]],[.,.]]] => 1
[.,[[[.,.],[.,.]],[.,.]]] => 1
[.,[[[.,[.,.]],.],[.,.]]] => 1
[.,[[[[.,.],.],.],[.,.]]] => 1
[.,[[.,[.,[.,[.,.]]]],.]] => 1
[.,[[.,[.,[[.,.],.]]],.]] => 1
[.,[[.,[[.,.],[.,.]]],.]] => 1
[.,[[.,[[.,[.,.]],.]],.]] => 1
[.,[[.,[[[.,.],.],.]],.]] => 1
[.,[[[.,.],[.,[.,.]]],.]] => 1
[.,[[[.,.],[[.,.],.]],.]] => 1
[.,[[[.,[.,.]],[.,.]],.]] => 1
[.,[[[[.,.],.],[.,.]],.]] => 1
[.,[[[.,[.,[.,.]]],.],.]] => 1
>>> Load all 195 entries. <<<
[.,[[[.,[[.,.],.]],.],.]] => 1
[.,[[[[.,.],[.,.]],.],.]] => 1
[.,[[[[.,[.,.]],.],.],.]] => 1
[.,[[[[[.,.],.],.],.],.]] => 1
[[.,.],[.,[.,[.,[.,.]]]]] => 2
[[.,.],[.,[.,[[.,.],.]]]] => 2
[[.,.],[.,[[.,.],[.,.]]]] => 2
[[.,.],[.,[[.,[.,.]],.]]] => 2
[[.,.],[.,[[[.,.],.],.]]] => 2
[[.,.],[[.,.],[.,[.,.]]]] => 2
[[.,.],[[.,.],[[.,.],.]]] => 2
[[.,.],[[.,[.,.]],[.,.]]] => 2
[[.,.],[[[.,.],.],[.,.]]] => 2
[[.,.],[[.,[.,[.,.]]],.]] => 2
[[.,.],[[.,[[.,.],.]],.]] => 2
[[.,.],[[[.,.],[.,.]],.]] => 2
[[.,.],[[[.,[.,.]],.],.]] => 2
[[.,.],[[[[.,.],.],.],.]] => 2
[[.,[.,.]],[.,[.,[.,.]]]] => 2
[[.,[.,.]],[.,[[.,.],.]]] => 2
[[.,[.,.]],[[.,.],[.,.]]] => 2
[[.,[.,.]],[[.,[.,.]],.]] => 2
[[.,[.,.]],[[[.,.],.],.]] => 2
[[[.,.],.],[.,[.,[.,.]]]] => 3
[[[.,.],.],[.,[[.,.],.]]] => 3
[[[.,.],.],[[.,.],[.,.]]] => 3
[[[.,.],.],[[.,[.,.]],.]] => 3
[[[.,.],.],[[[.,.],.],.]] => 3
[[.,[.,[.,.]]],[.,[.,.]]] => 2
[[.,[.,[.,.]]],[[.,.],.]] => 2
[[.,[[.,.],.]],[.,[.,.]]] => 2
[[.,[[.,.],.]],[[.,.],.]] => 2
[[[.,.],[.,.]],[.,[.,.]]] => 3
[[[.,.],[.,.]],[[.,.],.]] => 3
[[[.,[.,.]],.],[.,[.,.]]] => 3
[[[.,[.,.]],.],[[.,.],.]] => 3
[[[[.,.],.],.],[.,[.,.]]] => 4
[[[[.,.],.],.],[[.,.],.]] => 4
[[.,[.,[.,[.,.]]]],[.,.]] => 2
[[.,[.,[[.,.],.]]],[.,.]] => 2
[[.,[[.,.],[.,.]]],[.,.]] => 2
[[.,[[.,[.,.]],.]],[.,.]] => 2
[[.,[[[.,.],.],.]],[.,.]] => 2
[[[.,.],[.,[.,.]]],[.,.]] => 3
[[[.,.],[[.,.],.]],[.,.]] => 3
[[[.,[.,.]],[.,.]],[.,.]] => 3
[[[[.,.],.],[.,.]],[.,.]] => 4
[[[.,[.,[.,.]]],.],[.,.]] => 3
[[[.,[[.,.],.]],.],[.,.]] => 3
[[[[.,.],[.,.]],.],[.,.]] => 4
[[[[.,[.,.]],.],.],[.,.]] => 4
[[[[[.,.],.],.],.],[.,.]] => 5
[[.,[.,[.,[.,[.,.]]]]],.] => 2
[[.,[.,[.,[[.,.],.]]]],.] => 2
[[.,[.,[[.,.],[.,.]]]],.] => 2
[[.,[.,[[.,[.,.]],.]]],.] => 2
[[.,[.,[[[.,.],.],.]]],.] => 2
[[.,[[.,.],[.,[.,.]]]],.] => 2
[[.,[[.,.],[[.,.],.]]],.] => 2
[[.,[[.,[.,.]],[.,.]]],.] => 2
[[.,[[[.,.],.],[.,.]]],.] => 2
[[.,[[.,[.,[.,.]]],.]],.] => 2
[[.,[[.,[[.,.],.]],.]],.] => 2
[[.,[[[.,.],[.,.]],.]],.] => 2
[[.,[[[.,[.,.]],.],.]],.] => 2
[[.,[[[[.,.],.],.],.]],.] => 2
[[[.,.],[.,[.,[.,.]]]],.] => 3
[[[.,.],[.,[[.,.],.]]],.] => 3
[[[.,.],[[.,.],[.,.]]],.] => 3
[[[.,.],[[.,[.,.]],.]],.] => 3
[[[.,.],[[[.,.],.],.]],.] => 3
[[[.,[.,.]],[.,[.,.]]],.] => 3
[[[.,[.,.]],[[.,.],.]],.] => 3
[[[[.,.],.],[.,[.,.]]],.] => 4
[[[[.,.],.],[[.,.],.]],.] => 4
[[[.,[.,[.,.]]],[.,.]],.] => 3
[[[.,[[.,.],.]],[.,.]],.] => 3
[[[[.,.],[.,.]],[.,.]],.] => 4
[[[[.,[.,.]],.],[.,.]],.] => 4
[[[[[.,.],.],.],[.,.]],.] => 5
[[[.,[.,[.,[.,.]]]],.],.] => 3
[[[.,[.,[[.,.],.]]],.],.] => 3
[[[.,[[.,.],[.,.]]],.],.] => 3
[[[.,[[.,[.,.]],.]],.],.] => 3
[[[.,[[[.,.],.],.]],.],.] => 3
[[[[.,.],[.,[.,.]]],.],.] => 4
[[[[.,.],[[.,.],.]],.],.] => 4
[[[[.,[.,.]],[.,.]],.],.] => 4
[[[[[.,.],.],[.,.]],.],.] => 5
[[[[.,[.,[.,.]]],.],.],.] => 4
[[[[.,[[.,.],.]],.],.],.] => 4
[[[[[.,.],[.,.]],.],.],.] => 5
[[[[[.,[.,.]],.],.],.],.] => 5
[[[[[[.,.],.],.],.],.],.] => 6
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Description
The number of nodes on the left branch of a binary tree.
Also corresponds to ST000011 after applying the Tamari bijection between binary trees and Dyck path.
References
[1] Chapoton, F. Sur le nombre d'intervalles dans les treillis de Tamari MathSciNet:2264942
Code
def statistic(tree):
    if(not tree):
        return 0
    else: return 1 + statistic(tree[0])
Created
Apr 30, 2013 at 11:04 by Viviane Pons
Updated
Oct 17, 2015 at 10:47 by Christian Stump