edit this statistic or download as text // json
Identifier
Values
[.,[.,[.,.]]] => 1
[.,[[.,.],.]] => 1
[[.,.],[.,.]] => 2
[[.,[.,.]],.] => 1
[[[.,.],.],.] => 1
[.,[.,[.,[.,.]]]] => 1
[.,[.,[[.,.],.]]] => 1
[.,[[.,.],[.,.]]] => 2
[.,[[.,[.,.]],.]] => 1
[.,[[[.,.],.],.]] => 1
[[.,.],[.,[.,.]]] => 3
[[.,.],[[.,.],.]] => 3
[[.,[.,.]],[.,.]] => 3
[[[.,.],.],[.,.]] => 3
[[.,[.,[.,.]]],.] => 1
[[.,[[.,.],.]],.] => 1
[[[.,.],[.,.]],.] => 2
[[[.,[.,.]],.],.] => 1
[[[[.,.],.],.],.] => 1
[.,[.,[.,[.,[.,.]]]]] => 1
[.,[.,[.,[[.,.],.]]]] => 1
[.,[.,[[.,.],[.,.]]]] => 2
[.,[.,[[.,[.,.]],.]]] => 1
[.,[.,[[[.,.],.],.]]] => 1
[.,[[.,.],[.,[.,.]]]] => 3
[.,[[.,.],[[.,.],.]]] => 3
[.,[[.,[.,.]],[.,.]]] => 3
[.,[[[.,.],.],[.,.]]] => 3
[.,[[.,[.,[.,.]]],.]] => 1
[.,[[.,[[.,.],.]],.]] => 1
[.,[[[.,.],[.,.]],.]] => 2
[.,[[[.,[.,.]],.],.]] => 1
[.,[[[[.,.],.],.],.]] => 1
[[.,.],[.,[.,[.,.]]]] => 4
[[.,.],[.,[[.,.],.]]] => 4
[[.,.],[[.,.],[.,.]]] => 8
[[.,.],[[.,[.,.]],.]] => 4
[[.,.],[[[.,.],.],.]] => 4
[[.,[.,.]],[.,[.,.]]] => 6
[[.,[.,.]],[[.,.],.]] => 6
[[[.,.],.],[.,[.,.]]] => 6
[[[.,.],.],[[.,.],.]] => 6
[[.,[.,[.,.]]],[.,.]] => 4
[[.,[[.,.],.]],[.,.]] => 4
[[[.,.],[.,.]],[.,.]] => 8
[[[.,[.,.]],.],[.,.]] => 4
[[[[.,.],.],.],[.,.]] => 4
[[.,[.,[.,[.,.]]]],.] => 1
[[.,[.,[[.,.],.]]],.] => 1
[[.,[[.,.],[.,.]]],.] => 2
[[.,[[.,[.,.]],.]],.] => 1
[[.,[[[.,.],.],.]],.] => 1
[[[.,.],[.,[.,.]]],.] => 3
[[[.,.],[[.,.],.]],.] => 3
[[[.,[.,.]],[.,.]],.] => 3
[[[[.,.],.],[.,.]],.] => 3
[[[.,[.,[.,.]]],.],.] => 1
[[[.,[[.,.],.]],.],.] => 1
[[[[.,.],[.,.]],.],.] => 2
[[[[.,[.,.]],.],.],.] => 1
[[[[[.,.],.],.],.],.] => 1
[.,[.,[.,[.,[.,[.,.]]]]]] => 1
[.,[.,[.,[.,[[.,.],.]]]]] => 1
[.,[.,[.,[[.,.],[.,.]]]]] => 2
[.,[.,[.,[[.,[.,.]],.]]]] => 1
[.,[.,[.,[[[.,.],.],.]]]] => 1
[.,[.,[[.,.],[.,[.,.]]]]] => 3
[.,[.,[[.,.],[[.,.],.]]]] => 3
[.,[.,[[.,[.,.]],[.,.]]]] => 3
[.,[.,[[[.,.],.],[.,.]]]] => 3
[.,[.,[[.,[.,[.,.]]],.]]] => 1
[.,[.,[[.,[[.,.],.]],.]]] => 1
[.,[.,[[[.,.],[.,.]],.]]] => 2
[.,[.,[[[.,[.,.]],.],.]]] => 1
[.,[.,[[[[.,.],.],.],.]]] => 1
[.,[[.,.],[.,[.,[.,.]]]]] => 4
[.,[[.,.],[.,[[.,.],.]]]] => 4
[.,[[.,.],[[.,.],[.,.]]]] => 8
[.,[[.,.],[[.,[.,.]],.]]] => 4
[.,[[.,.],[[[.,.],.],.]]] => 4
[.,[[.,[.,.]],[.,[.,.]]]] => 6
[.,[[.,[.,.]],[[.,.],.]]] => 6
[.,[[[.,.],.],[.,[.,.]]]] => 6
[.,[[[.,.],.],[[.,.],.]]] => 6
[.,[[.,[.,[.,.]]],[.,.]]] => 4
[.,[[.,[[.,.],.]],[.,.]]] => 4
[.,[[[.,.],[.,.]],[.,.]]] => 8
[.,[[[.,[.,.]],.],[.,.]]] => 4
[.,[[[[.,.],.],.],[.,.]]] => 4
[.,[[.,[.,[.,[.,.]]]],.]] => 1
[.,[[.,[.,[[.,.],.]]],.]] => 1
[.,[[.,[[.,.],[.,.]]],.]] => 2
[.,[[.,[[.,[.,.]],.]],.]] => 1
[.,[[.,[[[.,.],.],.]],.]] => 1
[.,[[[.,.],[.,[.,.]]],.]] => 3
[.,[[[.,.],[[.,.],.]],.]] => 3
[.,[[[.,[.,.]],[.,.]],.]] => 3
[.,[[[[.,.],.],[.,.]],.]] => 3
[.,[[[.,[.,[.,.]]],.],.]] => 1
[.,[[[.,[[.,.],.]],.],.]] => 1
[.,[[[[.,.],[.,.]],.],.]] => 2
>>> Load all 193 entries. <<<
[.,[[[[.,[.,.]],.],.],.]] => 1
[.,[[[[[.,.],.],.],.],.]] => 1
[[.,.],[.,[.,[.,[.,.]]]]] => 5
[[.,.],[.,[.,[[.,.],.]]]] => 5
[[.,.],[.,[[.,.],[.,.]]]] => 10
[[.,.],[.,[[.,[.,.]],.]]] => 5
[[.,.],[.,[[[.,.],.],.]]] => 5
[[.,.],[[.,.],[.,[.,.]]]] => 15
[[.,.],[[.,.],[[.,.],.]]] => 15
[[.,.],[[.,[.,.]],[.,.]]] => 15
[[.,.],[[[.,.],.],[.,.]]] => 15
[[.,.],[[.,[.,[.,.]]],.]] => 5
[[.,.],[[.,[[.,.],.]],.]] => 5
[[.,.],[[[.,.],[.,.]],.]] => 10
[[.,.],[[[.,[.,.]],.],.]] => 5
[[.,.],[[[[.,.],.],.],.]] => 5
[[.,[.,.]],[.,[.,[.,.]]]] => 10
[[.,[.,.]],[.,[[.,.],.]]] => 10
[[.,[.,.]],[[.,.],[.,.]]] => 20
[[.,[.,.]],[[.,[.,.]],.]] => 10
[[.,[.,.]],[[[.,.],.],.]] => 10
[[[.,.],.],[.,[.,[.,.]]]] => 10
[[[.,.],.],[.,[[.,.],.]]] => 10
[[[.,.],.],[[.,.],[.,.]]] => 20
[[[.,.],.],[[.,[.,.]],.]] => 10
[[[.,.],.],[[[.,.],.],.]] => 10
[[.,[.,[.,.]]],[.,[.,.]]] => 10
[[.,[.,[.,.]]],[[.,.],.]] => 10
[[.,[[.,.],.]],[.,[.,.]]] => 10
[[.,[[.,.],.]],[[.,.],.]] => 10
[[[.,.],[.,.]],[.,[.,.]]] => 20
[[[.,.],[.,.]],[[.,.],.]] => 20
[[[.,[.,.]],.],[.,[.,.]]] => 10
[[[.,[.,.]],.],[[.,.],.]] => 10
[[[[.,.],.],.],[.,[.,.]]] => 10
[[[[.,.],.],.],[[.,.],.]] => 10
[[.,[.,[.,[.,.]]]],[.,.]] => 5
[[.,[.,[[.,.],.]]],[.,.]] => 5
[[.,[[.,.],[.,.]]],[.,.]] => 10
[[.,[[.,[.,.]],.]],[.,.]] => 5
[[.,[[[.,.],.],.]],[.,.]] => 5
[[[.,.],[.,[.,.]]],[.,.]] => 15
[[[.,.],[[.,.],.]],[.,.]] => 15
[[[.,[.,.]],[.,.]],[.,.]] => 15
[[[[.,.],.],[.,.]],[.,.]] => 15
[[[.,[.,[.,.]]],.],[.,.]] => 5
[[[.,[[.,.],.]],.],[.,.]] => 5
[[[[.,.],[.,.]],.],[.,.]] => 10
[[[[.,[.,.]],.],.],[.,.]] => 5
[[[[[.,.],.],.],.],[.,.]] => 5
[[.,[.,[.,[.,[.,.]]]]],.] => 1
[[.,[.,[.,[[.,.],.]]]],.] => 1
[[.,[.,[[.,.],[.,.]]]],.] => 2
[[.,[.,[[.,[.,.]],.]]],.] => 1
[[.,[.,[[[.,.],.],.]]],.] => 1
[[.,[[.,.],[.,[.,.]]]],.] => 3
[[.,[[.,.],[[.,.],.]]],.] => 3
[[.,[[.,[.,.]],[.,.]]],.] => 3
[[.,[[[.,.],.],[.,.]]],.] => 3
[[.,[[.,[.,[.,.]]],.]],.] => 1
[[.,[[.,[[.,.],.]],.]],.] => 1
[[.,[[[.,.],[.,.]],.]],.] => 2
[[.,[[[.,[.,.]],.],.]],.] => 1
[[.,[[[[.,.],.],.],.]],.] => 1
[[[.,.],[.,[.,[.,.]]]],.] => 4
[[[.,.],[.,[[.,.],.]]],.] => 4
[[[.,.],[[.,.],[.,.]]],.] => 8
[[[.,.],[[.,[.,.]],.]],.] => 4
[[[.,.],[[[.,.],.],.]],.] => 4
[[[.,[.,.]],[.,[.,.]]],.] => 6
[[[.,[.,.]],[[.,.],.]],.] => 6
[[[[.,.],.],[.,[.,.]]],.] => 6
[[[[.,.],.],[[.,.],.]],.] => 6
[[[.,[.,[.,.]]],[.,.]],.] => 4
[[[.,[[.,.],.]],[.,.]],.] => 4
[[[[.,.],[.,.]],[.,.]],.] => 8
[[[[.,[.,.]],.],[.,.]],.] => 4
[[[[[.,.],.],.],[.,.]],.] => 4
[[[.,[.,[.,[.,.]]]],.],.] => 1
[[[.,[.,[[.,.],.]]],.],.] => 1
[[[.,[[.,.],[.,.]]],.],.] => 2
[[[.,[[.,[.,.]],.]],.],.] => 1
[[[.,[[[.,.],.],.]],.],.] => 1
[[[[.,.],[.,[.,.]]],.],.] => 3
[[[[.,.],[[.,.],.]],.],.] => 3
[[[[.,[.,.]],[.,.]],.],.] => 3
[[[[[.,.],.],[.,.]],.],.] => 3
[[[[.,[.,[.,.]]],.],.],.] => 1
[[[[.,[[.,.],.]],.],.],.] => 1
[[[[[.,.],[.,.]],.],.],.] => 2
[[[[[.,[.,.]],.],.],.],.] => 1
[[[[[[.,.],.],.],.],.],.] => 1
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Description
The number of linear extensions of a binary tree.
Also, the number of increasing / decreasing binary trees labelled by $1, \dots, n$ of this shape.
Also, the size of the sylvester class corresponding to this tree when the Tamari order is seen as a quotient poset of the right weak order on permutations.
Also, the number of permutations which give this tree shape when inserted in a binary search tree.
Also, the number of permutations which increasing / decreasing tree is of this shape.
References
[1] Knuth, D. E. The art of computer programming. Volume 3 MathSciNet:0445948
[2] Björner, A., Wachs, M. L. $q$-hook length formulas for forests MathSciNet:1022316
[3] Hivert, F., Novelli, J.-C., Thibon, J.-Y. The algebra of binary search trees MathSciNet:2142078 arXiv:math/0401089
Code
def hook_product(tree):
    if(not tree):
        return 1
    hl = hook_product(tree[0])
    hr = hook_product(tree[1])
    return hl*hr*tree.node_number()
    
def statistic(tree):
    return factorial(tree.node_number()-1)/hook_product(tree[0])/hook_product(tree[1])
Created
Mar 12, 2013 at 19:04 by Viviane Pons
Updated
Oct 17, 2015 at 10:47 by Christian Stump