Identifier
Identifier
Values
[.,[.,[.,.]]] generating graphics... => 1
[.,[[.,.],.]] generating graphics... => 1
[[.,.],[.,.]] generating graphics... => 2
[[.,[.,.]],.] generating graphics... => 1
[[[.,.],.],.] generating graphics... => 1
[.,[.,[.,[.,.]]]] generating graphics... => 1
[.,[.,[[.,.],.]]] generating graphics... => 1
[.,[[.,.],[.,.]]] generating graphics... => 2
[.,[[.,[.,.]],.]] generating graphics... => 1
[.,[[[.,.],.],.]] generating graphics... => 1
[[.,.],[.,[.,.]]] generating graphics... => 3
[[.,.],[[.,.],.]] generating graphics... => 3
[[.,[.,.]],[.,.]] generating graphics... => 3
[[[.,.],.],[.,.]] generating graphics... => 3
[[.,[.,[.,.]]],.] generating graphics... => 1
[[.,[[.,.],.]],.] generating graphics... => 1
[[[.,.],[.,.]],.] generating graphics... => 2
[[[.,[.,.]],.],.] generating graphics... => 1
[[[[.,.],.],.],.] generating graphics... => 1
[.,[.,[.,[.,[.,.]]]]] generating graphics... => 1
[.,[.,[.,[[.,.],.]]]] generating graphics... => 1
[.,[.,[[.,.],[.,.]]]] generating graphics... => 2
[.,[.,[[.,[.,.]],.]]] generating graphics... => 1
[.,[.,[[[.,.],.],.]]] generating graphics... => 1
[.,[[.,.],[.,[.,.]]]] generating graphics... => 3
[.,[[.,.],[[.,.],.]]] generating graphics... => 3
[.,[[.,[.,.]],[.,.]]] generating graphics... => 3
[.,[[[.,.],.],[.,.]]] generating graphics... => 3
[.,[[.,[.,[.,.]]],.]] generating graphics... => 1
[.,[[.,[[.,.],.]],.]] generating graphics... => 1
[.,[[[.,.],[.,.]],.]] generating graphics... => 2
[.,[[[.,[.,.]],.],.]] generating graphics... => 1
[.,[[[[.,.],.],.],.]] generating graphics... => 1
[[.,.],[.,[.,[.,.]]]] generating graphics... => 4
[[.,.],[.,[[.,.],.]]] generating graphics... => 4
[[.,.],[[.,.],[.,.]]] generating graphics... => 8
[[.,.],[[.,[.,.]],.]] generating graphics... => 4
[[.,.],[[[.,.],.],.]] generating graphics... => 4
[[.,[.,.]],[.,[.,.]]] generating graphics... => 6
[[.,[.,.]],[[.,.],.]] generating graphics... => 6
[[[.,.],.],[.,[.,.]]] generating graphics... => 6
[[[.,.],.],[[.,.],.]] generating graphics... => 6
[[.,[.,[.,.]]],[.,.]] generating graphics... => 4
[[.,[[.,.],.]],[.,.]] generating graphics... => 4
[[[.,.],[.,.]],[.,.]] generating graphics... => 8
[[[.,[.,.]],.],[.,.]] generating graphics... => 4
[[[[.,.],.],.],[.,.]] generating graphics... => 4
[[.,[.,[.,[.,.]]]],.] generating graphics... => 1
[[.,[.,[[.,.],.]]],.] generating graphics... => 1
[[.,[[.,.],[.,.]]],.] generating graphics... => 2
[[.,[[.,[.,.]],.]],.] generating graphics... => 1
[[.,[[[.,.],.],.]],.] generating graphics... => 1
[[[.,.],[.,[.,.]]],.] generating graphics... => 3
[[[.,.],[[.,.],.]],.] generating graphics... => 3
[[[.,[.,.]],[.,.]],.] generating graphics... => 3
[[[[.,.],.],[.,.]],.] generating graphics... => 3
[[[.,[.,[.,.]]],.],.] generating graphics... => 1
[[[.,[[.,.],.]],.],.] generating graphics... => 1
[[[[.,.],[.,.]],.],.] generating graphics... => 2
[[[[.,[.,.]],.],.],.] generating graphics... => 1
[[[[[.,.],.],.],.],.] generating graphics... => 1
[.,[.,[.,[.,[.,[.,.]]]]]] generating graphics... => 1
[.,[.,[.,[.,[[.,.],.]]]]] generating graphics... => 1
[.,[.,[.,[[.,.],[.,.]]]]] generating graphics... => 2
[.,[.,[.,[[.,[.,.]],.]]]] generating graphics... => 1
[.,[.,[.,[[[.,.],.],.]]]] generating graphics... => 1
[.,[.,[[.,.],[.,[.,.]]]]] generating graphics... => 3
[.,[.,[[.,.],[[.,.],.]]]] generating graphics... => 3
[.,[.,[[.,[.,.]],[.,.]]]] generating graphics... => 3
[.,[.,[[[.,.],.],[.,.]]]] generating graphics... => 3
[.,[.,[[.,[.,[.,.]]],.]]] generating graphics... => 1
[.,[.,[[.,[[.,.],.]],.]]] generating graphics... => 1
[.,[.,[[[.,.],[.,.]],.]]] generating graphics... => 2
[.,[.,[[[.,[.,.]],.],.]]] generating graphics... => 1
[.,[.,[[[[.,.],.],.],.]]] generating graphics... => 1
[.,[[.,.],[.,[.,[.,.]]]]] generating graphics... => 4
[.,[[.,.],[.,[[.,.],.]]]] generating graphics... => 4
[.,[[.,.],[[.,.],[.,.]]]] generating graphics... => 8
[.,[[.,.],[[.,[.,.]],.]]] generating graphics... => 4
[.,[[.,.],[[[.,.],.],.]]] generating graphics... => 4
[.,[[.,[.,.]],[.,[.,.]]]] generating graphics... => 6
[.,[[.,[.,.]],[[.,.],.]]] generating graphics... => 6
[.,[[[.,.],.],[.,[.,.]]]] generating graphics... => 6
[.,[[[.,.],.],[[.,.],.]]] generating graphics... => 6
[.,[[.,[.,[.,.]]],[.,.]]] generating graphics... => 4
[.,[[.,[[.,.],.]],[.,.]]] generating graphics... => 4
[.,[[[.,.],[.,.]],[.,.]]] generating graphics... => 8
[.,[[[.,[.,.]],.],[.,.]]] generating graphics... => 4
[.,[[[[.,.],.],.],[.,.]]] generating graphics... => 4
[.,[[.,[.,[.,[.,.]]]],.]] generating graphics... => 1
[.,[[.,[.,[[.,.],.]]],.]] generating graphics... => 1
[.,[[.,[[.,.],[.,.]]],.]] generating graphics... => 2
[.,[[.,[[.,[.,.]],.]],.]] generating graphics... => 1
[.,[[.,[[[.,.],.],.]],.]] generating graphics... => 1
[.,[[[.,.],[.,[.,.]]],.]] generating graphics... => 3
[.,[[[.,.],[[.,.],.]],.]] generating graphics... => 3
[.,[[[.,[.,.]],[.,.]],.]] generating graphics... => 3
[.,[[[[.,.],.],[.,.]],.]] generating graphics... => 3
[.,[[[.,[.,[.,.]]],.],.]] generating graphics... => 1
[.,[[[.,[[.,.],.]],.],.]] generating graphics... => 1
[.,[[[[.,.],[.,.]],.],.]] generating graphics... => 2
[.,[[[[.,[.,.]],.],.],.]] generating graphics... => 1
[.,[[[[[.,.],.],.],.],.]] generating graphics... => 1
[[.,.],[.,[.,[.,[.,.]]]]] generating graphics... => 5
[[.,.],[.,[.,[[.,.],.]]]] generating graphics... => 5
[[.,.],[.,[[.,.],[.,.]]]] generating graphics... => 10
[[.,.],[.,[[.,[.,.]],.]]] generating graphics... => 5
[[.,.],[.,[[[.,.],.],.]]] generating graphics... => 5
[[.,.],[[.,.],[.,[.,.]]]] generating graphics... => 15
[[.,.],[[.,.],[[.,.],.]]] generating graphics... => 15
[[.,.],[[.,[.,.]],[.,.]]] generating graphics... => 15
[[.,.],[[[.,.],.],[.,.]]] generating graphics... => 15
[[.,.],[[.,[.,[.,.]]],.]] generating graphics... => 5
[[.,.],[[.,[[.,.],.]],.]] generating graphics... => 5
[[.,.],[[[.,.],[.,.]],.]] generating graphics... => 10
[[.,.],[[[.,[.,.]],.],.]] generating graphics... => 5
[[.,.],[[[[.,.],.],.],.]] generating graphics... => 5
[[.,[.,.]],[.,[.,[.,.]]]] generating graphics... => 10
[[.,[.,.]],[.,[[.,.],.]]] generating graphics... => 10
[[.,[.,.]],[[.,.],[.,.]]] generating graphics... => 20
[[.,[.,.]],[[.,[.,.]],.]] generating graphics... => 10
[[.,[.,.]],[[[.,.],.],.]] generating graphics... => 10
[[[.,.],.],[.,[.,[.,.]]]] generating graphics... => 10
[[[.,.],.],[.,[[.,.],.]]] generating graphics... => 10
[[[.,.],.],[[.,.],[.,.]]] generating graphics... => 20
[[[.,.],.],[[.,[.,.]],.]] generating graphics... => 10
[[[.,.],.],[[[.,.],.],.]] generating graphics... => 10
[[.,[.,[.,.]]],[.,[.,.]]] generating graphics... => 10
[[.,[.,[.,.]]],[[.,.],.]] generating graphics... => 10
[[.,[[.,.],.]],[.,[.,.]]] generating graphics... => 10
[[.,[[.,.],.]],[[.,.],.]] generating graphics... => 10
[[[.,.],[.,.]],[.,[.,.]]] generating graphics... => 20
[[[.,.],[.,.]],[[.,.],.]] generating graphics... => 20
[[[.,[.,.]],.],[.,[.,.]]] generating graphics... => 10
[[[.,[.,.]],.],[[.,.],.]] generating graphics... => 10
[[[[.,.],.],.],[.,[.,.]]] generating graphics... => 10
[[[[.,.],.],.],[[.,.],.]] generating graphics... => 10
[[.,[.,[.,[.,.]]]],[.,.]] generating graphics... => 5
[[.,[.,[[.,.],.]]],[.,.]] generating graphics... => 5
[[.,[[.,.],[.,.]]],[.,.]] generating graphics... => 10
[[.,[[.,[.,.]],.]],[.,.]] generating graphics... => 5
[[.,[[[.,.],.],.]],[.,.]] generating graphics... => 5
[[[.,.],[.,[.,.]]],[.,.]] generating graphics... => 15
[[[.,.],[[.,.],.]],[.,.]] generating graphics... => 15
[[[.,[.,.]],[.,.]],[.,.]] generating graphics... => 15
[[[[.,.],.],[.,.]],[.,.]] generating graphics... => 15
[[[.,[.,[.,.]]],.],[.,.]] generating graphics... => 5
[[[.,[[.,.],.]],.],[.,.]] generating graphics... => 5
[[[[.,.],[.,.]],.],[.,.]] generating graphics... => 10
[[[[.,[.,.]],.],.],[.,.]] generating graphics... => 5
[[[[[.,.],.],.],.],[.,.]] generating graphics... => 5
[[.,[.,[.,[.,[.,.]]]]],.] generating graphics... => 1
[[.,[.,[.,[[.,.],.]]]],.] generating graphics... => 1
[[.,[.,[[.,.],[.,.]]]],.] generating graphics... => 2
[[.,[.,[[.,[.,.]],.]]],.] generating graphics... => 1
[[.,[.,[[[.,.],.],.]]],.] generating graphics... => 1
[[.,[[.,.],[.,[.,.]]]],.] generating graphics... => 3
[[.,[[.,.],[[.,.],.]]],.] generating graphics... => 3
[[.,[[.,[.,.]],[.,.]]],.] generating graphics... => 3
[[.,[[[.,.],.],[.,.]]],.] generating graphics... => 3
[[.,[[.,[.,[.,.]]],.]],.] generating graphics... => 1
[[.,[[.,[[.,.],.]],.]],.] generating graphics... => 1
[[.,[[[.,.],[.,.]],.]],.] generating graphics... => 2
[[.,[[[.,[.,.]],.],.]],.] generating graphics... => 1
[[.,[[[[.,.],.],.],.]],.] generating graphics... => 1
[[[.,.],[.,[.,[.,.]]]],.] generating graphics... => 4
[[[.,.],[.,[[.,.],.]]],.] generating graphics... => 4
[[[.,.],[[.,.],[.,.]]],.] generating graphics... => 8
[[[.,.],[[.,[.,.]],.]],.] generating graphics... => 4
[[[.,.],[[[.,.],.],.]],.] generating graphics... => 4
[[[.,[.,.]],[.,[.,.]]],.] generating graphics... => 6
[[[.,[.,.]],[[.,.],.]],.] generating graphics... => 6
[[[[.,.],.],[.,[.,.]]],.] generating graphics... => 6
[[[[.,.],.],[[.,.],.]],.] generating graphics... => 6
[[[.,[.,[.,.]]],[.,.]],.] generating graphics... => 4
[[[.,[[.,.],.]],[.,.]],.] generating graphics... => 4
[[[[.,.],[.,.]],[.,.]],.] generating graphics... => 8
[[[[.,[.,.]],.],[.,.]],.] generating graphics... => 4
[[[[[.,.],.],.],[.,.]],.] generating graphics... => 4
[[[.,[.,[.,[.,.]]]],.],.] generating graphics... => 1
[[[.,[.,[[.,.],.]]],.],.] generating graphics... => 1
[[[.,[[.,.],[.,.]]],.],.] generating graphics... => 2
[[[.,[[.,[.,.]],.]],.],.] generating graphics... => 1
[[[.,[[[.,.],.],.]],.],.] generating graphics... => 1
[[[[.,.],[.,[.,.]]],.],.] generating graphics... => 3
[[[[.,.],[[.,.],.]],.],.] generating graphics... => 3
[[[[.,[.,.]],[.,.]],.],.] generating graphics... => 3
[[[[[.,.],.],[.,.]],.],.] generating graphics... => 3
[[[[.,[.,[.,.]]],.],.],.] generating graphics... => 1
[[[[.,[[.,.],.]],.],.],.] generating graphics... => 1
[[[[[.,.],[.,.]],.],.],.] generating graphics... => 2
[[[[[.,[.,.]],.],.],.],.] generating graphics... => 1
[[[[[[.,.],.],.],.],.],.] generating graphics... => 1
click to show generating function       
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