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Identifier
Values
=>
Cc0010;cc-rep
[.,[.,[.,.]]]=>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 [.,[[[[.,[.,.]],.],.],.]]=>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