Identifier
Values
[1] => [1,0] => [.,.] => 0
[1,1] => [1,0,1,0] => [.,[.,.]] => 0
[2] => [1,1,0,0] => [[.,.],.] => 0
[1,1,1] => [1,0,1,0,1,0] => [.,[.,[.,.]]] => 0
[1,2] => [1,0,1,1,0,0] => [.,[[.,.],.]] => 0
[2,1] => [1,1,0,0,1,0] => [[.,.],[.,.]] => 0
[3] => [1,1,1,0,0,0] => [[[.,.],.],.] => 0
[1,1,1,1] => [1,0,1,0,1,0,1,0] => [.,[.,[.,[.,.]]]] => 0
[1,1,2] => [1,0,1,0,1,1,0,0] => [.,[.,[[.,.],.]]] => 0
[1,2,1] => [1,0,1,1,0,0,1,0] => [.,[[.,.],[.,.]]] => 0
[1,3] => [1,0,1,1,1,0,0,0] => [.,[[[.,.],.],.]] => 0
[2,1,1] => [1,1,0,0,1,0,1,0] => [[.,.],[.,[.,.]]] => 0
[2,2] => [1,1,0,0,1,1,0,0] => [[.,.],[[.,.],.]] => 0
[3,1] => [1,1,1,0,0,0,1,0] => [[[.,.],.],[.,.]] => 0
[4] => [1,1,1,1,0,0,0,0] => [[[[.,.],.],.],.] => 0
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0] => [.,[.,[.,[.,[.,.]]]]] => 0
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0] => [.,[.,[.,[[.,.],.]]]] => 0
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => [.,[.,[[.,.],[.,.]]]] => 0
[1,1,3] => [1,0,1,0,1,1,1,0,0,0] => [.,[.,[[[.,.],.],.]]] => 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => [.,[[.,.],[.,[.,.]]]] => 0
[1,2,2] => [1,0,1,1,0,0,1,1,0,0] => [.,[[.,.],[[.,.],.]]] => 0
[1,3,1] => [1,0,1,1,1,0,0,0,1,0] => [.,[[[.,.],.],[.,.]]] => 0
[1,4] => [1,0,1,1,1,1,0,0,0,0] => [.,[[[[.,.],.],.],.]] => 0
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => [[.,.],[.,[.,[.,.]]]] => 0
[2,1,2] => [1,1,0,0,1,0,1,1,0,0] => [[.,.],[.,[[.,.],.]]] => 0
[2,2,1] => [1,1,0,0,1,1,0,0,1,0] => [[.,.],[[.,.],[.,.]]] => 0
[2,3] => [1,1,0,0,1,1,1,0,0,0] => [[.,.],[[[.,.],.],.]] => 0
[3,1,1] => [1,1,1,0,0,0,1,0,1,0] => [[[.,.],.],[.,[.,.]]] => 0
[3,2] => [1,1,1,0,0,0,1,1,0,0] => [[[.,.],.],[[.,.],.]] => 0
[4,1] => [1,1,1,1,0,0,0,0,1,0] => [[[[.,.],.],.],[.,.]] => 0
[5] => [1,1,1,1,1,0,0,0,0,0] => [[[[[.,.],.],.],.],.] => 0
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => [.,[.,[.,[.,[.,[.,.]]]]]] => 0
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0] => [.,[.,[.,[.,[[.,.],.]]]]] => 0
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0] => [.,[.,[.,[[.,.],[.,.]]]]] => 0
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0] => [.,[.,[.,[[[.,.],.],.]]]] => 1
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0] => [.,[.,[[.,.],[.,[.,.]]]]] => 0
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0] => [.,[.,[[.,.],[[.,.],.]]]] => 0
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0] => [.,[.,[[[.,.],.],[.,.]]]] => 1
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0] => [.,[.,[[[[.,.],.],.],.]]] => 1
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0] => [.,[[.,.],[.,[.,[.,.]]]]] => 0
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0] => [.,[[.,.],[.,[[.,.],.]]]] => 0
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0] => [.,[[.,.],[[.,.],[.,.]]]] => 0
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0] => [.,[[.,.],[[[.,.],.],.]]] => 1
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0] => [.,[[[.,.],.],[.,[.,.]]]] => 0
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0] => [.,[[[.,.],.],[[.,.],.]]] => 0
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0] => [.,[[[[.,.],.],.],[.,.]]] => 0
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0] => [.,[[[[[.,.],.],.],.],.]] => 0
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0] => [[.,.],[.,[.,[.,[.,.]]]]] => 0
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0] => [[.,.],[.,[.,[[.,.],.]]]] => 0
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0] => [[.,.],[.,[[.,.],[.,.]]]] => 0
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0] => [[.,.],[.,[[[.,.],.],.]]] => 1
[2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0] => [[.,.],[[.,.],[.,[.,.]]]] => 0
[2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0] => [[.,.],[[.,.],[[.,.],.]]] => 0
[2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0] => [[.,.],[[[.,.],.],[.,.]]] => 0
[2,4] => [1,1,0,0,1,1,1,1,0,0,0,0] => [[.,.],[[[[.,.],.],.],.]] => 0
[3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0] => [[[.,.],.],[.,[.,[.,.]]]] => 0
[3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0] => [[[.,.],.],[.,[[.,.],.]]] => 0
[3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0] => [[[.,.],.],[[.,.],[.,.]]] => 0
[3,3] => [1,1,1,0,0,0,1,1,1,0,0,0] => [[[.,.],.],[[[.,.],.],.]] => 0
[4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0] => [[[[.,.],.],.],[.,[.,.]]] => 0
[4,2] => [1,1,1,1,0,0,0,0,1,1,0,0] => [[[[.,.],.],.],[[.,.],.]] => 0
[5,1] => [1,1,1,1,1,0,0,0,0,0,1,0] => [[[[[.,.],.],.],.],[.,.]] => 0
[6] => [1,1,1,1,1,1,0,0,0,0,0,0] => [[[[[[.,.],.],.],.],.],.] => 0
[1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]] => 0
[1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0] => [.,[.,[.,[.,[.,[[.,.],.]]]]]] => 0
[1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0] => [.,[.,[.,[.,[[.,.],[.,.]]]]]] => 0
[1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0] => [.,[.,[.,[.,[[[.,.],.],.]]]]] => 1
[1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0] => [.,[.,[.,[[.,.],[.,[.,.]]]]]] => 0
[1,1,1,2,2] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0] => [.,[.,[.,[[.,.],[[.,.],.]]]]] => 0
[1,1,1,3,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0] => [.,[.,[.,[[[.,.],.],[.,.]]]]] => 1
[1,1,1,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0] => [.,[.,[.,[[[[.,.],.],.],.]]]] => 1
[1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0] => [.,[.,[[.,.],[.,[.,[.,.]]]]]] => 0
[1,1,2,1,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0] => [.,[.,[[.,.],[.,[[.,.],.]]]]] => 0
[1,1,2,2,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0] => [.,[.,[[.,.],[[.,.],[.,.]]]]] => 0
[1,1,2,3] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0] => [.,[.,[[.,.],[[[.,.],.],.]]]] => 1
[1,1,3,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0] => [.,[.,[[[.,.],.],[.,[.,.]]]]] => 1
[1,1,3,2] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0] => [.,[.,[[[.,.],.],[[.,.],.]]]] => 1
[1,1,4,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0] => [.,[.,[[[[.,.],.],.],[.,.]]]] => 1
[1,1,5] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0] => [.,[.,[[[[[.,.],.],.],.],.]]] => 1
[1,2,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0] => [.,[[.,.],[.,[.,[.,[.,.]]]]]] => 0
[1,2,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0] => [.,[[.,.],[.,[.,[[.,.],.]]]]] => 0
[1,2,1,2,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0] => [.,[[.,.],[.,[[.,.],[.,.]]]]] => 0
[1,2,1,3] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0] => [.,[[.,.],[.,[[[.,.],.],.]]]] => 1
[1,2,2,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0] => [.,[[.,.],[[.,.],[.,[.,.]]]]] => 0
[1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0] => [.,[[.,.],[[.,.],[[.,.],.]]]] => 0
[1,2,3,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0] => [.,[[.,.],[[[.,.],.],[.,.]]]] => 1
[1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0] => [.,[[.,.],[[[[.,.],.],.],.]]] => 1
[1,3,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0] => [.,[[[.,.],.],[.,[.,[.,.]]]]] => 0
[1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0] => [.,[[[.,.],.],[.,[[.,.],.]]]] => 0
[1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0] => [.,[[[.,.],.],[[.,.],[.,.]]]] => 0
[1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0] => [.,[[[.,.],.],[[[.,.],.],.]]] => 1
[1,4,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0] => [.,[[[[.,.],.],.],[.,[.,.]]]] => 0
[1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0] => [.,[[[[.,.],.],.],[[.,.],.]]] => 0
[1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0] => [.,[[[[[.,.],.],.],.],[.,.]]] => 0
[1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0] => [.,[[[[[[.,.],.],.],.],.],.]] => 0
[2,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0] => [[.,.],[.,[.,[.,[.,[.,.]]]]]] => 0
[2,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0] => [[.,.],[.,[.,[.,[[.,.],.]]]]] => 0
[2,1,1,2,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0] => [[.,.],[.,[.,[[.,.],[.,.]]]]] => 0
[2,1,1,3] => [1,1,0,0,1,0,1,0,1,1,1,0,0,0] => [[.,.],[.,[.,[[[.,.],.],.]]]] => 1
[2,1,2,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0] => [[.,.],[.,[[.,.],[.,[.,.]]]]] => 0
[2,1,2,2] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0] => [[.,.],[.,[[.,.],[[.,.],.]]]] => 0
>>> Load all 127 entries. <<<
[2,1,3,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0] => [[.,.],[.,[[[.,.],.],[.,.]]]] => 1
[2,1,4] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0] => [[.,.],[.,[[[[.,.],.],.],.]]] => 1
[2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0] => [[.,.],[[.,.],[.,[.,[.,.]]]]] => 0
[2,2,1,2] => [1,1,0,0,1,1,0,0,1,0,1,1,0,0] => [[.,.],[[.,.],[.,[[.,.],.]]]] => 0
[2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0] => [[.,.],[[.,.],[[.,.],[.,.]]]] => 0
[2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0] => [[.,.],[[.,.],[[[.,.],.],.]]] => 1
[2,3,1,1] => [1,1,0,0,1,1,1,0,0,0,1,0,1,0] => [[.,.],[[[.,.],.],[.,[.,.]]]] => 0
[2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0] => [[.,.],[[[.,.],.],[[.,.],.]]] => 0
[2,4,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0] => [[.,.],[[[[.,.],.],.],[.,.]]] => 0
[2,5] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0] => [[.,.],[[[[[.,.],.],.],.],.]] => 0
[3,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0] => [[[.,.],.],[.,[.,[.,[.,.]]]]] => 0
[3,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0] => [[[.,.],.],[.,[.,[[.,.],.]]]] => 0
[3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0] => [[[.,.],.],[.,[[.,.],[.,.]]]] => 0
[3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0] => [[[.,.],.],[.,[[[.,.],.],.]]] => 1
[3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0] => [[[.,.],.],[[.,.],[.,[.,.]]]] => 0
[3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0] => [[[.,.],.],[[.,.],[[.,.],.]]] => 0
[3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0] => [[[.,.],.],[[[.,.],.],[.,.]]] => 0
[3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0] => [[[.,.],.],[[[[.,.],.],.],.]] => 0
[4,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0] => [[[[.,.],.],.],[.,[.,[.,.]]]] => 0
[4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0] => [[[[.,.],.],.],[.,[[.,.],.]]] => 0
[4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0] => [[[[.,.],.],.],[[.,.],[.,.]]] => 0
[4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0] => [[[[.,.],.],.],[[[.,.],.],.]] => 0
[5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0] => [[[[[.,.],.],.],.],[.,[.,.]]] => 0
[5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0] => [[[[[.,.],.],.],.],[[.,.],.]] => 0
[6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => [[[[[[.,.],.],.],.],.],[.,.]] => 0
[7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0] => [[[[[[[.,.],.],.],.],.],.],.] => 0
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Description
The number of occurrences of the contiguous pattern [.,[.,[[[.,.],.],.]]] in a binary tree.
oeis:A159770 counts binary trees avoiding this pattern.
Map
bounce path
Description
The bounce path determined by an integer composition.
Map
to binary tree: up step, left tree, down step, right tree
Description
Return the binary tree corresponding to the Dyck path under the transformation up step - left tree - down step - right tree.
A Dyck path $D$ of semilength $n$ with $ n > 1$ may be uniquely decomposed into $1L0R$ for Dyck paths L,R of respective semilengths $n_1, n_2$ with $n_1 + n_2 = n-1$.
This map sends $D$ to the binary tree $T$ consisting of a root node with a left child according to $L$ and a right child according to $R$ and then recursively proceeds.
The base case of the unique Dyck path of semilength $1$ is sent to a single node.