- FindStat

Identifier

Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
St000196: Binary trees ⟶ ℤ (values match St000201The number of leaf nodes in a binary tree.)

Values

[.,.] => [1] => [.,.] => 0

[.,[.,.]] => [2,1] => [[.,.],.] => 0

[[.,.],.] => [1,2] => [.,[.,.]] => 0

[.,[.,[.,.]]] => [3,2,1] => [[[.,.],.],.] => 0

[.,[[.,.],.]] => [2,3,1] => [[.,[.,.]],.] => 0

[[.,.],[.,.]] => [3,1,2] => [[.,.],[.,.]] => 1

[[.,[.,.]],.] => [2,1,3] => [[.,.],[.,.]] => 1

[[[.,.],.],.] => [1,2,3] => [.,[.,[.,.]]] => 0

[.,[.,[.,[.,.]]]] => [4,3,2,1] => [[[[.,.],.],.],.] => 0

[.,[.,[[.,.],.]]] => [3,4,2,1] => [[[.,[.,.]],.],.] => 0

[.,[[.,.],[.,.]]] => [4,2,3,1] => [[[.,.],[.,.]],.] => 1

[.,[[.,[.,.]],.]] => [3,2,4,1] => [[[.,.],[.,.]],.] => 1

[.,[[[.,.],.],.]] => [2,3,4,1] => [[.,[.,[.,.]]],.] => 0

[[.,.],[.,[.,.]]] => [4,3,1,2] => [[[.,.],.],[.,.]] => 1

[[.,.],[[.,.],.]] => [3,4,1,2] => [[.,[.,.]],[.,.]] => 1

[[.,[.,.]],[.,.]] => [4,2,1,3] => [[[.,.],.],[.,.]] => 1

[[[.,.],.],[.,.]] => [4,1,2,3] => [[.,.],[.,[.,.]]] => 1

[[.,[.,[.,.]]],.] => [3,2,1,4] => [[[.,.],.],[.,.]] => 1

[[.,[[.,.],.]],.] => [2,3,1,4] => [[.,[.,.]],[.,.]] => 1

[[[.,.],[.,.]],.] => [3,1,2,4] => [[.,.],[.,[.,.]]] => 1

[[[.,[.,.]],.],.] => [2,1,3,4] => [[.,.],[.,[.,.]]] => 1

[[[[.,.],.],.],.] => [1,2,3,4] => [.,[.,[.,[.,.]]]] => 0

[.,[.,[.,[.,[.,.]]]]] => [5,4,3,2,1] => [[[[[.,.],.],.],.],.] => 0

[.,[.,[.,[[.,.],.]]]] => [4,5,3,2,1] => [[[[.,[.,.]],.],.],.] => 0

[.,[.,[[.,.],[.,.]]]] => [5,3,4,2,1] => [[[[.,.],[.,.]],.],.] => 1

[.,[.,[[.,[.,.]],.]]] => [4,3,5,2,1] => [[[[.,.],[.,.]],.],.] => 1

[.,[.,[[[.,.],.],.]]] => [3,4,5,2,1] => [[[.,[.,[.,.]]],.],.] => 0

[.,[[.,.],[.,[.,.]]]] => [5,4,2,3,1] => [[[[.,.],.],[.,.]],.] => 1

[.,[[.,.],[[.,.],.]]] => [4,5,2,3,1] => [[[.,[.,.]],[.,.]],.] => 1

[.,[[.,[.,.]],[.,.]]] => [5,3,2,4,1] => [[[[.,.],.],[.,.]],.] => 1

[.,[[[.,.],.],[.,.]]] => [5,2,3,4,1] => [[[.,.],[.,[.,.]]],.] => 1

[.,[[.,[.,[.,.]]],.]] => [4,3,2,5,1] => [[[[.,.],.],[.,.]],.] => 1

[.,[[.,[[.,.],.]],.]] => [3,4,2,5,1] => [[[.,[.,.]],[.,.]],.] => 1

[.,[[[.,.],[.,.]],.]] => [4,2,3,5,1] => [[[.,.],[.,[.,.]]],.] => 1

[.,[[[.,[.,.]],.],.]] => [3,2,4,5,1] => [[[.,.],[.,[.,.]]],.] => 1

[.,[[[[.,.],.],.],.]] => [2,3,4,5,1] => [[.,[.,[.,[.,.]]]],.] => 0

[[.,.],[.,[.,[.,.]]]] => [5,4,3,1,2] => [[[[.,.],.],.],[.,.]] => 1

[[.,.],[.,[[.,.],.]]] => [4,5,3,1,2] => [[[.,[.,.]],.],[.,.]] => 1

[[.,.],[[.,.],[.,.]]] => [5,3,4,1,2] => [[[.,.],[.,.]],[.,.]] => 2

[[.,.],[[.,[.,.]],.]] => [4,3,5,1,2] => [[[.,.],[.,.]],[.,.]] => 2

[[.,.],[[[.,.],.],.]] => [3,4,5,1,2] => [[.,[.,[.,.]]],[.,.]] => 1

[[.,[.,.]],[.,[.,.]]] => [5,4,2,1,3] => [[[[.,.],.],.],[.,.]] => 1

[[.,[.,.]],[[.,.],.]] => [4,5,2,1,3] => [[[.,[.,.]],.],[.,.]] => 1

[[[.,.],.],[.,[.,.]]] => [5,4,1,2,3] => [[[.,.],.],[.,[.,.]]] => 1

[[[.,.],.],[[.,.],.]] => [4,5,1,2,3] => [[.,[.,.]],[.,[.,.]]] => 1

[[.,[.,[.,.]]],[.,.]] => [5,3,2,1,4] => [[[[.,.],.],.],[.,.]] => 1

[[.,[[.,.],.]],[.,.]] => [5,2,3,1,4] => [[[.,.],[.,.]],[.,.]] => 2

[[[.,.],[.,.]],[.,.]] => [5,3,1,2,4] => [[[.,.],.],[.,[.,.]]] => 1

[[[.,[.,.]],.],[.,.]] => [5,2,1,3,4] => [[[.,.],.],[.,[.,.]]] => 1

[[[[.,.],.],.],[.,.]] => [5,1,2,3,4] => [[.,.],[.,[.,[.,.]]]] => 1

[[.,[.,[.,[.,.]]]],.] => [4,3,2,1,5] => [[[[.,.],.],.],[.,.]] => 1

[[.,[.,[[.,.],.]]],.] => [3,4,2,1,5] => [[[.,[.,.]],.],[.,.]] => 1

[[.,[[.,.],[.,.]]],.] => [4,2,3,1,5] => [[[.,.],[.,.]],[.,.]] => 2

[[.,[[.,[.,.]],.]],.] => [3,2,4,1,5] => [[[.,.],[.,.]],[.,.]] => 2

[[.,[[[.,.],.],.]],.] => [2,3,4,1,5] => [[.,[.,[.,.]]],[.,.]] => 1

[[[.,.],[.,[.,.]]],.] => [4,3,1,2,5] => [[[.,.],.],[.,[.,.]]] => 1

[[[.,.],[[.,.],.]],.] => [3,4,1,2,5] => [[.,[.,.]],[.,[.,.]]] => 1

[[[.,[.,.]],[.,.]],.] => [4,2,1,3,5] => [[[.,.],.],[.,[.,.]]] => 1

[[[[.,.],.],[.,.]],.] => [4,1,2,3,5] => [[.,.],[.,[.,[.,.]]]] => 1

[[[.,[.,[.,.]]],.],.] => [3,2,1,4,5] => [[[.,.],.],[.,[.,.]]] => 1

[[[.,[[.,.],.]],.],.] => [2,3,1,4,5] => [[.,[.,.]],[.,[.,.]]] => 1

[[[[.,.],[.,.]],.],.] => [3,1,2,4,5] => [[.,.],[.,[.,[.,.]]]] => 1

[[[[.,[.,.]],.],.],.] => [2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]] => 1

[[[[[.,.],.],.],.],.] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]] => 0

[.,[.,[.,[.,[.,[.,.]]]]]] => [6,5,4,3,2,1] => [[[[[[.,.],.],.],.],.],.] => 0

[.,[.,[.,[.,[[.,.],.]]]]] => [5,6,4,3,2,1] => [[[[[.,[.,.]],.],.],.],.] => 0

[.,[.,[.,[[.,.],[.,.]]]]] => [6,4,5,3,2,1] => [[[[[.,.],[.,.]],.],.],.] => 1

[.,[.,[.,[[.,[.,.]],.]]]] => [5,4,6,3,2,1] => [[[[[.,.],[.,.]],.],.],.] => 1

[.,[.,[.,[[[.,.],.],.]]]] => [4,5,6,3,2,1] => [[[[.,[.,[.,.]]],.],.],.] => 0

[.,[.,[[.,.],[.,[.,.]]]]] => [6,5,3,4,2,1] => [[[[[.,.],.],[.,.]],.],.] => 1

[.,[.,[[.,.],[[.,.],.]]]] => [5,6,3,4,2,1] => [[[[.,[.,.]],[.,.]],.],.] => 1

[.,[.,[[.,[.,.]],[.,.]]]] => [6,4,3,5,2,1] => [[[[[.,.],.],[.,.]],.],.] => 1

[.,[.,[[[.,.],.],[.,.]]]] => [6,3,4,5,2,1] => [[[[.,.],[.,[.,.]]],.],.] => 1

[.,[.,[[.,[.,[.,.]]],.]]] => [5,4,3,6,2,1] => [[[[[.,.],.],[.,.]],.],.] => 1

[.,[.,[[.,[[.,.],.]],.]]] => [4,5,3,6,2,1] => [[[[.,[.,.]],[.,.]],.],.] => 1

[.,[.,[[[.,.],[.,.]],.]]] => [5,3,4,6,2,1] => [[[[.,.],[.,[.,.]]],.],.] => 1

[.,[.,[[[.,[.,.]],.],.]]] => [4,3,5,6,2,1] => [[[[.,.],[.,[.,.]]],.],.] => 1

[.,[.,[[[[.,.],.],.],.]]] => [3,4,5,6,2,1] => [[[.,[.,[.,[.,.]]]],.],.] => 0

[.,[[.,.],[.,[.,[.,.]]]]] => [6,5,4,2,3,1] => [[[[[.,.],.],.],[.,.]],.] => 1

[.,[[.,.],[.,[[.,.],.]]]] => [5,6,4,2,3,1] => [[[[.,[.,.]],.],[.,.]],.] => 1

[.,[[.,.],[[.,.],[.,.]]]] => [6,4,5,2,3,1] => [[[[.,.],[.,.]],[.,.]],.] => 2

[.,[[.,.],[[.,[.,.]],.]]] => [5,4,6,2,3,1] => [[[[.,.],[.,.]],[.,.]],.] => 2

[.,[[.,.],[[[.,.],.],.]]] => [4,5,6,2,3,1] => [[[.,[.,[.,.]]],[.,.]],.] => 1

[.,[[.,[.,.]],[.,[.,.]]]] => [6,5,3,2,4,1] => [[[[[.,.],.],.],[.,.]],.] => 1

[.,[[.,[.,.]],[[.,.],.]]] => [5,6,3,2,4,1] => [[[[.,[.,.]],.],[.,.]],.] => 1

[.,[[[.,.],.],[.,[.,.]]]] => [6,5,2,3,4,1] => [[[[.,.],.],[.,[.,.]]],.] => 1

[.,[[[.,.],.],[[.,.],.]]] => [5,6,2,3,4,1] => [[[.,[.,.]],[.,[.,.]]],.] => 1

[.,[[.,[.,[.,.]]],[.,.]]] => [6,4,3,2,5,1] => [[[[[.,.],.],.],[.,.]],.] => 1

[.,[[.,[[.,.],.]],[.,.]]] => [6,3,4,2,5,1] => [[[[.,.],[.,.]],[.,.]],.] => 2

[.,[[[.,.],[.,.]],[.,.]]] => [6,4,2,3,5,1] => [[[[.,.],.],[.,[.,.]]],.] => 1

[.,[[[.,[.,.]],.],[.,.]]] => [6,3,2,4,5,1] => [[[[.,.],.],[.,[.,.]]],.] => 1

[.,[[[[.,.],.],.],[.,.]]] => [6,2,3,4,5,1] => [[[.,.],[.,[.,[.,.]]]],.] => 1

[.,[[.,[.,[.,[.,.]]]],.]] => [5,4,3,2,6,1] => [[[[[.,.],.],.],[.,.]],.] => 1

[.,[[.,[.,[[.,.],.]]],.]] => [4,5,3,2,6,1] => [[[[.,[.,.]],.],[.,.]],.] => 1

[.,[[.,[[.,.],[.,.]]],.]] => [5,3,4,2,6,1] => [[[[.,.],[.,.]],[.,.]],.] => 2

[.,[[.,[[.,[.,.]],.]],.]] => [4,3,5,2,6,1] => [[[[.,.],[.,.]],[.,.]],.] => 2

[.,[[.,[[[.,.],.],.]],.]] => [3,4,5,2,6,1] => [[[.,[.,[.,.]]],[.,.]],.] => 1

[.,[[[.,.],[.,[.,.]]],.]] => [5,4,2,3,6,1] => [[[[.,.],.],[.,[.,.]]],.] => 1

[.,[[[.,.],[[.,.],.]],.]] => [4,5,2,3,6,1] => [[[.,[.,.]],[.,[.,.]]],.] => 1

[.,[[[.,[.,.]],[.,.]],.]] => [5,3,2,4,6,1] => [[[[.,.],.],[.,[.,.]]],.] => 1

[.,[[[[.,.],.],[.,.]],.]] => [5,2,3,4,6,1] => [[[.,.],[.,[.,[.,.]]]],.] => 1

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Description

The number of occurrences of the contiguous pattern [[.,.],[.,.]] in a binary tree.
Equivalently, this is the number of branches in the tree, i.e. the number of nodes with two children. Binary trees avoiding this pattern are counted by $2^{n-2}$.

Map

to 132-avoiding permutation

Description

Return a 132-avoiding permutation corresponding to a binary tree.
The linear extensions of a binary tree form an interval of the weak order called the Sylvester class of the tree. This permutation is the maximal element of the Sylvester class.

Map

to increasing tree

Description

Sends a permutation to its associated increasing tree.
This tree is recursively obtained by sending the unique permutation of length $0$ to the empty tree, and sending a permutation $\sigma$ of length $n \geq 1$ to a root node with two subtrees $L$ and $R$ by splitting $\sigma$ at the index $\sigma^{-1}(1)$, normalizing both sides again to permutations and sending the permutations on the left and on the right of $\sigma^{-1}(1)$ to the trees $L$ and $R$, respectively.