- FindStat

Identifier

Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00018: Binary trees —left border symmetry⟶ Binary trees
St000118: Binary trees ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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Description

The number of occurrences of the contiguous pattern [.,[.,[.,.]]] in a binary tree.
oeis:A001006 counts binary trees avoiding this pattern.

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.

Map

left border symmetry

Description

Return the tree where a symmetry has been applied recursively on all left borders. If a tree is made of three trees $T_1, T_2, T_3$ on its left border, it becomes $T_3, T_2, T_1$ where same symmetry has been applied to $T_1, T_2, T_3$.