- FindStat

Identifier

Mp00048: Ordered trees —left-right symmetry⟶ Ordered trees
Mp00246: Ordered trees —rotate⟶ Ordered trees
St000974: Ordered trees ⟶ ℤ

Values

[[]] => [[]] => [[]] => 1

[[],[]] => [[],[]] => [[],[]] => 0

[[[]]] => [[[]]] => [[[]]] => 2

[[],[],[]] => [[],[],[]] => [[],[],[]] => 0

[[],[[]]] => [[[]],[]] => [[[],[]]] => 1

[[[]],[]] => [[],[[]]] => [[[]],[]] => 0

[[[],[]]] => [[[],[]]] => [[],[[]]] => 0

[[[[]]]] => [[[[]]]] => [[[[]]]] => 3

[[],[],[],[]] => [[],[],[],[]] => [[],[],[],[]] => 0

[[],[],[[]]] => [[[]],[],[]] => [[[],[],[]]] => 1

[[],[[]],[]] => [[],[[]],[]] => [[[]],[],[]] => 0

[[],[[],[]]] => [[[],[]],[]] => [[],[[],[]]] => 0

[[],[[[]]]] => [[[[]]],[]] => [[[[],[]]]] => 2

[[[]],[],[]] => [[],[],[[]]] => [[],[[]],[]] => 0

[[[]],[[]]] => [[[]],[[]]] => [[[[]],[]]] => 1

[[[],[]],[]] => [[],[[],[]]] => [[[],[]],[]] => 0

[[[[]]],[]] => [[],[[[]]]] => [[[[]]],[]] => 0

[[[],[],[]]] => [[[],[],[]]] => [[],[],[[]]] => 0

[[[],[[]]]] => [[[[]],[]]] => [[[],[[]]]] => 1

[[[[]],[]]] => [[[],[[]]]] => [[[]],[[]]] => 0

[[[[],[]]]] => [[[[],[]]]] => [[],[[[]]]] => 0

[[[[[]]]]] => [[[[[]]]]] => [[[[[]]]]] => 4

[[],[],[],[],[]] => [[],[],[],[],[]] => [[],[],[],[],[]] => 0

[[],[],[],[[]]] => [[[]],[],[],[]] => [[[],[],[],[]]] => 1

[[],[],[[]],[]] => [[],[[]],[],[]] => [[[]],[],[],[]] => 0

[[],[],[[],[]]] => [[[],[]],[],[]] => [[],[[],[],[]]] => 0

[[],[],[[[]]]] => [[[[]]],[],[]] => [[[[],[],[]]]] => 2

[[],[[]],[],[]] => [[],[],[[]],[]] => [[],[[]],[],[]] => 0

[[],[[]],[[]]] => [[[]],[[]],[]] => [[[[]],[],[]]] => 1

[[],[[],[]],[]] => [[],[[],[]],[]] => [[[],[]],[],[]] => 0

[[],[[[]]],[]] => [[],[[[]]],[]] => [[[[]]],[],[]] => 0

[[],[[],[],[]]] => [[[],[],[]],[]] => [[],[],[[],[]]] => 0

[[],[[],[[]]]] => [[[[]],[]],[]] => [[[],[[],[]]]] => 1

[[],[[[]],[]]] => [[[],[[]]],[]] => [[[]],[[],[]]] => 0

[[],[[[],[]]]] => [[[[],[]]],[]] => [[],[[[],[]]]] => 0

[[],[[[[]]]]] => [[[[[]]]],[]] => [[[[[],[]]]]] => 3

[[[]],[],[],[]] => [[],[],[],[[]]] => [[],[],[[]],[]] => 0

[[[]],[],[[]]] => [[[]],[],[[]]] => [[[],[[]],[]]] => 1

[[[]],[[]],[]] => [[],[[]],[[]]] => [[[]],[[]],[]] => 0

[[[]],[[],[]]] => [[[],[]],[[]]] => [[],[[[]],[]]] => 0

[[[]],[[[]]]] => [[[[]]],[[]]] => [[[[[]],[]]]] => 2

[[[],[]],[],[]] => [[],[],[[],[]]] => [[],[[],[]],[]] => 0

[[[[]]],[],[]] => [[],[],[[[]]]] => [[],[[[]]],[]] => 0

[[[],[]],[[]]] => [[[]],[[],[]]] => [[[[],[]],[]]] => 1

[[[[]]],[[]]] => [[[]],[[[]]]] => [[[[[]]],[]]] => 1

[[[],[],[]],[]] => [[],[[],[],[]]] => [[[],[],[]],[]] => 0

[[[],[[]]],[]] => [[],[[[]],[]]] => [[[[]],[]],[]] => 0

[[[[]],[]],[]] => [[],[[],[[]]]] => [[[],[[]]],[]] => 0

[[[[],[]]],[]] => [[],[[[],[]]]] => [[[[],[]]],[]] => 0

[[[[[]]]],[]] => [[],[[[[]]]]] => [[[[[]]]],[]] => 0

[[[],[],[],[]]] => [[[],[],[],[]]] => [[],[],[],[[]]] => 0

[[[],[],[[]]]] => [[[[]],[],[]]] => [[[],[],[[]]]] => 1

[[[],[[]],[]]] => [[[],[[]],[]]] => [[[]],[],[[]]] => 0

[[[],[[],[]]]] => [[[[],[]],[]]] => [[],[[],[[]]]] => 0

[[[],[[[]]]]] => [[[[[]]],[]]] => [[[[],[[]]]]] => 2

[[[[]],[],[]]] => [[[],[],[[]]]] => [[],[[]],[[]]] => 0

[[[[]],[[]]]] => [[[[]],[[]]]] => [[[[]],[[]]]] => 1

[[[[],[]],[]]] => [[[],[[],[]]]] => [[[],[]],[[]]] => 0

[[[[[]]],[]]] => [[[],[[[]]]]] => [[[[]]],[[]]] => 0

[[[[],[],[]]]] => [[[[],[],[]]]] => [[],[],[[[]]]] => 0

[[[[],[[]]]]] => [[[[[]],[]]]] => [[[],[[[]]]]] => 1

[[[[[]],[]]]] => [[[[],[[]]]]] => [[[]],[[[]]]] => 0

[[[[[],[]]]]] => [[[[[],[]]]]] => [[],[[[[]]]]] => 0

[[[[[[]]]]]] => [[[[[[]]]]]] => [[[[[[]]]]]] => 5

[[],[],[],[],[],[]] => [[],[],[],[],[],[]] => [[],[],[],[],[],[]] => 0

[[],[],[],[],[[]]] => [[[]],[],[],[],[]] => [[[],[],[],[],[]]] => 1

[[],[],[],[[]],[]] => [[],[[]],[],[],[]] => [[[]],[],[],[],[]] => 0

[[],[],[],[[],[]]] => [[[],[]],[],[],[]] => [[],[[],[],[],[]]] => 0

[[],[],[],[[[]]]] => [[[[]]],[],[],[]] => [[[[],[],[],[]]]] => 2

[[],[],[[]],[],[]] => [[],[],[[]],[],[]] => [[],[[]],[],[],[]] => 0

[[],[],[[]],[[]]] => [[[]],[[]],[],[]] => [[[[]],[],[],[]]] => 1

[[],[],[[],[]],[]] => [[],[[],[]],[],[]] => [[[],[]],[],[],[]] => 0

[[],[],[[[]]],[]] => [[],[[[]]],[],[]] => [[[[]]],[],[],[]] => 0

[[],[],[[],[],[]]] => [[[],[],[]],[],[]] => [[],[],[[],[],[]]] => 0

[[],[],[[],[[]]]] => [[[[]],[]],[],[]] => [[[],[[],[],[]]]] => 1

[[],[],[[[]],[]]] => [[[],[[]]],[],[]] => [[[]],[[],[],[]]] => 0

[[],[],[[[],[]]]] => [[[[],[]]],[],[]] => [[],[[[],[],[]]]] => 0

[[],[],[[[[]]]]] => [[[[[]]]],[],[]] => [[[[[],[],[]]]]] => 3

[[],[[]],[],[],[]] => [[],[],[],[[]],[]] => [[],[],[[]],[],[]] => 0

[[],[[]],[],[[]]] => [[[]],[],[[]],[]] => [[[],[[]],[],[]]] => 1

[[],[[]],[[]],[]] => [[],[[]],[[]],[]] => [[[]],[[]],[],[]] => 0

[[],[[]],[[],[]]] => [[[],[]],[[]],[]] => [[],[[[]],[],[]]] => 0

[[],[[]],[[[]]]] => [[[[]]],[[]],[]] => [[[[[]],[],[]]]] => 2

[[],[[],[]],[],[]] => [[],[],[[],[]],[]] => [[],[[],[]],[],[]] => 0

[[],[[[]]],[],[]] => [[],[],[[[]]],[]] => [[],[[[]]],[],[]] => 0

[[],[[],[]],[[]]] => [[[]],[[],[]],[]] => [[[[],[]],[],[]]] => 1

[[],[[[]]],[[]]] => [[[]],[[[]]],[]] => [[[[[]]],[],[]]] => 1

[[],[[],[],[]],[]] => [[],[[],[],[]],[]] => [[[],[],[]],[],[]] => 0

[[],[[],[[]]],[]] => [[],[[[]],[]],[]] => [[[[]],[]],[],[]] => 0

[[],[[[]],[]],[]] => [[],[[],[[]]],[]] => [[[],[[]]],[],[]] => 0

[[],[[[],[]]],[]] => [[],[[[],[]]],[]] => [[[[],[]]],[],[]] => 0

[[],[[[[]]]],[]] => [[],[[[[]]]],[]] => [[[[[]]]],[],[]] => 0

[[],[[],[],[],[]]] => [[[],[],[],[]],[]] => [[],[],[],[[],[]]] => 0

[[],[[],[],[[]]]] => [[[[]],[],[]],[]] => [[[],[],[[],[]]]] => 1

[[],[[],[[]],[]]] => [[[],[[]],[]],[]] => [[[]],[],[[],[]]] => 0

[[],[[],[[],[]]]] => [[[[],[]],[]],[]] => [[],[[],[[],[]]]] => 0

[[],[[],[[[]]]]] => [[[[[]]],[]],[]] => [[[[],[[],[]]]]] => 2

[[],[[[]],[],[]]] => [[[],[],[[]]],[]] => [[],[[]],[[],[]]] => 0

[[],[[[]],[[]]]] => [[[[]],[[]]],[]] => [[[[]],[[],[]]]] => 1

[[],[[[],[]],[]]] => [[[],[[],[]]],[]] => [[[],[]],[[],[]]] => 0

[[],[[[[]]],[]]] => [[[],[[[]]]],[]] => [[[[]]],[[],[]]] => 0

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Description

The length of the trunk of an ordered tree.
This is the length of the path from the root to the first vertex which has not exactly one child.

Map

rotate

Description

The rotation of an ordered tree.
Let $T$ be a rooted ordered tree with root $r$ and left most leaf $\ell$. It's rotation is obtained by attaching a leaf to $r$, removing the leaf $\ell$ and letting the vertex attached to $\ell$ be the new root.
By duality, this operation corresponds to rotation of polygon dissections.

Map

left-right symmetry

Description

Return the left-right symmetrized tree of an ordered tree.