- FindStat

Identifier

Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
St000920: Dyck paths ⟶ ℤ

Values

[] => [] => 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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Description

The logarithmic height of a Dyck path.
This is the floor of the binary logarithm of the usual height increased by one:
$$ \lfloor\log_2(1+height(D))\rfloor $$

Map

to Dyck path

Description

Return the Dyck path of the corresponding ordered tree induced by the recurrence of the Catalan numbers, see wikipedia:Catalan_number.
This sends the maximal height of the Dyck path to the depth of the tree.