- FindStat

Identifier

Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
St001107: Dyck paths ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[[[[[]]]]] => [1,1,1,1,0,0,0,0] => 4

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[[[[[[]]]]]] => [1,1,1,1,1,0,0,0,0,0] => 5

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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Description

The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path.
In other words, this is the lowest height of a valley of a Dyck path, or its semilength in case of the unique path without valleys.

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.