- FindStat

Identifier

Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
St000013: Dyck paths ⟶ ℤ (values match St000442The maximal area to the right of an up step of a Dyck path.)

Values

[] => [] => 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 1386 entries. <<<

search for individual values

searching the database for the individual values of this statistic

/ search for generating function

searching the database for statistics with the same generating function

click to show known generating functions

Description

The height of a Dyck path.
The height of a Dyck path $D$ of semilength $n$ is defined as the maximal height of a peak of $D$. The height of $D$ at position $i$ is the number of up-steps minus the number of down-steps before position $i$.

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.