- FindStat

Identifier

Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
St000024: Dyck paths ⟶ ℤ (values match St000443The number of long tunnels of a Dyck path., St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path., St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra., St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra.)

Values

[] => [] => 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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Description

The number of double up and double down steps of a Dyck path.
In other words, this is the number of double rises (and, equivalently, the number of double falls) of a Dyck path.

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.