- FindStat

Identifier

Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
St001007: Dyck paths ⟶ ℤ (values match St000024The number of double up and double down steps of a Dyck path., St000443The number of long tunnels of a 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

[[]] => [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] => 3

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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Description

Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the 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.