- FindStat

Identifier

Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
St001185: Dyck paths ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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$F_{2} = 1$

$F_{3} = 1 + q$

$F_{4} = 2 + 2\ q + q^{2}$

$F_{5} = 5 + 5\ q + 3\ q^{2} + q^{3}$

$F_{6} = 14 + 14\ q + 9\ q^{2} + 4\ q^{3} + q^{4}$

$F_{7} = 42 + 42\ q + 28\ q^{2} + 14\ q^{3} + 5\ q^{4} + q^{5}$

Description

The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra.

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.