- FindStat

Identifier

Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
St000386: Dyck paths ⟶ ℤ

Values

[] => [] => 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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$F_{1} = 1$

$F_{2} = 1$

$F_{3} = 2$

$F_{4} = 4 + q$

$F_{5} = 8 + 6\ q$

$F_{6} = 16 + 24\ q + 2\ q^{2}$

$F_{7} = 32 + 80\ q + 20\ q^{2}$

$F_{8} = 64 + 240\ q + 120\ q^{2} + 5\ q^{3}$

Description

The number of factors DDU in 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.