Identifier
-
Mp00051:
Ordered trees
—to Dyck path⟶
Dyck paths
St000015: Dyck paths ⟶ ℤ (values match St000053The number of valleys of the Dyck path., St001068Number of torsionless simple modules in the corresponding Nakayama algebra., St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra.)
Values
[[]] => [1,0] => 1
[[],[]] => [1,0,1,0] => 2
[[[]]] => [1,1,0,0] => 1
[[],[],[]] => [1,0,1,0,1,0] => 3
[[],[[]]] => [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] => 1
[[],[],[],[]] => [1,0,1,0,1,0,1,0] => 4
[[],[],[[]]] => [1,0,1,0,1,1,0,0] => 3
[[],[[]],[]] => [1,0,1,1,0,0,1,0] => 3
[[],[[],[]]] => [1,0,1,1,0,1,0,0] => 3
[[],[[[]]]] => [1,0,1,1,1,0,0,0] => 2
[[[]],[],[]] => [1,1,0,0,1,0,1,0] => 3
[[[]],[[]]] => [1,1,0,0,1,1,0,0] => 2
[[[],[]],[]] => [1,1,0,1,0,0,1,0] => 3
[[[[]]],[]] => [1,1,1,0,0,0,1,0] => 2
[[[],[],[]]] => [1,1,0,1,0,1,0,0] => 3
[[[],[[]]]] => [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] => 1
[[],[],[],[],[]] => [1,0,1,0,1,0,1,0,1,0] => 5
[[],[],[],[[]]] => [1,0,1,0,1,0,1,1,0,0] => 4
[[],[],[[]],[]] => [1,0,1,0,1,1,0,0,1,0] => 4
[[],[],[[],[]]] => [1,0,1,0,1,1,0,1,0,0] => 4
[[],[],[[[]]]] => [1,0,1,0,1,1,1,0,0,0] => 3
[[],[[]],[],[]] => [1,0,1,1,0,0,1,0,1,0] => 4
[[],[[]],[[]]] => [1,0,1,1,0,0,1,1,0,0] => 3
[[],[[],[]],[]] => [1,0,1,1,0,1,0,0,1,0] => 4
[[],[[[]]],[]] => [1,0,1,1,1,0,0,0,1,0] => 3
[[],[[],[],[]]] => [1,0,1,1,0,1,0,1,0,0] => 4
[[],[[],[[]]]] => [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] => 2
[[[]],[],[],[]] => [1,1,0,0,1,0,1,0,1,0] => 4
[[[]],[],[[]]] => [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] => 2
[[[],[]],[],[]] => [1,1,0,1,0,0,1,0,1,0] => 4
[[[[]]],[],[]] => [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] => 2
[[[],[],[]],[]] => [1,1,0,1,0,1,0,0,1,0] => 4
[[[],[[]]],[]] => [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] => 2
[[[],[],[],[]]] => [1,1,0,1,0,1,0,1,0,0] => 4
[[[],[],[[]]]] => [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] => 2
[[[[]],[],[]]] => [1,1,1,0,0,1,0,1,0,0] => 3
[[[[]],[[]]]] => [1,1,1,0,0,1,1,0,0,0] => 2
[[[[],[]],[]]] => [1,1,1,0,1,0,0,1,0,0] => 3
[[[[[]]],[]]] => [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] => 2
[[[[[]],[]]]] => [1,1,1,1,0,0,1,0,0,0] => 2
[[[[[],[]]]]] => [1,1,1,1,0,1,0,0,0,0] => 2
[[[[[[]]]]]] => [1,1,1,1,1,0,0,0,0,0] => 1
[[],[],[],[],[],[]] => [1,0,1,0,1,0,1,0,1,0,1,0] => 6
[[],[],[],[],[[]]] => [1,0,1,0,1,0,1,0,1,1,0,0] => 5
[[],[],[],[[]],[]] => [1,0,1,0,1,0,1,1,0,0,1,0] => 5
[[],[],[],[[],[]]] => [1,0,1,0,1,0,1,1,0,1,0,0] => 5
[[],[],[],[[[]]]] => [1,0,1,0,1,0,1,1,1,0,0,0] => 4
[[],[],[[]],[],[]] => [1,0,1,0,1,1,0,0,1,0,1,0] => 5
[[],[],[[]],[[]]] => [1,0,1,0,1,1,0,0,1,1,0,0] => 4
[[],[],[[],[]],[]] => [1,0,1,0,1,1,0,1,0,0,1,0] => 5
[[],[],[[[]]],[]] => [1,0,1,0,1,1,1,0,0,0,1,0] => 4
[[],[],[[],[],[]]] => [1,0,1,0,1,1,0,1,0,1,0,0] => 5
[[],[],[[],[[]]]] => [1,0,1,0,1,1,0,1,1,0,0,0] => 4
[[],[],[[[]],[]]] => [1,0,1,0,1,1,1,0,0,1,0,0] => 4
[[],[],[[[],[]]]] => [1,0,1,0,1,1,1,0,1,0,0,0] => 4
[[],[],[[[[]]]]] => [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] => 5
[[],[[]],[],[[]]] => [1,0,1,1,0,0,1,0,1,1,0,0] => 4
[[],[[]],[[]],[]] => [1,0,1,1,0,0,1,1,0,0,1,0] => 4
[[],[[]],[[],[]]] => [1,0,1,1,0,0,1,1,0,1,0,0] => 4
[[],[[]],[[[]]]] => [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] => 5
[[],[[[]]],[],[]] => [1,0,1,1,1,0,0,0,1,0,1,0] => 4
[[],[[],[]],[[]]] => [1,0,1,1,0,1,0,0,1,1,0,0] => 4
[[],[[[]]],[[]]] => [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] => 5
[[],[[],[[]]],[]] => [1,0,1,1,0,1,1,0,0,0,1,0] => 4
[[],[[[]],[]],[]] => [1,0,1,1,1,0,0,1,0,0,1,0] => 4
[[],[[[],[]]],[]] => [1,0,1,1,1,0,1,0,0,0,1,0] => 4
[[],[[[[]]]],[]] => [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] => 5
[[],[[],[],[[]]]] => [1,0,1,1,0,1,0,1,1,0,0,0] => 4
[[],[[],[[]],[]]] => [1,0,1,1,0,1,1,0,0,1,0,0] => 4
[[],[[],[[],[]]]] => [1,0,1,1,0,1,1,0,1,0,0,0] => 4
[[],[[],[[[]]]]] => [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] => 4
[[],[[[]],[[]]]] => [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] => 4
[[],[[[[]]],[]]] => [1,0,1,1,1,1,0,0,0,1,0,0] => 3
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Description
The number of peaks 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.
This sends the maximal height of the Dyck path to the depth of the tree.
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