Identifier
-
Mp00051:
Ordered trees
—to Dyck path⟶
Dyck paths
Mp00102: Dyck paths —rise composition⟶ Integer compositions
St001235: Integer compositions ⟶ ℤ
Values
[[]] => [1,0] => [1] => 1
[[],[]] => [1,0,1,0] => [1,1] => 2
[[[]]] => [1,1,0,0] => [2] => 1
[[],[],[]] => [1,0,1,0,1,0] => [1,1,1] => 3
[[],[[]]] => [1,0,1,1,0,0] => [1,2] => 2
[[[]],[]] => [1,1,0,0,1,0] => [2,1] => 2
[[[],[]]] => [1,1,0,1,0,0] => [2,1] => 2
[[[[]]]] => [1,1,1,0,0,0] => [3] => 1
[[],[],[],[]] => [1,0,1,0,1,0,1,0] => [1,1,1,1] => 4
[[],[],[[]]] => [1,0,1,0,1,1,0,0] => [1,1,2] => 3
[[],[[]],[]] => [1,0,1,1,0,0,1,0] => [1,2,1] => 2
[[],[[],[]]] => [1,0,1,1,0,1,0,0] => [1,2,1] => 2
[[],[[[]]]] => [1,0,1,1,1,0,0,0] => [1,3] => 2
[[[]],[],[]] => [1,1,0,0,1,0,1,0] => [2,1,1] => 3
[[[]],[[]]] => [1,1,0,0,1,1,0,0] => [2,2] => 2
[[[],[]],[]] => [1,1,0,1,0,0,1,0] => [2,1,1] => 3
[[[[]]],[]] => [1,1,1,0,0,0,1,0] => [3,1] => 2
[[[],[],[]]] => [1,1,0,1,0,1,0,0] => [2,1,1] => 3
[[[],[[]]]] => [1,1,0,1,1,0,0,0] => [2,2] => 2
[[[[]],[]]] => [1,1,1,0,0,1,0,0] => [3,1] => 2
[[[[],[]]]] => [1,1,1,0,1,0,0,0] => [3,1] => 2
[[[[[]]]]] => [1,1,1,1,0,0,0,0] => [4] => 1
[[],[],[],[],[]] => [1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1] => 5
[[],[],[],[[]]] => [1,0,1,0,1,0,1,1,0,0] => [1,1,1,2] => 4
[[],[],[[]],[]] => [1,0,1,0,1,1,0,0,1,0] => [1,1,2,1] => 3
[[],[],[[],[]]] => [1,0,1,0,1,1,0,1,0,0] => [1,1,2,1] => 3
[[],[],[[[]]]] => [1,0,1,0,1,1,1,0,0,0] => [1,1,3] => 3
[[],[[]],[],[]] => [1,0,1,1,0,0,1,0,1,0] => [1,2,1,1] => 3
[[],[[]],[[]]] => [1,0,1,1,0,0,1,1,0,0] => [1,2,2] => 2
[[],[[],[]],[]] => [1,0,1,1,0,1,0,0,1,0] => [1,2,1,1] => 3
[[],[[[]]],[]] => [1,0,1,1,1,0,0,0,1,0] => [1,3,1] => 2
[[],[[],[],[]]] => [1,0,1,1,0,1,0,1,0,0] => [1,2,1,1] => 3
[[],[[],[[]]]] => [1,0,1,1,0,1,1,0,0,0] => [1,2,2] => 2
[[],[[[]],[]]] => [1,0,1,1,1,0,0,1,0,0] => [1,3,1] => 2
[[],[[[],[]]]] => [1,0,1,1,1,0,1,0,0,0] => [1,3,1] => 2
[[],[[[[]]]]] => [1,0,1,1,1,1,0,0,0,0] => [1,4] => 2
[[[]],[],[],[]] => [1,1,0,0,1,0,1,0,1,0] => [2,1,1,1] => 4
[[[]],[],[[]]] => [1,1,0,0,1,0,1,1,0,0] => [2,1,2] => 3
[[[]],[[]],[]] => [1,1,0,0,1,1,0,0,1,0] => [2,2,1] => 2
[[[]],[[],[]]] => [1,1,0,0,1,1,0,1,0,0] => [2,2,1] => 2
[[[]],[[[]]]] => [1,1,0,0,1,1,1,0,0,0] => [2,3] => 2
[[[],[]],[],[]] => [1,1,0,1,0,0,1,0,1,0] => [2,1,1,1] => 4
[[[[]]],[],[]] => [1,1,1,0,0,0,1,0,1,0] => [3,1,1] => 3
[[[],[]],[[]]] => [1,1,0,1,0,0,1,1,0,0] => [2,1,2] => 3
[[[[]]],[[]]] => [1,1,1,0,0,0,1,1,0,0] => [3,2] => 2
[[[],[],[]],[]] => [1,1,0,1,0,1,0,0,1,0] => [2,1,1,1] => 4
[[[],[[]]],[]] => [1,1,0,1,1,0,0,0,1,0] => [2,2,1] => 2
[[[[]],[]],[]] => [1,1,1,0,0,1,0,0,1,0] => [3,1,1] => 3
[[[[],[]]],[]] => [1,1,1,0,1,0,0,0,1,0] => [3,1,1] => 3
[[[[[]]]],[]] => [1,1,1,1,0,0,0,0,1,0] => [4,1] => 2
[[[],[],[],[]]] => [1,1,0,1,0,1,0,1,0,0] => [2,1,1,1] => 4
[[[],[],[[]]]] => [1,1,0,1,0,1,1,0,0,0] => [2,1,2] => 3
[[[],[[]],[]]] => [1,1,0,1,1,0,0,1,0,0] => [2,2,1] => 2
[[[],[[],[]]]] => [1,1,0,1,1,0,1,0,0,0] => [2,2,1] => 2
[[[],[[[]]]]] => [1,1,0,1,1,1,0,0,0,0] => [2,3] => 2
[[[[]],[],[]]] => [1,1,1,0,0,1,0,1,0,0] => [3,1,1] => 3
[[[[]],[[]]]] => [1,1,1,0,0,1,1,0,0,0] => [3,2] => 2
[[[[],[]],[]]] => [1,1,1,0,1,0,0,1,0,0] => [3,1,1] => 3
[[[[[]]],[]]] => [1,1,1,1,0,0,0,1,0,0] => [4,1] => 2
[[[[],[],[]]]] => [1,1,1,0,1,0,1,0,0,0] => [3,1,1] => 3
[[[[],[[]]]]] => [1,1,1,0,1,1,0,0,0,0] => [3,2] => 2
[[[[[]],[]]]] => [1,1,1,1,0,0,1,0,0,0] => [4,1] => 2
[[[[[],[]]]]] => [1,1,1,1,0,1,0,0,0,0] => [4,1] => 2
[[[[[[]]]]]] => [1,1,1,1,1,0,0,0,0,0] => [5] => 1
[[],[],[],[],[],[]] => [1,0,1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,1] => 6
[[],[],[],[],[[]]] => [1,0,1,0,1,0,1,0,1,1,0,0] => [1,1,1,1,2] => 5
[[],[],[],[[]],[]] => [1,0,1,0,1,0,1,1,0,0,1,0] => [1,1,1,2,1] => 4
[[],[],[],[[],[]]] => [1,0,1,0,1,0,1,1,0,1,0,0] => [1,1,1,2,1] => 4
[[],[],[],[[[]]]] => [1,0,1,0,1,0,1,1,1,0,0,0] => [1,1,1,3] => 4
[[],[],[[]],[],[]] => [1,0,1,0,1,1,0,0,1,0,1,0] => [1,1,2,1,1] => 3
[[],[],[[]],[[]]] => [1,0,1,0,1,1,0,0,1,1,0,0] => [1,1,2,2] => 3
[[],[],[[],[]],[]] => [1,0,1,0,1,1,0,1,0,0,1,0] => [1,1,2,1,1] => 3
[[],[],[[[]]],[]] => [1,0,1,0,1,1,1,0,0,0,1,0] => [1,1,3,1] => 3
[[],[],[[],[],[]]] => [1,0,1,0,1,1,0,1,0,1,0,0] => [1,1,2,1,1] => 3
[[],[],[[],[[]]]] => [1,0,1,0,1,1,0,1,1,0,0,0] => [1,1,2,2] => 3
[[],[],[[[]],[]]] => [1,0,1,0,1,1,1,0,0,1,0,0] => [1,1,3,1] => 3
[[],[],[[[],[]]]] => [1,0,1,0,1,1,1,0,1,0,0,0] => [1,1,3,1] => 3
[[],[],[[[[]]]]] => [1,0,1,0,1,1,1,1,0,0,0,0] => [1,1,4] => 3
[[],[[]],[],[],[]] => [1,0,1,1,0,0,1,0,1,0,1,0] => [1,2,1,1,1] => 4
[[],[[]],[],[[]]] => [1,0,1,1,0,0,1,0,1,1,0,0] => [1,2,1,2] => 3
[[],[[]],[[]],[]] => [1,0,1,1,0,0,1,1,0,0,1,0] => [1,2,2,1] => 2
[[],[[]],[[],[]]] => [1,0,1,1,0,0,1,1,0,1,0,0] => [1,2,2,1] => 2
[[],[[]],[[[]]]] => [1,0,1,1,0,0,1,1,1,0,0,0] => [1,2,3] => 2
[[],[[],[]],[],[]] => [1,0,1,1,0,1,0,0,1,0,1,0] => [1,2,1,1,1] => 4
[[],[[[]]],[],[]] => [1,0,1,1,1,0,0,0,1,0,1,0] => [1,3,1,1] => 3
[[],[[],[]],[[]]] => [1,0,1,1,0,1,0,0,1,1,0,0] => [1,2,1,2] => 3
[[],[[[]]],[[]]] => [1,0,1,1,1,0,0,0,1,1,0,0] => [1,3,2] => 2
[[],[[],[],[]],[]] => [1,0,1,1,0,1,0,1,0,0,1,0] => [1,2,1,1,1] => 4
[[],[[],[[]]],[]] => [1,0,1,1,0,1,1,0,0,0,1,0] => [1,2,2,1] => 2
[[],[[[]],[]],[]] => [1,0,1,1,1,0,0,1,0,0,1,0] => [1,3,1,1] => 3
[[],[[[],[]]],[]] => [1,0,1,1,1,0,1,0,0,0,1,0] => [1,3,1,1] => 3
[[],[[[[]]]],[]] => [1,0,1,1,1,1,0,0,0,0,1,0] => [1,4,1] => 2
[[],[[],[],[],[]]] => [1,0,1,1,0,1,0,1,0,1,0,0] => [1,2,1,1,1] => 4
[[],[[],[],[[]]]] => [1,0,1,1,0,1,0,1,1,0,0,0] => [1,2,1,2] => 3
[[],[[],[[]],[]]] => [1,0,1,1,0,1,1,0,0,1,0,0] => [1,2,2,1] => 2
[[],[[],[[],[]]]] => [1,0,1,1,0,1,1,0,1,0,0,0] => [1,2,2,1] => 2
[[],[[],[[[]]]]] => [1,0,1,1,0,1,1,1,0,0,0,0] => [1,2,3] => 2
[[],[[[]],[],[]]] => [1,0,1,1,1,0,0,1,0,1,0,0] => [1,3,1,1] => 3
[[],[[[]],[[]]]] => [1,0,1,1,1,0,0,1,1,0,0,0] => [1,3,2] => 2
[[],[[[],[]],[]]] => [1,0,1,1,1,0,1,0,0,1,0,0] => [1,3,1,1] => 3
[[],[[[[]]],[]]] => [1,0,1,1,1,1,0,0,0,1,0,0] => [1,4,1] => 2
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Description
The global dimension of the corresponding Comp-Nakayama algebra.
We identify the composition [n1-1,n2-1,...,nr-1] with the Nakayama algebra with Kupisch series [n1,n1-1,...,2,n2,n2-1,...,2,...,nr,nr-1,...,3,2,1]. We call such Nakayama algebras with Kupisch series corresponding to a integer composition "Comp-Nakayama algebra".
We identify the composition [n1-1,n2-1,...,nr-1] with the Nakayama algebra with Kupisch series [n1,n1-1,...,2,n2,n2-1,...,2,...,nr,nr-1,...,3,2,1]. We call such Nakayama algebras with Kupisch series corresponding to a integer composition "Comp-Nakayama algebra".
Map
rise composition
Description
Send a Dyck path to the composition of sizes of its rises.
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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