Identifier
-
Mp00051:
Ordered trees
—to Dyck path⟶
Dyck paths
St001226: Dyck paths ⟶ ℤ
Values
[[]] => [1,0] => 2
[[],[]] => [1,0,1,0] => 2
[[[]]] => [1,1,0,0] => 3
[[],[],[]] => [1,0,1,0,1,0] => 2
[[],[[]]] => [1,0,1,1,0,0] => 3
[[[]],[]] => [1,1,0,0,1,0] => 3
[[[],[]]] => [1,1,0,1,0,0] => 2
[[[[]]]] => [1,1,1,0,0,0] => 4
[[],[],[],[]] => [1,0,1,0,1,0,1,0] => 2
[[],[],[[]]] => [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] => 2
[[],[[[]]]] => [1,0,1,1,1,0,0,0] => 4
[[[]],[],[]] => [1,1,0,0,1,0,1,0] => 3
[[[]],[[]]] => [1,1,0,0,1,1,0,0] => 4
[[[],[]],[]] => [1,1,0,1,0,0,1,0] => 2
[[[[]]],[]] => [1,1,1,0,0,0,1,0] => 4
[[[],[],[]]] => [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] => 2
[[[[[]]]]] => [1,1,1,1,0,0,0,0] => 5
[[],[],[],[],[]] => [1,0,1,0,1,0,1,0,1,0] => 2
[[],[],[],[[]]] => [1,0,1,0,1,0,1,1,0,0] => 3
[[],[],[[]],[]] => [1,0,1,0,1,1,0,0,1,0] => 3
[[],[],[[],[]]] => [1,0,1,0,1,1,0,1,0,0] => 2
[[],[],[[[]]]] => [1,0,1,0,1,1,1,0,0,0] => 4
[[],[[]],[],[]] => [1,0,1,1,0,0,1,0,1,0] => 3
[[],[[]],[[]]] => [1,0,1,1,0,0,1,1,0,0] => 4
[[],[[],[]],[]] => [1,0,1,1,0,1,0,0,1,0] => 2
[[],[[[]]],[]] => [1,0,1,1,1,0,0,0,1,0] => 4
[[],[[],[],[]]] => [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] => 2
[[],[[[[]]]]] => [1,0,1,1,1,1,0,0,0,0] => 5
[[[]],[],[],[]] => [1,1,0,0,1,0,1,0,1,0] => 3
[[[]],[],[[]]] => [1,1,0,0,1,0,1,1,0,0] => 4
[[[]],[[]],[]] => [1,1,0,0,1,1,0,0,1,0] => 4
[[[]],[[],[]]] => [1,1,0,0,1,1,0,1,0,0] => 3
[[[]],[[[]]]] => [1,1,0,0,1,1,1,0,0,0] => 5
[[[],[]],[],[]] => [1,1,0,1,0,0,1,0,1,0] => 2
[[[[]]],[],[]] => [1,1,1,0,0,0,1,0,1,0] => 4
[[[],[]],[[]]] => [1,1,0,1,0,0,1,1,0,0] => 3
[[[[]]],[[]]] => [1,1,1,0,0,0,1,1,0,0] => 5
[[[],[],[]],[]] => [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] => 2
[[[[[]]]],[]] => [1,1,1,1,0,0,0,0,1,0] => 5
[[[],[],[],[]]] => [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] => 2
[[[],[[],[]]]] => [1,1,0,1,1,0,1,0,0,0] => 2
[[[],[[[]]]]] => [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] => 2
[[[[[]]],[]]] => [1,1,1,1,0,0,0,1,0,0] => 4
[[[[],[],[]]]] => [1,1,1,0,1,0,1,0,0,0] => 2
[[[[],[[]]]]] => [1,1,1,0,1,1,0,0,0,0] => 3
[[[[[]],[]]]] => [1,1,1,1,0,0,1,0,0,0] => 3
[[[[[],[]]]]] => [1,1,1,1,0,1,0,0,0,0] => 2
[[[[[[]]]]]] => [1,1,1,1,1,0,0,0,0,0] => 6
[[],[],[],[],[],[]] => [1,0,1,0,1,0,1,0,1,0,1,0] => 2
[[],[],[],[],[[]]] => [1,0,1,0,1,0,1,0,1,1,0,0] => 3
[[],[],[],[[]],[]] => [1,0,1,0,1,0,1,1,0,0,1,0] => 3
[[],[],[],[[],[]]] => [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] => 4
[[],[],[[]],[],[]] => [1,0,1,0,1,1,0,0,1,0,1,0] => 3
[[],[],[[]],[[]]] => [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] => 2
[[],[],[[[]]],[]] => [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] => 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] => 2
[[],[],[[[[]]]]] => [1,0,1,0,1,1,1,1,0,0,0,0] => 5
[[],[[]],[],[],[]] => [1,0,1,1,0,0,1,0,1,0,1,0] => 3
[[],[[]],[],[[]]] => [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] => 3
[[],[[]],[[[]]]] => [1,0,1,1,0,0,1,1,1,0,0,0] => 5
[[],[[],[]],[],[]] => [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] => 4
[[],[[],[]],[[]]] => [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] => 5
[[],[[],[],[]],[]] => [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] => 2
[[],[[[[]]]],[]] => [1,0,1,1,1,1,0,0,0,0,1,0] => 5
[[],[[],[],[],[]]] => [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] => 2
[[],[[],[[],[]]]] => [1,0,1,1,0,1,1,0,1,0,0,0] => 2
[[],[[],[[[]]]]] => [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] => 2
[[],[[[[]]],[]]] => [1,0,1,1,1,1,0,0,0,1,0,0] => 4
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Description
The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra.
That is the number of i such that $Ext_A^1(J,e_i J)=0$.
That is the number of i such that $Ext_A^1(J,e_i J)=0$.
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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