Identifier
Values
[[]] => [1,0] => 0
[[],[]] => [1,0,1,0] => 0
[[[]]] => [1,1,0,0] => 1
[[],[],[]] => [1,0,1,0,1,0] => 0
[[],[[]]] => [1,0,1,1,0,0] => 1
[[[]],[]] => [1,1,0,0,1,0] => 1
[[[],[]]] => [1,1,0,1,0,0] => 2
[[[[]]]] => [1,1,1,0,0,0] => 1
[[],[],[],[]] => [1,0,1,0,1,0,1,0] => 0
[[],[],[[]]] => [1,0,1,0,1,1,0,0] => 1
[[],[[]],[]] => [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] => 1
[[[]],[],[]] => [1,1,0,0,1,0,1,0] => 1
[[[]],[[]]] => [1,1,0,0,1,1,0,0] => 2
[[[],[]],[]] => [1,1,0,1,0,0,1,0] => 2
[[[[]]],[]] => [1,1,1,0,0,0,1,0] => 1
[[[],[],[]]] => [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] => 1
[[[[[]]]]] => [1,1,1,1,0,0,0,0] => 1
[[],[],[],[],[]] => [1,0,1,0,1,0,1,0,1,0] => 0
[[],[],[],[[]]] => [1,0,1,0,1,0,1,1,0,0] => 1
[[],[],[[]],[]] => [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] => 1
[[],[[]],[],[]] => [1,0,1,1,0,0,1,0,1,0] => 1
[[],[[]],[[]]] => [1,0,1,1,0,0,1,1,0,0] => 2
[[],[[],[]],[]] => [1,0,1,1,0,1,0,0,1,0] => 2
[[],[[[]]],[]] => [1,0,1,1,1,0,0,0,1,0] => 1
[[],[[],[],[]]] => [1,0,1,1,0,1,0,1,0,0] => 3
[[],[[],[[]]]] => [1,0,1,1,0,1,1,0,0,0] => 2
[[],[[[]],[]]] => [1,0,1,1,1,0,0,1,0,0] => 2
[[],[[[],[]]]] => [1,0,1,1,1,0,1,0,0,0] => 1
[[],[[[[]]]]] => [1,0,1,1,1,1,0,0,0,0] => 1
[[[]],[],[],[]] => [1,1,0,0,1,0,1,0,1,0] => 1
[[[]],[],[[]]] => [1,1,0,0,1,0,1,1,0,0] => 2
[[[]],[[]],[]] => [1,1,0,0,1,1,0,0,1,0] => 2
[[[]],[[],[]]] => [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] => 2
[[[[]]],[],[]] => [1,1,1,0,0,0,1,0,1,0] => 1
[[[],[]],[[]]] => [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] => 3
[[[],[[]]],[]] => [1,1,0,1,1,0,0,0,1,0] => 2
[[[[]],[]],[]] => [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] => 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] => 2
[[[],[[[]]]]] => [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] => 2
[[[[[]]],[]]] => [1,1,1,1,0,0,0,1,0,0] => 2
[[[[],[],[]]]] => [1,1,1,0,1,0,1,0,0,0] => 1
[[[[],[[]]]]] => [1,1,1,0,1,1,0,0,0,0] => 1
[[[[[]],[]]]] => [1,1,1,1,0,0,1,0,0,0] => 1
[[[[[],[]]]]] => [1,1,1,1,0,1,0,0,0,0] => 1
[[[[[[]]]]]] => [1,1,1,1,1,0,0,0,0,0] => 1
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Description
The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path.
The statistic returns zero in case that bimodule is the zero module.
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.