Identifier
Values
[1] => [1] => [1,0] => [1,1,0,0] => 1
[1,1] => [1,1] => [1,0,1,0] => [1,1,0,1,0,0] => 2
[2] => [2] => [1,1,0,0] => [1,1,1,0,0,0] => 1
[1,1,1] => [1,1,1] => [1,0,1,0,1,0] => [1,1,0,1,0,1,0,0] => 3
[1,2] => [2,1] => [1,1,0,0,1,0] => [1,1,1,0,0,1,0,0] => 2
[2,1] => [1,2] => [1,0,1,1,0,0] => [1,1,0,1,1,0,0,0] => 2
[3] => [3] => [1,1,1,0,0,0] => [1,1,1,1,0,0,0,0] => 1
[1,1,1,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,0] => 4
[1,1,2] => [1,2,1] => [1,0,1,1,0,0,1,0] => [1,1,0,1,1,0,0,1,0,0] => 2
[1,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0] => [1,1,1,0,0,1,0,1,0,0] => 3
[1,3] => [3,1] => [1,1,1,0,0,0,1,0] => [1,1,1,1,0,0,0,1,0,0] => 2
[2,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0] => [1,1,0,1,0,1,1,0,0,0] => 3
[2,2] => [2,2] => [1,1,0,0,1,1,0,0] => [1,1,1,0,0,1,1,0,0,0] => 2
[3,1] => [1,3] => [1,0,1,1,1,0,0,0] => [1,1,0,1,1,1,0,0,0,0] => 2
[4] => [4] => [1,1,1,1,0,0,0,0] => [1,1,1,1,1,0,0,0,0,0] => 1
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Description
Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path.
The modified algebra B is obtained from the stable Auslander algebra kQ/I by deleting all relations which contain walks of length at least three (conjectural this step of deletion is not necessary as the stable higher Auslander algebras might be quadratic) and taking as B then the algebra kQ^(op)/J when J is the quadratic perp of the ideal I.
See www.findstat.org/DyckPaths/NakayamaAlgebras for the definition of Loewy length and Nakayama algebras associated to Dyck paths.
Map
prime Dyck path
Description
Return the Dyck path obtained by adding an initial up and a final down step.
Map
bounce path
Description
The bounce path determined by an integer composition.
Map
rotate front to back
Description
The front to back rotation of the entries of an integer composition.