Identifier
Values
[1] => [1,0] => [1,0] => 1
[1,1] => [1,0,1,0] => [1,1,0,0] => 2
[2] => [1,1,0,0] => [1,0,1,0] => 2
[1,1,1] => [1,0,1,0,1,0] => [1,1,1,0,0,0] => 3
[1,2] => [1,0,1,1,0,0] => [1,1,0,0,1,0] => 2
[2,1] => [1,1,0,0,1,0] => [1,0,1,1,0,0] => 2
[3] => [1,1,1,0,0,0] => [1,1,0,1,0,0] => 3
[1,1,1,1] => [1,0,1,0,1,0,1,0] => [1,1,1,1,0,0,0,0] => 4
[1,1,2] => [1,0,1,0,1,1,0,0] => [1,1,1,0,0,0,1,0] => 3
[1,2,1] => [1,0,1,1,0,0,1,0] => [1,1,0,0,1,1,0,0] => 2
[1,3] => [1,0,1,1,1,0,0,0] => [1,1,1,0,0,1,0,0] => 3
[2,1,1] => [1,1,0,0,1,0,1,0] => [1,0,1,1,1,0,0,0] => 3
[2,2] => [1,1,0,0,1,1,0,0] => [1,0,1,1,0,0,1,0] => 2
[3,1] => [1,1,1,0,0,0,1,0] => [1,1,0,1,1,0,0,0] => 3
[4] => [1,1,1,1,0,0,0,0] => [1,1,1,0,1,0,0,0] => 4
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,0,0,0,0,0] => 5
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0] => [1,1,1,1,0,0,0,0,1,0] => 4
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => [1,1,1,0,0,0,1,1,0,0] => 3
[1,1,3] => [1,0,1,0,1,1,1,0,0,0] => [1,1,1,1,0,0,0,1,0,0] => 4
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => [1,1,0,0,1,1,1,0,0,0] => 3
[1,2,2] => [1,0,1,1,0,0,1,1,0,0] => [1,1,0,0,1,1,0,0,1,0] => 2
[1,3,1] => [1,0,1,1,1,0,0,0,1,0] => [1,1,1,0,0,1,1,0,0,0] => 3
[1,4] => [1,0,1,1,1,1,0,0,0,0] => [1,1,1,1,0,0,1,0,0,0] => 4
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => [1,0,1,1,1,1,0,0,0,0] => 4
[2,1,2] => [1,1,0,0,1,0,1,1,0,0] => [1,0,1,1,1,0,0,0,1,0] => 3
[2,2,1] => [1,1,0,0,1,1,0,0,1,0] => [1,0,1,1,0,0,1,1,0,0] => 2
[2,3] => [1,1,0,0,1,1,1,0,0,0] => [1,0,1,1,1,0,0,1,0,0] => 3
[3,1,1] => [1,1,1,0,0,0,1,0,1,0] => [1,1,0,1,1,1,0,0,0,0] => 4
[3,2] => [1,1,1,0,0,0,1,1,0,0] => [1,1,0,1,1,0,0,0,1,0] => 3
[4,1] => [1,1,1,1,0,0,0,0,1,0] => [1,1,1,0,1,1,0,0,0,0] => 4
[5] => [1,1,1,1,1,0,0,0,0,0] => [1,1,1,1,0,1,0,0,0,0] => 5
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Description
The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path.
Map
peaks-to-valleys
Description
Return the path that has a valley wherever the original path has a peak of height at least one.
More precisely, the height of a valley in the image is the height of the corresponding peak minus $2$.
This is also (the inverse of) rowmotion on Dyck paths regarded as order ideals in the triangular poset.
Map
bounce path
Description
The bounce path determined by an integer composition.