Identifier
Values
[1] => [1,0] => [1,0] => 1
[2] => [1,0,1,0] => [1,1,0,0] => 1
[1,1] => [1,1,0,0] => [1,0,1,0] => 0
[3] => [1,0,1,0,1,0] => [1,1,1,0,0,0] => 1
[2,1] => [1,0,1,1,0,0] => [1,1,0,0,1,0] => 1
[1,1,1] => [1,1,0,1,0,0] => [1,0,1,0,1,0] => 0
[4] => [1,0,1,0,1,0,1,0] => [1,1,1,1,0,0,0,0] => 1
[3,1] => [1,0,1,0,1,1,0,0] => [1,1,1,0,0,0,1,0] => 1
[2,2] => [1,1,1,0,0,0] => [1,1,0,1,0,0] => 0
[2,1,1] => [1,0,1,1,0,1,0,0] => [1,1,0,0,1,0,1,0] => 1
[1,1,1,1] => [1,1,0,1,0,1,0,0] => [1,0,1,0,1,0,1,0] => 0
[5] => [1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,0,0,0,0,0] => 1
[4,1] => [1,0,1,0,1,0,1,1,0,0] => [1,1,1,1,0,0,0,0,1,0] => 1
[3,2] => [1,0,1,1,1,0,0,0] => [1,1,1,0,0,1,0,0] => 1
[3,1,1] => [1,0,1,0,1,1,0,1,0,0] => [1,1,1,0,0,0,1,0,1,0] => 1
[2,2,1] => [1,1,1,0,0,1,0,0] => [1,1,0,1,0,0,1,0] => 0
[2,1,1,1] => [1,0,1,1,0,1,0,1,0,0] => [1,1,0,0,1,0,1,0,1,0] => 1
[1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,0] => [1,0,1,0,1,0,1,0,1,0] => 0
[6] => [1,0,1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,1,0,0,0,0,0,0] => 1
[5,1] => [1,0,1,0,1,0,1,0,1,1,0,0] => [1,1,1,1,1,0,0,0,0,0,1,0] => 1
[4,2] => [1,0,1,0,1,1,1,0,0,0] => [1,1,1,1,0,0,0,1,0,0] => 1
[4,1,1] => [1,0,1,0,1,0,1,1,0,1,0,0] => [1,1,1,1,0,0,0,0,1,0,1,0] => 1
[3,3] => [1,1,1,0,1,0,0,0] => [1,1,0,1,0,1,0,0] => 0
[3,2,1] => [1,0,1,1,1,0,0,1,0,0] => [1,1,1,0,0,1,0,0,1,0] => 1
[3,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,0] => [1,1,1,0,0,0,1,0,1,0,1,0] => 1
[2,2,2] => [1,1,1,1,0,0,0,0] => [1,1,1,0,1,0,0,0] => 0
[2,2,1,1] => [1,1,1,0,0,1,0,1,0,0] => [1,1,0,1,0,0,1,0,1,0] => 0
[2,1,1,1,1] => [1,0,1,1,0,1,0,1,0,1,0,0] => [1,1,0,0,1,0,1,0,1,0,1,0] => 1
[1,1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,1,0,0] => [1,0,1,0,1,0,1,0,1,0,1,0] => 0
[5,2] => [1,0,1,0,1,0,1,1,1,0,0,0] => [1,1,1,1,1,0,0,0,0,1,0,0] => 1
[4,3] => [1,0,1,1,1,0,1,0,0,0] => [1,1,1,0,0,1,0,1,0,0] => 1
[4,2,1] => [1,0,1,0,1,1,1,0,0,1,0,0] => [1,1,1,1,0,0,0,1,0,0,1,0] => 1
[3,3,1] => [1,1,1,0,1,0,0,1,0,0] => [1,1,0,1,0,1,0,0,1,0] => 0
[3,2,2] => [1,0,1,1,1,1,0,0,0,0] => [1,1,1,1,0,0,1,0,0,0] => 1
[3,2,1,1] => [1,0,1,1,1,0,0,1,0,1,0,0] => [1,1,1,0,0,1,0,0,1,0,1,0] => 1
[2,2,2,1] => [1,1,1,1,0,0,0,1,0,0] => [1,1,1,0,1,0,0,0,1,0] => 0
[2,2,1,1,1] => [1,1,1,0,0,1,0,1,0,1,0,0] => [1,1,0,1,0,0,1,0,1,0,1,0] => 0
[5,3] => [1,0,1,0,1,1,1,0,1,0,0,0] => [1,1,1,1,0,0,0,1,0,1,0,0] => 1
[4,4] => [1,1,1,0,1,0,1,0,0,0] => [1,1,0,1,0,1,0,1,0,0] => 0
[4,3,1] => [1,0,1,1,1,0,1,0,0,1,0,0] => [1,1,1,0,0,1,0,1,0,0,1,0] => 1
[4,2,2] => [1,0,1,0,1,1,1,1,0,0,0,0] => [1,1,1,1,1,0,0,0,1,0,0,0] => 1
[3,3,2] => [1,1,1,0,1,1,0,0,0,0] => [1,1,0,1,1,0,1,0,0,0] => 0
[3,3,1,1] => [1,1,1,0,1,0,0,1,0,1,0,0] => [1,1,0,1,0,1,0,0,1,0,1,0] => 0
[3,2,2,1] => [1,0,1,1,1,1,0,0,0,1,0,0] => [1,1,1,1,0,0,1,0,0,0,1,0] => 1
[2,2,2,2] => [1,1,1,1,0,1,0,0,0,0] => [1,1,1,0,1,0,1,0,0,0] => 0
[2,2,2,1,1] => [1,1,1,1,0,0,0,1,0,1,0,0] => [1,1,1,0,1,0,0,0,1,0,1,0] => 0
[5,4] => [1,0,1,1,1,0,1,0,1,0,0,0] => [1,1,1,0,0,1,0,1,0,1,0,0] => 1
[4,4,1] => [1,1,1,0,1,0,1,0,0,1,0,0] => [1,1,0,1,0,1,0,1,0,0,1,0] => 0
[4,3,2] => [1,0,1,1,1,0,1,1,0,0,0,0] => [1,1,1,0,0,1,1,0,1,0,0,0] => 1
[3,3,3] => [1,1,1,1,1,0,0,0,0,0] => [1,1,1,1,0,1,0,0,0,0] => 0
[3,3,2,1] => [1,1,1,0,1,1,0,0,0,1,0,0] => [1,1,0,1,1,0,1,0,0,0,1,0] => 0
[3,2,2,2] => [1,0,1,1,1,1,0,1,0,0,0,0] => [1,1,1,1,0,0,1,0,1,0,0,0] => 1
[2,2,2,2,1] => [1,1,1,1,0,1,0,0,0,1,0,0] => [1,1,1,0,1,0,1,0,0,0,1,0] => 0
[5,5] => [1,1,1,0,1,0,1,0,1,0,0,0] => [1,1,0,1,0,1,0,1,0,1,0,0] => 0
[4,4,2] => [1,1,1,0,1,0,1,1,0,0,0,0] => [1,1,0,1,0,1,1,0,1,0,0,0] => 0
[4,3,3] => [1,0,1,1,1,1,1,0,0,0,0,0] => [1,1,1,1,1,0,0,1,0,0,0,0] => 1
[3,3,3,1] => [1,1,1,1,1,0,0,0,0,1,0,0] => [1,1,1,1,0,1,0,0,0,0,1,0] => 0
[3,3,2,2] => [1,1,1,0,1,1,0,1,0,0,0,0] => [1,1,0,1,1,0,1,0,1,0,0,0] => 0
[2,2,2,2,2] => [1,1,1,1,0,1,0,1,0,0,0,0] => [1,1,1,0,1,0,1,0,1,0,0,0] => 0
[4,4,3] => [1,1,1,0,1,1,1,0,0,0,0,0] => [1,1,0,1,1,1,0,1,0,0,0,0] => 0
[3,3,3,2] => [1,1,1,1,1,0,0,1,0,0,0,0] => [1,1,1,1,0,1,0,0,1,0,0,0] => 0
[4,4,4] => [1,1,1,1,1,0,1,0,0,0,0,0] => [1,1,1,1,0,1,0,1,0,0,0,0] => 0
[3,3,3,3] => [1,1,1,1,1,1,0,0,0,0,0,0] => [1,1,1,1,1,0,1,0,0,0,0,0] => 0
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Description
The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1.
Map
parallelogram polyomino
Description
Return the Dyck path corresponding to the partition interpreted as a parallogram polyomino.
The Ferrers diagram of an integer partition can be interpreted as a parallogram polyomino, such that each part corresponds to a column.
This map returns the corresponding 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.