Identifier
Values
[1] => [1,0] => [1,0] => [1,0] => 1
[2] => [1,0,1,0] => [1,0,1,0] => [1,1,0,0] => 2
[1,1] => [1,1,0,0] => [1,1,0,0] => [1,0,1,0] => 2
[3] => [1,0,1,0,1,0] => [1,0,1,0,1,0] => [1,1,1,0,0,0] => 2
[2,1] => [1,0,1,1,0,0] => [1,1,0,1,0,0] => [1,0,1,0,1,0] => 3
[1,1,1] => [1,1,0,1,0,0] => [1,0,1,1,0,0] => [1,1,0,0,1,0] => 3
[4] => [1,0,1,0,1,0,1,0] => [1,0,1,0,1,0,1,0] => [1,1,1,1,0,0,0,0] => 2
[3,1] => [1,0,1,0,1,1,0,0] => [1,1,0,1,0,1,0,0] => [1,0,1,0,1,0,1,0] => 4
[2,2] => [1,1,1,0,0,0] => [1,1,1,0,0,0] => [1,1,0,1,0,0] => 2
[2,1,1] => [1,0,1,1,0,1,0,0] => [1,0,1,1,0,1,0,0] => [1,1,0,0,1,0,1,0] => 4
[1,1,1,1] => [1,1,0,1,0,1,0,0] => [1,0,1,0,1,1,0,0] => [1,1,1,0,0,0,1,0] => 3
[5] => [1,0,1,0,1,0,1,0,1,0] => [1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,0,0,0,0,0] => 2
[4,1] => [1,0,1,0,1,0,1,1,0,0] => [1,1,0,1,0,1,0,1,0,0] => [1,0,1,0,1,0,1,0,1,0] => 5
[3,2] => [1,0,1,1,1,0,0,0] => [1,1,1,0,1,0,0,0] => [1,1,0,1,0,1,0,0] => 3
[3,1,1] => [1,0,1,0,1,1,0,1,0,0] => [1,0,1,1,0,1,0,1,0,0] => [1,1,0,0,1,0,1,0,1,0] => 5
[2,2,1] => [1,1,1,0,0,1,0,0] => [1,1,0,0,1,1,0,0] => [1,0,1,1,0,0,1,0] => 4
[2,1,1,1] => [1,0,1,1,0,1,0,1,0,0] => [1,0,1,0,1,1,0,1,0,0] => [1,1,1,0,0,0,1,0,1,0] => 4
[1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,0] => [1,0,1,0,1,0,1,1,0,0] => [1,1,1,1,0,0,0,0,1,0] => 3
[6] => [1,0,1,0,1,0,1,0,1,0,1,0] => [1,0,1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,1,0,0,0,0,0,0] => 2
[5,1] => [1,0,1,0,1,0,1,0,1,1,0,0] => [1,1,0,1,0,1,0,1,0,1,0,0] => [1,0,1,0,1,0,1,0,1,0,1,0] => 6
[4,2] => [1,0,1,0,1,1,1,0,0,0] => [1,1,1,0,1,0,1,0,0,0] => [1,1,0,1,0,1,0,1,0,0] => 4
[4,1,1] => [1,0,1,0,1,0,1,1,0,1,0,0] => [1,0,1,1,0,1,0,1,0,1,0,0] => [1,1,0,0,1,0,1,0,1,0,1,0] => 6
[3,3] => [1,1,1,0,1,0,0,0] => [1,1,0,1,1,0,0,0] => [1,0,1,1,0,1,0,0] => 3
[3,2,1] => [1,0,1,1,1,0,0,1,0,0] => [1,1,0,1,1,0,0,1,0,0] => [1,0,1,1,0,1,0,0,1,0] => 4
[3,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,0] => [1,0,1,0,1,1,0,1,0,1,0,0] => [1,1,1,0,0,0,1,0,1,0,1,0] => 5
[2,2,2] => [1,1,1,1,0,0,0,0] => [1,1,1,1,0,0,0,0] => [1,1,1,0,1,0,0,0] => 2
[2,2,1,1] => [1,1,1,0,0,1,0,1,0,0] => [1,1,0,0,1,0,1,1,0,0] => [1,0,1,1,1,0,0,0,1,0] => 4
[2,1,1,1,1] => [1,0,1,1,0,1,0,1,0,1,0,0] => [1,0,1,0,1,0,1,1,0,1,0,0] => [1,1,1,1,0,0,0,0,1,0,1,0] => 4
[1,1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,1,0,0] => [1,0,1,0,1,0,1,0,1,1,0,0] => [1,1,1,1,1,0,0,0,0,0,1,0] => 3
[5,2] => [1,0,1,0,1,0,1,1,1,0,0,0] => [1,1,1,0,1,0,1,0,1,0,0,0] => [1,1,0,1,0,1,0,1,0,1,0,0] => 4
[4,3] => [1,0,1,1,1,0,1,0,0,0] => [1,1,0,1,1,0,1,0,0,0] => [1,0,1,1,0,1,0,1,0,0] => 4
[4,2,1] => [1,0,1,0,1,1,1,0,0,1,0,0] => [1,1,0,1,1,0,1,0,0,1,0,0] => [1,0,1,1,0,1,0,1,0,0,1,0] => 5
[3,3,1] => [1,1,1,0,1,0,0,1,0,0] => [1,1,0,1,0,0,1,1,0,0] => [1,0,1,0,1,1,0,0,1,0] => 5
[3,2,2] => [1,0,1,1,1,1,0,0,0,0] => [1,1,1,1,0,1,0,0,0,0] => [1,1,1,0,1,0,1,0,0,0] => 3
[3,2,1,1] => [1,0,1,1,1,0,0,1,0,1,0,0] => [1,1,0,1,0,0,1,1,0,1,0,0] => [1,0,1,0,1,1,0,0,1,0,1,0] => 6
[2,2,2,1] => [1,1,1,1,0,0,0,1,0,0] => [1,1,1,0,0,0,1,1,0,0] => [1,1,0,1,1,0,0,0,1,0] => 4
[2,2,1,1,1] => [1,1,1,0,0,1,0,1,0,1,0,0] => [1,1,0,0,1,0,1,0,1,1,0,0] => [1,0,1,1,1,1,0,0,0,0,1,0] => 4
[5,3] => [1,0,1,0,1,1,1,0,1,0,0,0] => [1,1,0,1,1,0,1,0,1,0,0,0] => [1,0,1,1,0,1,0,1,0,1,0,0] => 5
[4,4] => [1,1,1,0,1,0,1,0,0,0] => [1,1,0,1,0,1,1,0,0,0] => [1,0,1,0,1,1,0,1,0,0] => 4
[4,3,1] => [1,0,1,1,1,0,1,0,0,1,0,0] => [1,1,0,1,0,1,1,0,0,1,0,0] => [1,0,1,0,1,1,0,1,0,0,1,0] => 5
[4,2,2] => [1,0,1,0,1,1,1,1,0,0,0,0] => [1,1,1,1,0,1,0,1,0,0,0,0] => [1,1,1,0,1,0,1,0,1,0,0,0] => 4
[3,3,2] => [1,1,1,0,1,1,0,0,0,0] => [1,1,0,1,1,1,0,0,0,0] => [1,0,1,1,1,0,1,0,0,0] => 3
[3,3,1,1] => [1,1,1,0,1,0,0,1,0,1,0,0] => [1,1,0,1,0,0,1,0,1,1,0,0] => [1,0,1,0,1,1,1,0,0,0,1,0] => 5
[3,2,2,1] => [1,0,1,1,1,1,0,0,0,1,0,0] => [1,1,1,0,1,1,0,0,0,1,0,0] => [1,1,0,1,1,0,1,0,0,0,1,0] => 4
[2,2,2,2] => [1,1,1,1,0,1,0,0,0,0] => [1,1,1,0,1,1,0,0,0,0] => [1,1,0,1,1,0,1,0,0,0] => 3
[2,2,2,1,1] => [1,1,1,1,0,0,0,1,0,1,0,0] => [1,1,1,0,0,0,1,0,1,1,0,0] => [1,1,0,1,1,1,0,0,0,0,1,0] => 4
[5,4] => [1,0,1,1,1,0,1,0,1,0,0,0] => [1,1,0,1,0,1,1,0,1,0,0,0] => [1,0,1,0,1,1,0,1,0,1,0,0] => 5
[4,4,1] => [1,1,1,0,1,0,1,0,0,1,0,0] => [1,1,0,1,0,1,0,0,1,1,0,0] => [1,0,1,0,1,0,1,1,0,0,1,0] => 6
[4,3,2] => [1,0,1,1,1,0,1,1,0,0,0,0] => [1,1,0,1,1,1,0,1,0,0,0,0] => [1,0,1,1,1,0,1,0,1,0,0,0] => 4
[3,3,3] => [1,1,1,1,1,0,0,0,0,0] => [1,1,1,1,1,0,0,0,0,0] => [1,1,1,1,0,1,0,0,0,0] => 2
[3,3,2,1] => [1,1,1,0,1,1,0,0,0,1,0,0] => [1,1,0,1,1,0,0,0,1,1,0,0] => [1,0,1,1,0,1,1,0,0,0,1,0] => 5
[3,2,2,2] => [1,0,1,1,1,1,0,1,0,0,0,0] => [1,1,1,0,1,1,0,1,0,0,0,0] => [1,1,0,1,1,0,1,0,1,0,0,0] => 4
[2,2,2,2,1] => [1,1,1,1,0,1,0,0,0,1,0,0] => [1,1,1,0,1,0,0,0,1,1,0,0] => [1,1,0,1,0,1,1,0,0,0,1,0] => 5
[5,5] => [1,1,1,0,1,0,1,0,1,0,0,0] => [1,1,0,1,0,1,0,1,1,0,0,0] => [1,0,1,0,1,0,1,1,0,1,0,0] => 5
[4,4,2] => [1,1,1,0,1,0,1,1,0,0,0,0] => [1,1,0,1,0,1,1,1,0,0,0,0] => [1,0,1,0,1,1,1,0,1,0,0,0] => 4
[4,3,3] => [1,0,1,1,1,1,1,0,0,0,0,0] => [1,1,1,1,1,0,1,0,0,0,0,0] => [1,1,1,1,0,1,0,1,0,0,0,0] => 3
[3,3,3,1] => [1,1,1,1,1,0,0,0,0,1,0,0] => [1,1,1,1,0,0,0,0,1,1,0,0] => [1,1,1,0,1,1,0,0,0,0,1,0] => 4
[3,3,2,2] => [1,1,1,0,1,1,0,1,0,0,0,0] => [1,1,0,1,1,0,1,1,0,0,0,0] => [1,0,1,1,0,1,1,0,1,0,0,0] => 4
[2,2,2,2,2] => [1,1,1,1,0,1,0,1,0,0,0,0] => [1,1,1,0,1,0,1,1,0,0,0,0] => [1,1,0,1,0,1,1,0,1,0,0,0] => 4
[4,4,3] => [1,1,1,0,1,1,1,0,0,0,0,0] => [1,1,0,1,1,1,1,0,0,0,0,0] => [1,0,1,1,1,1,0,1,0,0,0,0] => 3
[3,3,3,2] => [1,1,1,1,1,0,0,1,0,0,0,0] => [1,1,1,1,0,0,1,1,0,0,0,0] => [1,1,1,0,1,1,0,0,1,0,0,0] => 4
[4,4,4] => [1,1,1,1,1,0,1,0,0,0,0,0] => [1,1,1,1,0,1,1,0,0,0,0,0] => [1,1,1,0,1,1,0,1,0,0,0,0] => 3
[3,3,3,3] => [1,1,1,1,1,1,0,0,0,0,0,0] => [1,1,1,1,1,1,0,0,0,0,0,0] => [1,1,1,1,1,0,1,0,0,0,0,0] => 2
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Description
Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra.
For at most 6 simple modules this statistic coincides with the injective dimension of the regular module as a bimodule.
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.
Map
Adin-Bagno-Roichman transformation
Description
The Adin-Bagno-Roichman transformation of a Dyck path.
This is a bijection preserving the number of up steps before each peak and sending the number of returns to the number of up steps after the last double up step.