Identifier
Values
[1,0] => [1,0] => [1,1,0,0] => 1
[1,0,1,0] => [1,1,0,0] => [1,1,1,0,0,0] => 1
[1,1,0,0] => [1,0,1,0] => [1,1,0,1,0,0] => 4
[1,0,1,0,1,0] => [1,1,1,0,0,0] => [1,1,1,1,0,0,0,0] => 1
[1,0,1,1,0,0] => [1,1,0,0,1,0] => [1,1,1,0,0,1,0,0] => 4
[1,1,0,0,1,0] => [1,0,1,1,0,0] => [1,1,0,1,1,0,0,0] => 4
[1,1,0,1,0,0] => [1,0,1,0,1,0] => [1,1,0,1,0,1,0,0] => 5
[1,1,1,0,0,0] => [1,1,0,1,0,0] => [1,1,1,0,1,0,0,0] => 5
[1,0,1,0,1,0,1,0] => [1,1,1,1,0,0,0,0] => [1,1,1,1,1,0,0,0,0,0] => 1
[1,0,1,0,1,1,0,0] => [1,1,1,0,0,0,1,0] => [1,1,1,1,0,0,0,1,0,0] => 4
[1,0,1,1,0,0,1,0] => [1,1,0,0,1,1,0,0] => [1,1,1,0,0,1,1,0,0,0] => 4
[1,0,1,1,0,1,0,0] => [1,1,0,0,1,0,1,0] => [1,1,1,0,0,1,0,1,0,0] => 5
[1,0,1,1,1,0,0,0] => [1,1,1,0,0,1,0,0] => [1,1,1,1,0,0,1,0,0,0] => 5
[1,1,0,0,1,0,1,0] => [1,0,1,1,1,0,0,0] => [1,1,0,1,1,1,0,0,0,0] => 4
[1,1,0,0,1,1,0,0] => [1,0,1,1,0,0,1,0] => [1,1,0,1,1,0,0,1,0,0] => 6
[1,1,0,1,0,0,1,0] => [1,0,1,0,1,1,0,0] => [1,1,0,1,0,1,1,0,0,0] => 5
[1,1,0,1,0,1,0,0] => [1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,0] => 6
[1,1,0,1,1,0,0,0] => [1,0,1,1,0,1,0,0] => [1,1,0,1,1,0,1,0,0,0] => 6
[1,1,1,0,0,0,1,0] => [1,1,0,1,1,0,0,0] => [1,1,1,0,1,1,0,0,0,0] => 5
[1,1,1,0,0,1,0,0] => [1,1,0,1,0,0,1,0] => [1,1,1,0,1,0,0,1,0,0] => 6
[1,1,1,0,1,0,0,0] => [1,1,0,1,0,1,0,0] => [1,1,1,0,1,0,1,0,0,0] => 6
[1,1,1,1,0,0,0,0] => [1,1,1,0,1,0,0,0] => [1,1,1,1,0,1,0,0,0,0] => 6
[1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,0,0,0,0,0] => [1,1,1,1,1,1,0,0,0,0,0,0] => 1
[1,0,1,0,1,0,1,1,0,0] => [1,1,1,1,0,0,0,0,1,0] => [1,1,1,1,1,0,0,0,0,1,0,0] => 4
[1,0,1,0,1,1,0,0,1,0] => [1,1,1,0,0,0,1,1,0,0] => [1,1,1,1,0,0,0,1,1,0,0,0] => 4
[1,0,1,0,1,1,0,1,0,0] => [1,1,1,0,0,0,1,0,1,0] => [1,1,1,1,0,0,0,1,0,1,0,0] => 5
[1,0,1,0,1,1,1,0,0,0] => [1,1,1,1,0,0,0,1,0,0] => [1,1,1,1,1,0,0,0,1,0,0,0] => 5
[1,0,1,1,0,0,1,0,1,0] => [1,1,0,0,1,1,1,0,0,0] => [1,1,1,0,0,1,1,1,0,0,0,0] => 4
[1,0,1,1,0,0,1,1,0,0] => [1,1,0,0,1,1,0,0,1,0] => [1,1,1,0,0,1,1,0,0,1,0,0] => 6
[1,0,1,1,0,1,0,0,1,0] => [1,1,0,0,1,0,1,1,0,0] => [1,1,1,0,0,1,0,1,1,0,0,0] => 5
[1,0,1,1,0,1,0,1,0,0] => [1,1,0,0,1,0,1,0,1,0] => [1,1,1,0,0,1,0,1,0,1,0,0] => 6
[1,0,1,1,0,1,1,0,0,0] => [1,1,0,0,1,1,0,1,0,0] => [1,1,1,0,0,1,1,0,1,0,0,0] => 6
[1,0,1,1,1,0,0,0,1,0] => [1,1,1,0,0,1,1,0,0,0] => [1,1,1,1,0,0,1,1,0,0,0,0] => 5
[1,0,1,1,1,0,0,1,0,0] => [1,1,1,0,0,1,0,0,1,0] => [1,1,1,1,0,0,1,0,0,1,0,0] => 6
[1,0,1,1,1,0,1,0,0,0] => [1,1,1,0,0,1,0,1,0,0] => [1,1,1,1,0,0,1,0,1,0,0,0] => 6
[1,0,1,1,1,1,0,0,0,0] => [1,1,1,1,0,0,1,0,0,0] => [1,1,1,1,1,0,0,1,0,0,0,0] => 6
[1,1,0,0,1,0,1,0,1,0] => [1,0,1,1,1,1,0,0,0,0] => [1,1,0,1,1,1,1,0,0,0,0,0] => 4
[1,1,0,0,1,0,1,1,0,0] => [1,0,1,1,1,0,0,0,1,0] => [1,1,0,1,1,1,0,0,0,1,0,0] => 7
[1,1,0,0,1,1,0,0,1,0] => [1,0,1,1,0,0,1,1,0,0] => [1,1,0,1,1,0,0,1,1,0,0,0] => 6
[1,1,0,0,1,1,0,1,0,0] => [1,0,1,1,0,0,1,0,1,0] => [1,1,0,1,1,0,0,1,0,1,0,0] => 7
[1,1,0,0,1,1,1,0,0,0] => [1,0,1,1,1,0,0,1,0,0] => [1,1,0,1,1,1,0,0,1,0,0,0] => 7
[1,1,0,1,0,0,1,0,1,0] => [1,0,1,0,1,1,1,0,0,0] => [1,1,0,1,0,1,1,1,0,0,0,0] => 5
[1,1,0,1,0,0,1,1,0,0] => [1,0,1,0,1,1,0,0,1,0] => [1,1,0,1,0,1,1,0,0,1,0,0] => 7
[1,1,0,1,0,1,0,0,1,0] => [1,0,1,0,1,0,1,1,0,0] => [1,1,0,1,0,1,0,1,1,0,0,0] => 6
[1,1,0,1,0,1,0,1,0,0] => [1,0,1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,1,0,0] => 7
[1,1,0,1,0,1,1,0,0,0] => [1,0,1,0,1,1,0,1,0,0] => [1,1,0,1,0,1,1,0,1,0,0,0] => 7
[1,1,0,1,1,0,0,0,1,0] => [1,0,1,1,0,1,1,0,0,0] => [1,1,0,1,1,0,1,1,0,0,0,0] => 6
[1,1,0,1,1,0,0,1,0,0] => [1,0,1,1,0,1,0,0,1,0] => [1,1,0,1,1,0,1,0,0,1,0,0] => 7
[1,1,0,1,1,0,1,0,0,0] => [1,0,1,1,0,1,0,1,0,0] => [1,1,0,1,1,0,1,0,1,0,0,0] => 7
[1,1,0,1,1,1,0,0,0,0] => [1,0,1,1,1,0,1,0,0,0] => [1,1,0,1,1,1,0,1,0,0,0,0] => 7
[1,1,1,0,0,0,1,0,1,0] => [1,1,0,1,1,1,0,0,0,0] => [1,1,1,0,1,1,1,0,0,0,0,0] => 5
[1,1,1,0,0,0,1,1,0,0] => [1,1,0,1,1,0,0,0,1,0] => [1,1,1,0,1,1,0,0,0,1,0,0] => 7
[1,1,1,0,0,1,0,0,1,0] => [1,1,0,1,0,0,1,1,0,0] => [1,1,1,0,1,0,0,1,1,0,0,0] => 6
[1,1,1,0,0,1,0,1,0,0] => [1,1,0,1,0,0,1,0,1,0] => [1,1,1,0,1,0,0,1,0,1,0,0] => 7
[1,1,1,0,0,1,1,0,0,0] => [1,1,0,1,1,0,0,1,0,0] => [1,1,1,0,1,1,0,0,1,0,0,0] => 7
[1,1,1,0,1,0,0,0,1,0] => [1,1,0,1,0,1,1,0,0,0] => [1,1,1,0,1,0,1,1,0,0,0,0] => 6
[1,1,1,0,1,0,0,1,0,0] => [1,1,0,1,0,1,0,0,1,0] => [1,1,1,0,1,0,1,0,0,1,0,0] => 7
[1,1,1,0,1,0,1,0,0,0] => [1,1,0,1,0,1,0,1,0,0] => [1,1,1,0,1,0,1,0,1,0,0,0] => 7
[1,1,1,0,1,1,0,0,0,0] => [1,1,0,1,1,0,1,0,0,0] => [1,1,1,0,1,1,0,1,0,0,0,0] => 7
[1,1,1,1,0,0,0,0,1,0] => [1,1,1,0,1,1,0,0,0,0] => [1,1,1,1,0,1,1,0,0,0,0,0] => 6
[1,1,1,1,0,0,0,1,0,0] => [1,1,1,0,1,0,0,0,1,0] => [1,1,1,1,0,1,0,0,0,1,0,0] => 7
[1,1,1,1,0,0,1,0,0,0] => [1,1,1,0,1,0,0,1,0,0] => [1,1,1,1,0,1,0,0,1,0,0,0] => 7
[1,1,1,1,0,1,0,0,0,0] => [1,1,1,0,1,0,1,0,0,0] => [1,1,1,1,0,1,0,1,0,0,0,0] => 7
[1,1,1,1,1,0,0,0,0,0] => [1,1,1,1,0,1,0,0,0,0] => [1,1,1,1,1,0,1,0,0,0,0,0] => 7
[] => [] => [1,0] => 1
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Description
The vector space dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J.
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
prime Dyck path
Description
Return the Dyck path obtained by adding an initial up and a final down step.