Identifier
Values
[1] => [1,0,1,0] => [1,0,1,0] => [1,1,0,1,0,0] => 1
[2] => [1,1,0,0,1,0] => [1,1,0,0,1,0] => [1,1,1,0,0,1,0,0] => 2
[1,1] => [1,0,1,1,0,0] => [1,1,0,1,0,0] => [1,1,1,0,1,0,0,0] => 1
[3] => [1,1,1,0,0,0,1,0] => [1,1,1,0,0,1,0,0] => [1,1,1,1,0,0,1,0,0,0] => 2
[2,1] => [1,0,1,0,1,0] => [1,0,1,0,1,0] => [1,1,0,1,0,1,0,0] => 2
[1,1,1] => [1,0,1,1,1,0,0,0] => [1,1,1,0,1,0,0,0] => [1,1,1,1,0,1,0,0,0,0] => 1
[3,1] => [1,1,0,1,0,0,1,0] => [1,0,1,1,0,0,1,0] => [1,1,0,1,1,0,0,1,0,0] => 3
[2,2] => [1,1,0,0,1,1,0,0] => [1,1,0,0,1,1,0,0] => [1,1,1,0,0,1,1,0,0,0] => 3
[2,1,1] => [1,0,1,1,0,1,0,0] => [1,0,1,1,0,1,0,0] => [1,1,0,1,1,0,1,0,0,0] => 2
[3,2] => [1,1,0,0,1,0,1,0] => [1,1,0,0,1,0,1,0] => [1,1,1,0,0,1,0,1,0,0] => 3
[3,1,1] => [1,0,1,1,0,0,1,0] => [1,1,0,1,0,0,1,0] => [1,1,1,0,1,0,0,1,0,0] => 2
[2,2,1] => [1,0,1,0,1,1,0,0] => [1,1,0,1,0,1,0,0] => [1,1,1,0,1,0,1,0,0,0] => 2
[3,2,1] => [1,0,1,0,1,0,1,0] => [1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,0] => 3
[] => [] => [] => [1,0] => 1
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Description
The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule.
Map
Cori-Le Borgne involution
Description
The Cori-Le Borgne involution on Dyck paths.
Append an additional down step to the Dyck path and consider its (literal) reversal. The image of the involution is then the unique rotation of this word which is a Dyck word followed by an additional down step. Alternatively, it is the composite $\zeta\circ\mathrm{rev}\circ\zeta^{(-1)}$, where $\zeta$ is Mp00030zeta map.
Map
to Dyck path
Description
Sends a partition to the shortest Dyck path tracing the shape of its Ferrers diagram.
Map
prime Dyck path
Description
Return the Dyck path obtained by adding an initial up and a final down step.