Identifier
Values
[.,.] => [1,0] => [1,1,0,0] => [1,1,1,0,0,0] => 4
[.,[.,.]] => [1,1,0,0] => [1,1,1,0,0,0] => [1,1,1,1,0,0,0,0] => 5
[[.,.],.] => [1,0,1,0] => [1,1,0,1,0,0] => [1,1,1,0,1,0,0,0] => 5
[.,[.,[.,.]]] => [1,1,1,0,0,0] => [1,1,1,1,0,0,0,0] => [1,1,1,1,1,0,0,0,0,0] => 6
[.,[[.,.],.]] => [1,1,0,1,0,0] => [1,1,1,0,1,0,0,0] => [1,1,1,1,0,1,0,0,0,0] => 6
[[.,.],[.,.]] => [1,0,1,1,0,0] => [1,1,0,1,1,0,0,0] => [1,1,1,0,1,1,0,0,0,0] => 6
[[.,[.,.]],.] => [1,1,0,0,1,0] => [1,1,1,0,0,1,0,0] => [1,1,1,1,0,0,1,0,0,0] => 5
[[[.,.],.],.] => [1,0,1,0,1,0] => [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] => [1,1,1,1,1,0,0,0,0,0] => [1,1,1,1,1,1,0,0,0,0,0,0] => 7
[.,[.,[[.,.],.]]] => [1,1,1,0,1,0,0,0] => [1,1,1,1,0,1,0,0,0,0] => [1,1,1,1,1,0,1,0,0,0,0,0] => 7
[.,[[.,.],[.,.]]] => [1,1,0,1,1,0,0,0] => [1,1,1,0,1,1,0,0,0,0] => [1,1,1,1,0,1,1,0,0,0,0,0] => 7
[.,[[.,[.,.]],.]] => [1,1,1,0,0,1,0,0] => [1,1,1,1,0,0,1,0,0,0] => [1,1,1,1,1,0,0,1,0,0,0,0] => 6
[.,[[[.,.],.],.]] => [1,1,0,1,0,1,0,0] => [1,1,1,0,1,0,1,0,0,0] => [1,1,1,1,0,1,0,1,0,0,0,0] => 7
[[.,.],[.,[.,.]]] => [1,0,1,1,1,0,0,0] => [1,1,0,1,1,1,0,0,0,0] => [1,1,1,0,1,1,1,0,0,0,0,0] => 7
[[.,.],[[.,.],.]] => [1,0,1,1,0,1,0,0] => [1,1,0,1,1,0,1,0,0,0] => [1,1,1,0,1,1,0,1,0,0,0,0] => 7
[[.,[.,.]],[.,.]] => [1,1,0,0,1,1,0,0] => [1,1,1,0,0,1,1,0,0,0] => [1,1,1,1,0,0,1,1,0,0,0,0] => 6
[[[.,.],.],[.,.]] => [1,0,1,0,1,1,0,0] => [1,1,0,1,0,1,1,0,0,0] => [1,1,1,0,1,0,1,1,0,0,0,0] => 7
[[.,[.,[.,.]]],.] => [1,1,1,0,0,0,1,0] => [1,1,1,1,0,0,0,1,0,0] => [1,1,1,1,1,0,0,0,1,0,0,0] => 6
[[.,[[.,.],.]],.] => [1,1,0,1,0,0,1,0] => [1,1,1,0,1,0,0,1,0,0] => [1,1,1,1,0,1,0,0,1,0,0,0] => 6
[[[.,.],[.,.]],.] => [1,0,1,1,0,0,1,0] => [1,1,0,1,1,0,0,1,0,0] => [1,1,1,0,1,1,0,0,1,0,0,0] => 6
[[[.,[.,.]],.],.] => [1,1,0,0,1,0,1,0] => [1,1,1,0,0,1,0,1,0,0] => [1,1,1,1,0,0,1,0,1,0,0,0] => 6
[[[[.,.],.],.],.] => [1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,0] => [1,1,1,0,1,0,1,0,1,0,0,0] => 7
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Description
The number of simple modules with injective dimension at most one or dominant dimension at least one.
Map
prime Dyck path
Description
Return the Dyck path obtained by adding an initial up and a final down step.
Map
to Tamari-corresponding Dyck path
Description
Return the Dyck path associated with a binary tree in consistency with the Tamari order on Dyck words and binary trees.
The bijection is defined recursively as follows:
  • a leaf is associated with an empty Dyck path,
  • a tree with children $l,r$ is associated with the Dyck word $T(l) 1 T(r) 0$ where $T(l)$ and $T(r)$ are the images of this bijection to $l$ and $r$.