Identifier
Values
[.,.] => [1,0] => 2
[.,[.,.]] => [1,1,0,0] => 3
[[.,.],.] => [1,0,1,0] => 3
[.,[.,[.,.]]] => [1,1,1,0,0,0] => 4
[.,[[.,.],.]] => [1,1,0,1,0,0] => 4
[[.,.],[.,.]] => [1,0,1,1,0,0] => 4
[[.,[.,.]],.] => [1,1,0,0,1,0] => 3
[[[.,.],.],.] => [1,0,1,0,1,0] => 4
[.,[.,[.,[.,.]]]] => [1,1,1,1,0,0,0,0] => 5
[.,[.,[[.,.],.]]] => [1,1,1,0,1,0,0,0] => 5
[.,[[.,.],[.,.]]] => [1,1,0,1,1,0,0,0] => 5
[.,[[.,[.,.]],.]] => [1,1,1,0,0,1,0,0] => 4
[.,[[[.,.],.],.]] => [1,1,0,1,0,1,0,0] => 5
[[.,.],[.,[.,.]]] => [1,0,1,1,1,0,0,0] => 5
[[.,.],[[.,.],.]] => [1,0,1,1,0,1,0,0] => 5
[[.,[.,.]],[.,.]] => [1,1,0,0,1,1,0,0] => 4
[[[.,.],.],[.,.]] => [1,0,1,0,1,1,0,0] => 5
[[.,[.,[.,.]]],.] => [1,1,1,0,0,0,1,0] => 4
[[.,[[.,.],.]],.] => [1,1,0,1,0,0,1,0] => 4
[[[.,.],[.,.]],.] => [1,0,1,1,0,0,1,0] => 4
[[[.,[.,.]],.],.] => [1,1,0,0,1,0,1,0] => 4
[[[[.,.],.],.],.] => [1,0,1,0,1,0,1,0] => 5
[.,[.,[.,[.,[.,.]]]]] => [1,1,1,1,1,0,0,0,0,0] => 6
[.,[.,[.,[[.,.],.]]]] => [1,1,1,1,0,1,0,0,0,0] => 6
[.,[.,[[.,.],[.,.]]]] => [1,1,1,0,1,1,0,0,0,0] => 6
[.,[.,[[.,[.,.]],.]]] => [1,1,1,1,0,0,1,0,0,0] => 5
[.,[.,[[[.,.],.],.]]] => [1,1,1,0,1,0,1,0,0,0] => 6
[.,[[.,.],[.,[.,.]]]] => [1,1,0,1,1,1,0,0,0,0] => 6
[.,[[.,.],[[.,.],.]]] => [1,1,0,1,1,0,1,0,0,0] => 6
[.,[[.,[.,.]],[.,.]]] => [1,1,1,0,0,1,1,0,0,0] => 5
[.,[[[.,.],.],[.,.]]] => [1,1,0,1,0,1,1,0,0,0] => 6
[.,[[.,[.,[.,.]]],.]] => [1,1,1,1,0,0,0,1,0,0] => 5
[.,[[.,[[.,.],.]],.]] => [1,1,1,0,1,0,0,1,0,0] => 5
[.,[[[.,.],[.,.]],.]] => [1,1,0,1,1,0,0,1,0,0] => 5
[.,[[[.,[.,.]],.],.]] => [1,1,1,0,0,1,0,1,0,0] => 5
[.,[[[[.,.],.],.],.]] => [1,1,0,1,0,1,0,1,0,0] => 6
[[.,.],[.,[.,[.,.]]]] => [1,0,1,1,1,1,0,0,0,0] => 6
[[.,.],[.,[[.,.],.]]] => [1,0,1,1,1,0,1,0,0,0] => 6
[[.,.],[[.,.],[.,.]]] => [1,0,1,1,0,1,1,0,0,0] => 6
[[.,.],[[.,[.,.]],.]] => [1,0,1,1,1,0,0,1,0,0] => 5
[[.,.],[[[.,.],.],.]] => [1,0,1,1,0,1,0,1,0,0] => 6
[[.,[.,.]],[.,[.,.]]] => [1,1,0,0,1,1,1,0,0,0] => 5
[[.,[.,.]],[[.,.],.]] => [1,1,0,0,1,1,0,1,0,0] => 5
[[[.,.],.],[.,[.,.]]] => [1,0,1,0,1,1,1,0,0,0] => 6
[[[.,.],.],[[.,.],.]] => [1,0,1,0,1,1,0,1,0,0] => 6
[[.,[.,[.,.]]],[.,.]] => [1,1,1,0,0,0,1,1,0,0] => 5
[[.,[[.,.],.]],[.,.]] => [1,1,0,1,0,0,1,1,0,0] => 5
[[[.,.],[.,.]],[.,.]] => [1,0,1,1,0,0,1,1,0,0] => 5
[[[.,[.,.]],.],[.,.]] => [1,1,0,0,1,0,1,1,0,0] => 5
[[[[.,.],.],.],[.,.]] => [1,0,1,0,1,0,1,1,0,0] => 6
[[.,[.,[.,[.,.]]]],.] => [1,1,1,1,0,0,0,0,1,0] => 5
[[.,[.,[[.,.],.]]],.] => [1,1,1,0,1,0,0,0,1,0] => 5
[[.,[[.,.],[.,.]]],.] => [1,1,0,1,1,0,0,0,1,0] => 5
[[.,[[.,[.,.]],.]],.] => [1,1,1,0,0,1,0,0,1,0] => 4
[[.,[[[.,.],.],.]],.] => [1,1,0,1,0,1,0,0,1,0] => 5
[[[.,.],[.,[.,.]]],.] => [1,0,1,1,1,0,0,0,1,0] => 5
[[[.,.],[[.,.],.]],.] => [1,0,1,1,0,1,0,0,1,0] => 5
[[[.,[.,.]],[.,.]],.] => [1,1,0,0,1,1,0,0,1,0] => 4
[[[[.,.],.],[.,.]],.] => [1,0,1,0,1,1,0,0,1,0] => 5
[[[.,[.,[.,.]]],.],.] => [1,1,1,0,0,0,1,0,1,0] => 5
[[[.,[[.,.],.]],.],.] => [1,1,0,1,0,0,1,0,1,0] => 5
[[[[.,.],[.,.]],.],.] => [1,0,1,1,0,0,1,0,1,0] => 5
[[[[.,[.,.]],.],.],.] => [1,1,0,0,1,0,1,0,1,0] => 5
[[[[[.,.],.],.],.],.] => [1,0,1,0,1,0,1,0,1,0] => 6
[.,[.,[.,[.,[.,[.,.]]]]]] => [1,1,1,1,1,1,0,0,0,0,0,0] => 7
[.,[.,[.,[.,[[.,.],.]]]]] => [1,1,1,1,1,0,1,0,0,0,0,0] => 7
[.,[.,[.,[[.,.],[.,.]]]]] => [1,1,1,1,0,1,1,0,0,0,0,0] => 7
[.,[.,[.,[[.,[.,.]],.]]]] => [1,1,1,1,1,0,0,1,0,0,0,0] => 6
[.,[.,[.,[[[.,.],.],.]]]] => [1,1,1,1,0,1,0,1,0,0,0,0] => 7
[.,[.,[[.,.],[.,[.,.]]]]] => [1,1,1,0,1,1,1,0,0,0,0,0] => 7
[.,[.,[[.,.],[[.,.],.]]]] => [1,1,1,0,1,1,0,1,0,0,0,0] => 7
[.,[.,[[.,[.,.]],[.,.]]]] => [1,1,1,1,0,0,1,1,0,0,0,0] => 6
[.,[.,[[[.,.],.],[.,.]]]] => [1,1,1,0,1,0,1,1,0,0,0,0] => 7
[.,[.,[[.,[.,[.,.]]],.]]] => [1,1,1,1,1,0,0,0,1,0,0,0] => 6
[.,[.,[[.,[[.,.],.]],.]]] => [1,1,1,1,0,1,0,0,1,0,0,0] => 6
[.,[.,[[[.,.],[.,.]],.]]] => [1,1,1,0,1,1,0,0,1,0,0,0] => 6
[.,[.,[[[.,[.,.]],.],.]]] => [1,1,1,1,0,0,1,0,1,0,0,0] => 6
[.,[.,[[[[.,.],.],.],.]]] => [1,1,1,0,1,0,1,0,1,0,0,0] => 7
[.,[[.,.],[.,[.,[.,.]]]]] => [1,1,0,1,1,1,1,0,0,0,0,0] => 7
[.,[[.,.],[.,[[.,.],.]]]] => [1,1,0,1,1,1,0,1,0,0,0,0] => 7
[.,[[.,.],[[.,.],[.,.]]]] => [1,1,0,1,1,0,1,1,0,0,0,0] => 7
[.,[[.,.],[[.,[.,.]],.]]] => [1,1,0,1,1,1,0,0,1,0,0,0] => 6
[.,[[.,.],[[[.,.],.],.]]] => [1,1,0,1,1,0,1,0,1,0,0,0] => 7
[.,[[.,[.,.]],[.,[.,.]]]] => [1,1,1,0,0,1,1,1,0,0,0,0] => 6
[.,[[.,[.,.]],[[.,.],.]]] => [1,1,1,0,0,1,1,0,1,0,0,0] => 6
[.,[[[.,.],.],[.,[.,.]]]] => [1,1,0,1,0,1,1,1,0,0,0,0] => 7
[.,[[[.,.],.],[[.,.],.]]] => [1,1,0,1,0,1,1,0,1,0,0,0] => 7
[.,[[.,[.,[.,.]]],[.,.]]] => [1,1,1,1,0,0,0,1,1,0,0,0] => 6
[.,[[.,[[.,.],.]],[.,.]]] => [1,1,1,0,1,0,0,1,1,0,0,0] => 6
[.,[[[.,.],[.,.]],[.,.]]] => [1,1,0,1,1,0,0,1,1,0,0,0] => 6
[.,[[[.,[.,.]],.],[.,.]]] => [1,1,1,0,0,1,0,1,1,0,0,0] => 6
[.,[[[[.,.],.],.],[.,.]]] => [1,1,0,1,0,1,0,1,1,0,0,0] => 7
[.,[[.,[.,[.,[.,.]]]],.]] => [1,1,1,1,1,0,0,0,0,1,0,0] => 6
[.,[[.,[.,[[.,.],.]]],.]] => [1,1,1,1,0,1,0,0,0,1,0,0] => 6
[.,[[.,[[.,.],[.,.]]],.]] => [1,1,1,0,1,1,0,0,0,1,0,0] => 6
[.,[[.,[[.,[.,.]],.]],.]] => [1,1,1,1,0,0,1,0,0,1,0,0] => 5
[.,[[.,[[[.,.],.],.]],.]] => [1,1,1,0,1,0,1,0,0,1,0,0] => 6
[.,[[[.,.],[.,[.,.]]],.]] => [1,1,0,1,1,1,0,0,0,1,0,0] => 6
[.,[[[.,.],[[.,.],.]],.]] => [1,1,0,1,1,0,1,0,0,1,0,0] => 6
[.,[[[.,[.,.]],[.,.]],.]] => [1,1,1,0,0,1,1,0,0,1,0,0] => 5
[.,[[[[.,.],.],[.,.]],.]] => [1,1,0,1,0,1,1,0,0,1,0,0] => 6
>>> Load all 196 entries. <<<
[.,[[[.,[.,[.,.]]],.],.]] => [1,1,1,1,0,0,0,1,0,1,0,0] => 6
[.,[[[.,[[.,.],.]],.],.]] => [1,1,1,0,1,0,0,1,0,1,0,0] => 6
[.,[[[[.,.],[.,.]],.],.]] => [1,1,0,1,1,0,0,1,0,1,0,0] => 6
[.,[[[[.,[.,.]],.],.],.]] => [1,1,1,0,0,1,0,1,0,1,0,0] => 6
[.,[[[[[.,.],.],.],.],.]] => [1,1,0,1,0,1,0,1,0,1,0,0] => 7
[[.,.],[.,[.,[.,[.,.]]]]] => [1,0,1,1,1,1,1,0,0,0,0,0] => 7
[[.,.],[.,[.,[[.,.],.]]]] => [1,0,1,1,1,1,0,1,0,0,0,0] => 7
[[.,.],[.,[[.,.],[.,.]]]] => [1,0,1,1,1,0,1,1,0,0,0,0] => 7
[[.,.],[.,[[.,[.,.]],.]]] => [1,0,1,1,1,1,0,0,1,0,0,0] => 6
[[.,.],[.,[[[.,.],.],.]]] => [1,0,1,1,1,0,1,0,1,0,0,0] => 7
[[.,.],[[.,.],[.,[.,.]]]] => [1,0,1,1,0,1,1,1,0,0,0,0] => 7
[[.,.],[[.,.],[[.,.],.]]] => [1,0,1,1,0,1,1,0,1,0,0,0] => 7
[[.,.],[[.,[.,.]],[.,.]]] => [1,0,1,1,1,0,0,1,1,0,0,0] => 6
[[.,.],[[[.,.],.],[.,.]]] => [1,0,1,1,0,1,0,1,1,0,0,0] => 7
[[.,.],[[.,[.,[.,.]]],.]] => [1,0,1,1,1,1,0,0,0,1,0,0] => 6
[[.,.],[[.,[[.,.],.]],.]] => [1,0,1,1,1,0,1,0,0,1,0,0] => 6
[[.,.],[[[.,.],[.,.]],.]] => [1,0,1,1,0,1,1,0,0,1,0,0] => 6
[[.,.],[[[.,[.,.]],.],.]] => [1,0,1,1,1,0,0,1,0,1,0,0] => 6
[[.,.],[[[[.,.],.],.],.]] => [1,0,1,1,0,1,0,1,0,1,0,0] => 7
[[.,[.,.]],[.,[.,[.,.]]]] => [1,1,0,0,1,1,1,1,0,0,0,0] => 6
[[.,[.,.]],[.,[[.,.],.]]] => [1,1,0,0,1,1,1,0,1,0,0,0] => 6
[[.,[.,.]],[[.,.],[.,.]]] => [1,1,0,0,1,1,0,1,1,0,0,0] => 6
[[.,[.,.]],[[.,[.,.]],.]] => [1,1,0,0,1,1,1,0,0,1,0,0] => 5
[[.,[.,.]],[[[.,.],.],.]] => [1,1,0,0,1,1,0,1,0,1,0,0] => 6
[[[.,.],.],[.,[.,[.,.]]]] => [1,0,1,0,1,1,1,1,0,0,0,0] => 7
[[[.,.],.],[.,[[.,.],.]]] => [1,0,1,0,1,1,1,0,1,0,0,0] => 7
[[[.,.],.],[[.,.],[.,.]]] => [1,0,1,0,1,1,0,1,1,0,0,0] => 7
[[[.,.],.],[[.,[.,.]],.]] => [1,0,1,0,1,1,1,0,0,1,0,0] => 6
[[[.,.],.],[[[.,.],.],.]] => [1,0,1,0,1,1,0,1,0,1,0,0] => 7
[[.,[.,[.,.]]],[.,[.,.]]] => [1,1,1,0,0,0,1,1,1,0,0,0] => 6
[[.,[.,[.,.]]],[[.,.],.]] => [1,1,1,0,0,0,1,1,0,1,0,0] => 6
[[.,[[.,.],.]],[.,[.,.]]] => [1,1,0,1,0,0,1,1,1,0,0,0] => 6
[[.,[[.,.],.]],[[.,.],.]] => [1,1,0,1,0,0,1,1,0,1,0,0] => 6
[[[.,.],[.,.]],[.,[.,.]]] => [1,0,1,1,0,0,1,1,1,0,0,0] => 6
[[[.,.],[.,.]],[[.,.],.]] => [1,0,1,1,0,0,1,1,0,1,0,0] => 6
[[[.,[.,.]],.],[.,[.,.]]] => [1,1,0,0,1,0,1,1,1,0,0,0] => 6
[[[.,[.,.]],.],[[.,.],.]] => [1,1,0,0,1,0,1,1,0,1,0,0] => 6
[[[[.,.],.],.],[.,[.,.]]] => [1,0,1,0,1,0,1,1,1,0,0,0] => 7
[[[[.,.],.],.],[[.,.],.]] => [1,0,1,0,1,0,1,1,0,1,0,0] => 7
[[.,[.,[.,[.,.]]]],[.,.]] => [1,1,1,1,0,0,0,0,1,1,0,0] => 6
[[.,[.,[[.,.],.]]],[.,.]] => [1,1,1,0,1,0,0,0,1,1,0,0] => 6
[[.,[[.,.],[.,.]]],[.,.]] => [1,1,0,1,1,0,0,0,1,1,0,0] => 6
[[.,[[.,[.,.]],.]],[.,.]] => [1,1,1,0,0,1,0,0,1,1,0,0] => 5
[[.,[[[.,.],.],.]],[.,.]] => [1,1,0,1,0,1,0,0,1,1,0,0] => 6
[[[.,.],[.,[.,.]]],[.,.]] => [1,0,1,1,1,0,0,0,1,1,0,0] => 6
[[[.,.],[[.,.],.]],[.,.]] => [1,0,1,1,0,1,0,0,1,1,0,0] => 6
[[[.,[.,.]],[.,.]],[.,.]] => [1,1,0,0,1,1,0,0,1,1,0,0] => 5
[[[[.,.],.],[.,.]],[.,.]] => [1,0,1,0,1,1,0,0,1,1,0,0] => 6
[[[.,[.,[.,.]]],.],[.,.]] => [1,1,1,0,0,0,1,0,1,1,0,0] => 6
[[[.,[[.,.],.]],.],[.,.]] => [1,1,0,1,0,0,1,0,1,1,0,0] => 6
[[[[.,.],[.,.]],.],[.,.]] => [1,0,1,1,0,0,1,0,1,1,0,0] => 6
[[[[.,[.,.]],.],.],[.,.]] => [1,1,0,0,1,0,1,0,1,1,0,0] => 6
[[[[[.,.],.],.],.],[.,.]] => [1,0,1,0,1,0,1,0,1,1,0,0] => 7
[[.,[.,[.,[.,[.,.]]]]],.] => [1,1,1,1,1,0,0,0,0,0,1,0] => 6
[[.,[.,[.,[[.,.],.]]]],.] => [1,1,1,1,0,1,0,0,0,0,1,0] => 6
[[.,[.,[[.,.],[.,.]]]],.] => [1,1,1,0,1,1,0,0,0,0,1,0] => 6
[[.,[.,[[.,[.,.]],.]]],.] => [1,1,1,1,0,0,1,0,0,0,1,0] => 5
[[.,[.,[[[.,.],.],.]]],.] => [1,1,1,0,1,0,1,0,0,0,1,0] => 6
[[.,[[.,.],[.,[.,.]]]],.] => [1,1,0,1,1,1,0,0,0,0,1,0] => 6
[[.,[[.,.],[[.,.],.]]],.] => [1,1,0,1,1,0,1,0,0,0,1,0] => 6
[[.,[[.,[.,.]],[.,.]]],.] => [1,1,1,0,0,1,1,0,0,0,1,0] => 5
[[.,[[[.,.],.],[.,.]]],.] => [1,1,0,1,0,1,1,0,0,0,1,0] => 6
[[.,[[.,[.,[.,.]]],.]],.] => [1,1,1,1,0,0,0,1,0,0,1,0] => 5
[[.,[[.,[[.,.],.]],.]],.] => [1,1,1,0,1,0,0,1,0,0,1,0] => 5
[[.,[[[.,.],[.,.]],.]],.] => [1,1,0,1,1,0,0,1,0,0,1,0] => 5
[[.,[[[.,[.,.]],.],.]],.] => [1,1,1,0,0,1,0,1,0,0,1,0] => 5
[[.,[[[[.,.],.],.],.]],.] => [1,1,0,1,0,1,0,1,0,0,1,0] => 6
[[[.,.],[.,[.,[.,.]]]],.] => [1,0,1,1,1,1,0,0,0,0,1,0] => 6
[[[.,.],[.,[[.,.],.]]],.] => [1,0,1,1,1,0,1,0,0,0,1,0] => 6
[[[.,.],[[.,.],[.,.]]],.] => [1,0,1,1,0,1,1,0,0,0,1,0] => 6
[[[.,.],[[.,[.,.]],.]],.] => [1,0,1,1,1,0,0,1,0,0,1,0] => 5
[[[.,.],[[[.,.],.],.]],.] => [1,0,1,1,0,1,0,1,0,0,1,0] => 6
[[[.,[.,.]],[.,[.,.]]],.] => [1,1,0,0,1,1,1,0,0,0,1,0] => 5
[[[.,[.,.]],[[.,.],.]],.] => [1,1,0,0,1,1,0,1,0,0,1,0] => 5
[[[[.,.],.],[.,[.,.]]],.] => [1,0,1,0,1,1,1,0,0,0,1,0] => 6
[[[[.,.],.],[[.,.],.]],.] => [1,0,1,0,1,1,0,1,0,0,1,0] => 6
[[[.,[.,[.,.]]],[.,.]],.] => [1,1,1,0,0,0,1,1,0,0,1,0] => 5
[[[.,[[.,.],.]],[.,.]],.] => [1,1,0,1,0,0,1,1,0,0,1,0] => 5
[[[[.,.],[.,.]],[.,.]],.] => [1,0,1,1,0,0,1,1,0,0,1,0] => 5
[[[[.,[.,.]],.],[.,.]],.] => [1,1,0,0,1,0,1,1,0,0,1,0] => 5
[[[[[.,.],.],.],[.,.]],.] => [1,0,1,0,1,0,1,1,0,0,1,0] => 6
[[[.,[.,[.,[.,.]]]],.],.] => [1,1,1,1,0,0,0,0,1,0,1,0] => 6
[[[.,[.,[[.,.],.]]],.],.] => [1,1,1,0,1,0,0,0,1,0,1,0] => 6
[[[.,[[.,.],[.,.]]],.],.] => [1,1,0,1,1,0,0,0,1,0,1,0] => 6
[[[.,[[.,[.,.]],.]],.],.] => [1,1,1,0,0,1,0,0,1,0,1,0] => 5
[[[.,[[[.,.],.],.]],.],.] => [1,1,0,1,0,1,0,0,1,0,1,0] => 6
[[[[.,.],[.,[.,.]]],.],.] => [1,0,1,1,1,0,0,0,1,0,1,0] => 6
[[[[.,.],[[.,.],.]],.],.] => [1,0,1,1,0,1,0,0,1,0,1,0] => 6
[[[[.,[.,.]],[.,.]],.],.] => [1,1,0,0,1,1,0,0,1,0,1,0] => 5
[[[[[.,.],.],[.,.]],.],.] => [1,0,1,0,1,1,0,0,1,0,1,0] => 6
[[[[.,[.,[.,.]]],.],.],.] => [1,1,1,0,0,0,1,0,1,0,1,0] => 6
[[[[.,[[.,.],.]],.],.],.] => [1,1,0,1,0,0,1,0,1,0,1,0] => 6
[[[[[.,.],[.,.]],.],.],.] => [1,0,1,1,0,0,1,0,1,0,1,0] => 6
[[[[[.,[.,.]],.],.],.],.] => [1,1,0,0,1,0,1,0,1,0,1,0] => 6
[[[[[[.,.],.],.],.],.],.] => [1,0,1,0,1,0,1,0,1,0,1,0] => 7
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Description
The number of simple modules with injective dimension at most one or dominant dimension at least one.
Map
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 $r$.