Identifier
Values
[.,.] => [1,0] => 0
[.,[.,.]] => [1,0,1,0] => 1
[[.,.],.] => [1,1,0,0] => 0
[.,[.,[.,.]]] => [1,0,1,0,1,0] => 2
[.,[[.,.],.]] => [1,0,1,1,0,0] => 1
[[.,.],[.,.]] => [1,1,0,0,1,0] => 1
[[.,[.,.]],.] => [1,1,0,1,0,0] => 1
[[[.,.],.],.] => [1,1,1,0,0,0] => 0
[.,[.,[.,[.,.]]]] => [1,0,1,0,1,0,1,0] => 3
[.,[.,[[.,.],.]]] => [1,0,1,0,1,1,0,0] => 2
[.,[[.,.],[.,.]]] => [1,0,1,1,0,0,1,0] => 2
[.,[[.,[.,.]],.]] => [1,0,1,1,0,1,0,0] => 2
[.,[[[.,.],.],.]] => [1,0,1,1,1,0,0,0] => 1
[[.,.],[.,[.,.]]] => [1,1,0,0,1,0,1,0] => 2
[[.,.],[[.,.],.]] => [1,1,0,0,1,1,0,0] => 1
[[.,[.,.]],[.,.]] => [1,1,0,1,0,0,1,0] => 2
[[[.,.],.],[.,.]] => [1,1,1,0,0,0,1,0] => 1
[[.,[.,[.,.]]],.] => [1,1,0,1,0,1,0,0] => 2
[[.,[[.,.],.]],.] => [1,1,0,1,1,0,0,0] => 1
[[[.,.],[.,.]],.] => [1,1,1,0,0,1,0,0] => 1
[[[.,[.,.]],.],.] => [1,1,1,0,1,0,0,0] => 1
[[[[.,.],.],.],.] => [1,1,1,1,0,0,0,0] => 0
[.,[.,[.,[.,[.,.]]]]] => [1,0,1,0,1,0,1,0,1,0] => 4
[.,[.,[.,[[.,.],.]]]] => [1,0,1,0,1,0,1,1,0,0] => 3
[.,[.,[[.,.],[.,.]]]] => [1,0,1,0,1,1,0,0,1,0] => 3
[.,[.,[[.,[.,.]],.]]] => [1,0,1,0,1,1,0,1,0,0] => 3
[.,[.,[[[.,.],.],.]]] => [1,0,1,0,1,1,1,0,0,0] => 2
[.,[[.,.],[.,[.,.]]]] => [1,0,1,1,0,0,1,0,1,0] => 3
[.,[[.,.],[[.,.],.]]] => [1,0,1,1,0,0,1,1,0,0] => 2
[.,[[.,[.,.]],[.,.]]] => [1,0,1,1,0,1,0,0,1,0] => 3
[.,[[[.,.],.],[.,.]]] => [1,0,1,1,1,0,0,0,1,0] => 2
[.,[[.,[.,[.,.]]],.]] => [1,0,1,1,0,1,0,1,0,0] => 3
[.,[[.,[[.,.],.]],.]] => [1,0,1,1,0,1,1,0,0,0] => 2
[.,[[[.,.],[.,.]],.]] => [1,0,1,1,1,0,0,1,0,0] => 2
[.,[[[.,[.,.]],.],.]] => [1,0,1,1,1,0,1,0,0,0] => 2
[.,[[[[.,.],.],.],.]] => [1,0,1,1,1,1,0,0,0,0] => 1
[[.,.],[.,[.,[.,.]]]] => [1,1,0,0,1,0,1,0,1,0] => 3
[[.,.],[.,[[.,.],.]]] => [1,1,0,0,1,0,1,1,0,0] => 2
[[.,.],[[.,.],[.,.]]] => [1,1,0,0,1,1,0,0,1,0] => 2
[[.,.],[[.,[.,.]],.]] => [1,1,0,0,1,1,0,1,0,0] => 2
[[.,.],[[[.,.],.],.]] => [1,1,0,0,1,1,1,0,0,0] => 1
[[.,[.,.]],[.,[.,.]]] => [1,1,0,1,0,0,1,0,1,0] => 3
[[.,[.,.]],[[.,.],.]] => [1,1,0,1,0,0,1,1,0,0] => 2
[[[.,.],.],[.,[.,.]]] => [1,1,1,0,0,0,1,0,1,0] => 2
[[[.,.],.],[[.,.],.]] => [1,1,1,0,0,0,1,1,0,0] => 1
[[.,[.,[.,.]]],[.,.]] => [1,1,0,1,0,1,0,0,1,0] => 3
[[.,[[.,.],.]],[.,.]] => [1,1,0,1,1,0,0,0,1,0] => 2
[[[.,.],[.,.]],[.,.]] => [1,1,1,0,0,1,0,0,1,0] => 2
[[[.,[.,.]],.],[.,.]] => [1,1,1,0,1,0,0,0,1,0] => 2
[[[[.,.],.],.],[.,.]] => [1,1,1,1,0,0,0,0,1,0] => 1
[[.,[.,[.,[.,.]]]],.] => [1,1,0,1,0,1,0,1,0,0] => 3
[[.,[.,[[.,.],.]]],.] => [1,1,0,1,0,1,1,0,0,0] => 2
[[.,[[.,.],[.,.]]],.] => [1,1,0,1,1,0,0,1,0,0] => 2
[[.,[[.,[.,.]],.]],.] => [1,1,0,1,1,0,1,0,0,0] => 2
[[.,[[[.,.],.],.]],.] => [1,1,0,1,1,1,0,0,0,0] => 1
[[[.,.],[.,[.,.]]],.] => [1,1,1,0,0,1,0,1,0,0] => 2
[[[.,.],[[.,.],.]],.] => [1,1,1,0,0,1,1,0,0,0] => 1
[[[.,[.,.]],[.,.]],.] => [1,1,1,0,1,0,0,1,0,0] => 2
[[[[.,.],.],[.,.]],.] => [1,1,1,1,0,0,0,1,0,0] => 1
[[[.,[.,[.,.]]],.],.] => [1,1,1,0,1,0,1,0,0,0] => 2
[[[.,[[.,.],.]],.],.] => [1,1,1,0,1,1,0,0,0,0] => 1
[[[[.,.],[.,.]],.],.] => [1,1,1,1,0,0,1,0,0,0] => 1
[[[[.,[.,.]],.],.],.] => [1,1,1,1,0,1,0,0,0,0] => 1
[[[[[.,.],.],.],.],.] => [1,1,1,1,1,0,0,0,0,0] => 0
[.,[.,[.,[.,[.,[.,.]]]]]] => [1,0,1,0,1,0,1,0,1,0,1,0] => 5
[.,[.,[.,[.,[[.,.],.]]]]] => [1,0,1,0,1,0,1,0,1,1,0,0] => 4
[.,[.,[.,[[.,.],[.,.]]]]] => [1,0,1,0,1,0,1,1,0,0,1,0] => 4
[.,[.,[.,[[.,[.,.]],.]]]] => [1,0,1,0,1,0,1,1,0,1,0,0] => 4
[.,[.,[.,[[[.,.],.],.]]]] => [1,0,1,0,1,0,1,1,1,0,0,0] => 3
[.,[.,[[.,.],[.,[.,.]]]]] => [1,0,1,0,1,1,0,0,1,0,1,0] => 4
[.,[.,[[.,.],[[.,.],.]]]] => [1,0,1,0,1,1,0,0,1,1,0,0] => 3
[.,[.,[[.,[.,.]],[.,.]]]] => [1,0,1,0,1,1,0,1,0,0,1,0] => 4
[.,[.,[[[.,.],.],[.,.]]]] => [1,0,1,0,1,1,1,0,0,0,1,0] => 3
[.,[.,[[.,[.,[.,.]]],.]]] => [1,0,1,0,1,1,0,1,0,1,0,0] => 4
[.,[.,[[.,[[.,.],.]],.]]] => [1,0,1,0,1,1,0,1,1,0,0,0] => 3
[.,[.,[[[.,.],[.,.]],.]]] => [1,0,1,0,1,1,1,0,0,1,0,0] => 3
[.,[.,[[[.,[.,.]],.],.]]] => [1,0,1,0,1,1,1,0,1,0,0,0] => 3
[.,[.,[[[[.,.],.],.],.]]] => [1,0,1,0,1,1,1,1,0,0,0,0] => 2
[.,[[.,.],[.,[.,[.,.]]]]] => [1,0,1,1,0,0,1,0,1,0,1,0] => 4
[.,[[.,.],[.,[[.,.],.]]]] => [1,0,1,1,0,0,1,0,1,1,0,0] => 3
[.,[[.,.],[[.,.],[.,.]]]] => [1,0,1,1,0,0,1,1,0,0,1,0] => 3
[.,[[.,.],[[.,[.,.]],.]]] => [1,0,1,1,0,0,1,1,0,1,0,0] => 3
[.,[[.,.],[[[.,.],.],.]]] => [1,0,1,1,0,0,1,1,1,0,0,0] => 2
[.,[[.,[.,.]],[.,[.,.]]]] => [1,0,1,1,0,1,0,0,1,0,1,0] => 4
[.,[[.,[.,.]],[[.,.],.]]] => [1,0,1,1,0,1,0,0,1,1,0,0] => 3
[.,[[[.,.],.],[.,[.,.]]]] => [1,0,1,1,1,0,0,0,1,0,1,0] => 3
[.,[[[.,.],.],[[.,.],.]]] => [1,0,1,1,1,0,0,0,1,1,0,0] => 2
[.,[[.,[.,[.,.]]],[.,.]]] => [1,0,1,1,0,1,0,1,0,0,1,0] => 4
[.,[[.,[[.,.],.]],[.,.]]] => [1,0,1,1,0,1,1,0,0,0,1,0] => 3
[.,[[[.,.],[.,.]],[.,.]]] => [1,0,1,1,1,0,0,1,0,0,1,0] => 3
[.,[[[.,[.,.]],.],[.,.]]] => [1,0,1,1,1,0,1,0,0,0,1,0] => 3
[.,[[[[.,.],.],.],[.,.]]] => [1,0,1,1,1,1,0,0,0,0,1,0] => 2
[.,[[.,[.,[.,[.,.]]]],.]] => [1,0,1,1,0,1,0,1,0,1,0,0] => 4
[.,[[.,[.,[[.,.],.]]],.]] => [1,0,1,1,0,1,0,1,1,0,0,0] => 3
[.,[[.,[[.,.],[.,.]]],.]] => [1,0,1,1,0,1,1,0,0,1,0,0] => 3
[.,[[.,[[.,[.,.]],.]],.]] => [1,0,1,1,0,1,1,0,1,0,0,0] => 3
[.,[[.,[[[.,.],.],.]],.]] => [1,0,1,1,0,1,1,1,0,0,0,0] => 2
[.,[[[.,.],[.,[.,.]]],.]] => [1,0,1,1,1,0,0,1,0,1,0,0] => 3
[.,[[[.,.],[[.,.],.]],.]] => [1,0,1,1,1,0,0,1,1,0,0,0] => 2
[.,[[[.,[.,.]],[.,.]],.]] => [1,0,1,1,1,0,1,0,0,1,0,0] => 3
[.,[[[[.,.],.],[.,.]],.]] => [1,0,1,1,1,1,0,0,0,1,0,0] => 2
>>> Load all 196 entries. <<<
[.,[[[.,[.,[.,.]]],.],.]] => [1,0,1,1,1,0,1,0,1,0,0,0] => 3
[.,[[[.,[[.,.],.]],.],.]] => [1,0,1,1,1,0,1,1,0,0,0,0] => 2
[.,[[[[.,.],[.,.]],.],.]] => [1,0,1,1,1,1,0,0,1,0,0,0] => 2
[.,[[[[.,[.,.]],.],.],.]] => [1,0,1,1,1,1,0,1,0,0,0,0] => 2
[.,[[[[[.,.],.],.],.],.]] => [1,0,1,1,1,1,1,0,0,0,0,0] => 1
[[.,.],[.,[.,[.,[.,.]]]]] => [1,1,0,0,1,0,1,0,1,0,1,0] => 4
[[.,.],[.,[.,[[.,.],.]]]] => [1,1,0,0,1,0,1,0,1,1,0,0] => 3
[[.,.],[.,[[.,.],[.,.]]]] => [1,1,0,0,1,0,1,1,0,0,1,0] => 3
[[.,.],[.,[[.,[.,.]],.]]] => [1,1,0,0,1,0,1,1,0,1,0,0] => 3
[[.,.],[.,[[[.,.],.],.]]] => [1,1,0,0,1,0,1,1,1,0,0,0] => 2
[[.,.],[[.,.],[.,[.,.]]]] => [1,1,0,0,1,1,0,0,1,0,1,0] => 3
[[.,.],[[.,.],[[.,.],.]]] => [1,1,0,0,1,1,0,0,1,1,0,0] => 2
[[.,.],[[.,[.,.]],[.,.]]] => [1,1,0,0,1,1,0,1,0,0,1,0] => 3
[[.,.],[[[.,.],.],[.,.]]] => [1,1,0,0,1,1,1,0,0,0,1,0] => 2
[[.,.],[[.,[.,[.,.]]],.]] => [1,1,0,0,1,1,0,1,0,1,0,0] => 3
[[.,.],[[.,[[.,.],.]],.]] => [1,1,0,0,1,1,0,1,1,0,0,0] => 2
[[.,.],[[[.,.],[.,.]],.]] => [1,1,0,0,1,1,1,0,0,1,0,0] => 2
[[.,.],[[[.,[.,.]],.],.]] => [1,1,0,0,1,1,1,0,1,0,0,0] => 2
[[.,.],[[[[.,.],.],.],.]] => [1,1,0,0,1,1,1,1,0,0,0,0] => 1
[[.,[.,.]],[.,[.,[.,.]]]] => [1,1,0,1,0,0,1,0,1,0,1,0] => 4
[[.,[.,.]],[.,[[.,.],.]]] => [1,1,0,1,0,0,1,0,1,1,0,0] => 3
[[.,[.,.]],[[.,.],[.,.]]] => [1,1,0,1,0,0,1,1,0,0,1,0] => 3
[[.,[.,.]],[[.,[.,.]],.]] => [1,1,0,1,0,0,1,1,0,1,0,0] => 3
[[.,[.,.]],[[[.,.],.],.]] => [1,1,0,1,0,0,1,1,1,0,0,0] => 2
[[[.,.],.],[.,[.,[.,.]]]] => [1,1,1,0,0,0,1,0,1,0,1,0] => 3
[[[.,.],.],[.,[[.,.],.]]] => [1,1,1,0,0,0,1,0,1,1,0,0] => 2
[[[.,.],.],[[.,.],[.,.]]] => [1,1,1,0,0,0,1,1,0,0,1,0] => 2
[[[.,.],.],[[.,[.,.]],.]] => [1,1,1,0,0,0,1,1,0,1,0,0] => 2
[[[.,.],.],[[[.,.],.],.]] => [1,1,1,0,0,0,1,1,1,0,0,0] => 1
[[.,[.,[.,.]]],[.,[.,.]]] => [1,1,0,1,0,1,0,0,1,0,1,0] => 4
[[.,[.,[.,.]]],[[.,.],.]] => [1,1,0,1,0,1,0,0,1,1,0,0] => 3
[[.,[[.,.],.]],[.,[.,.]]] => [1,1,0,1,1,0,0,0,1,0,1,0] => 3
[[.,[[.,.],.]],[[.,.],.]] => [1,1,0,1,1,0,0,0,1,1,0,0] => 2
[[[.,.],[.,.]],[.,[.,.]]] => [1,1,1,0,0,1,0,0,1,0,1,0] => 3
[[[.,.],[.,.]],[[.,.],.]] => [1,1,1,0,0,1,0,0,1,1,0,0] => 2
[[[.,[.,.]],.],[.,[.,.]]] => [1,1,1,0,1,0,0,0,1,0,1,0] => 3
[[[.,[.,.]],.],[[.,.],.]] => [1,1,1,0,1,0,0,0,1,1,0,0] => 2
[[[[.,.],.],.],[.,[.,.]]] => [1,1,1,1,0,0,0,0,1,0,1,0] => 2
[[[[.,.],.],.],[[.,.],.]] => [1,1,1,1,0,0,0,0,1,1,0,0] => 1
[[.,[.,[.,[.,.]]]],[.,.]] => [1,1,0,1,0,1,0,1,0,0,1,0] => 4
[[.,[.,[[.,.],.]]],[.,.]] => [1,1,0,1,0,1,1,0,0,0,1,0] => 3
[[.,[[.,.],[.,.]]],[.,.]] => [1,1,0,1,1,0,0,1,0,0,1,0] => 3
[[.,[[.,[.,.]],.]],[.,.]] => [1,1,0,1,1,0,1,0,0,0,1,0] => 3
[[.,[[[.,.],.],.]],[.,.]] => [1,1,0,1,1,1,0,0,0,0,1,0] => 2
[[[.,.],[.,[.,.]]],[.,.]] => [1,1,1,0,0,1,0,1,0,0,1,0] => 3
[[[.,.],[[.,.],.]],[.,.]] => [1,1,1,0,0,1,1,0,0,0,1,0] => 2
[[[.,[.,.]],[.,.]],[.,.]] => [1,1,1,0,1,0,0,1,0,0,1,0] => 3
[[[[.,.],.],[.,.]],[.,.]] => [1,1,1,1,0,0,0,1,0,0,1,0] => 2
[[[.,[.,[.,.]]],.],[.,.]] => [1,1,1,0,1,0,1,0,0,0,1,0] => 3
[[[.,[[.,.],.]],.],[.,.]] => [1,1,1,0,1,1,0,0,0,0,1,0] => 2
[[[[.,.],[.,.]],.],[.,.]] => [1,1,1,1,0,0,1,0,0,0,1,0] => 2
[[[[.,[.,.]],.],.],[.,.]] => [1,1,1,1,0,1,0,0,0,0,1,0] => 2
[[[[[.,.],.],.],.],[.,.]] => [1,1,1,1,1,0,0,0,0,0,1,0] => 1
[[.,[.,[.,[.,[.,.]]]]],.] => [1,1,0,1,0,1,0,1,0,1,0,0] => 4
[[.,[.,[.,[[.,.],.]]]],.] => [1,1,0,1,0,1,0,1,1,0,0,0] => 3
[[.,[.,[[.,.],[.,.]]]],.] => [1,1,0,1,0,1,1,0,0,1,0,0] => 3
[[.,[.,[[.,[.,.]],.]]],.] => [1,1,0,1,0,1,1,0,1,0,0,0] => 3
[[.,[.,[[[.,.],.],.]]],.] => [1,1,0,1,0,1,1,1,0,0,0,0] => 2
[[.,[[.,.],[.,[.,.]]]],.] => [1,1,0,1,1,0,0,1,0,1,0,0] => 3
[[.,[[.,.],[[.,.],.]]],.] => [1,1,0,1,1,0,0,1,1,0,0,0] => 2
[[.,[[.,[.,.]],[.,.]]],.] => [1,1,0,1,1,0,1,0,0,1,0,0] => 3
[[.,[[[.,.],.],[.,.]]],.] => [1,1,0,1,1,1,0,0,0,1,0,0] => 2
[[.,[[.,[.,[.,.]]],.]],.] => [1,1,0,1,1,0,1,0,1,0,0,0] => 3
[[.,[[.,[[.,.],.]],.]],.] => [1,1,0,1,1,0,1,1,0,0,0,0] => 2
[[.,[[[.,.],[.,.]],.]],.] => [1,1,0,1,1,1,0,0,1,0,0,0] => 2
[[.,[[[.,[.,.]],.],.]],.] => [1,1,0,1,1,1,0,1,0,0,0,0] => 2
[[.,[[[[.,.],.],.],.]],.] => [1,1,0,1,1,1,1,0,0,0,0,0] => 1
[[[.,.],[.,[.,[.,.]]]],.] => [1,1,1,0,0,1,0,1,0,1,0,0] => 3
[[[.,.],[.,[[.,.],.]]],.] => [1,1,1,0,0,1,0,1,1,0,0,0] => 2
[[[.,.],[[.,.],[.,.]]],.] => [1,1,1,0,0,1,1,0,0,1,0,0] => 2
[[[.,.],[[.,[.,.]],.]],.] => [1,1,1,0,0,1,1,0,1,0,0,0] => 2
[[[.,.],[[[.,.],.],.]],.] => [1,1,1,0,0,1,1,1,0,0,0,0] => 1
[[[.,[.,.]],[.,[.,.]]],.] => [1,1,1,0,1,0,0,1,0,1,0,0] => 3
[[[.,[.,.]],[[.,.],.]],.] => [1,1,1,0,1,0,0,1,1,0,0,0] => 2
[[[[.,.],.],[.,[.,.]]],.] => [1,1,1,1,0,0,0,1,0,1,0,0] => 2
[[[[.,.],.],[[.,.],.]],.] => [1,1,1,1,0,0,0,1,1,0,0,0] => 1
[[[.,[.,[.,.]]],[.,.]],.] => [1,1,1,0,1,0,1,0,0,1,0,0] => 3
[[[.,[[.,.],.]],[.,.]],.] => [1,1,1,0,1,1,0,0,0,1,0,0] => 2
[[[[.,.],[.,.]],[.,.]],.] => [1,1,1,1,0,0,1,0,0,1,0,0] => 2
[[[[.,[.,.]],.],[.,.]],.] => [1,1,1,1,0,1,0,0,0,1,0,0] => 2
[[[[[.,.],.],.],[.,.]],.] => [1,1,1,1,1,0,0,0,0,1,0,0] => 1
[[[.,[.,[.,[.,.]]]],.],.] => [1,1,1,0,1,0,1,0,1,0,0,0] => 3
[[[.,[.,[[.,.],.]]],.],.] => [1,1,1,0,1,0,1,1,0,0,0,0] => 2
[[[.,[[.,.],[.,.]]],.],.] => [1,1,1,0,1,1,0,0,1,0,0,0] => 2
[[[.,[[.,[.,.]],.]],.],.] => [1,1,1,0,1,1,0,1,0,0,0,0] => 2
[[[.,[[[.,.],.],.]],.],.] => [1,1,1,0,1,1,1,0,0,0,0,0] => 1
[[[[.,.],[.,[.,.]]],.],.] => [1,1,1,1,0,0,1,0,1,0,0,0] => 2
[[[[.,.],[[.,.],.]],.],.] => [1,1,1,1,0,0,1,1,0,0,0,0] => 1
[[[[.,[.,.]],[.,.]],.],.] => [1,1,1,1,0,1,0,0,1,0,0,0] => 2
[[[[[.,.],.],[.,.]],.],.] => [1,1,1,1,1,0,0,0,1,0,0,0] => 1
[[[[.,[.,[.,.]]],.],.],.] => [1,1,1,1,0,1,0,1,0,0,0,0] => 2
[[[[.,[[.,.],.]],.],.],.] => [1,1,1,1,0,1,1,0,0,0,0,0] => 1
[[[[[.,.],[.,.]],.],.],.] => [1,1,1,1,1,0,0,1,0,0,0,0] => 1
[[[[[.,[.,.]],.],.],.],.] => [1,1,1,1,1,0,1,0,0,0,0,0] => 1
[[[[[[.,.],.],.],.],.],.] => [1,1,1,1,1,1,0,0,0,0,0,0] => 0
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Description
Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra.
Map
to Dyck path: up step, left tree, down step, right tree
Description
Return the associated Dyck path, using the bijection 1L0R.
This is given recursively as follows:
  • a leaf is associated to the empty Dyck Word
  • a tree with children $l,r$ is associated with the Dyck path described by 1L0R where $L$ and $R$ are respectively the Dyck words associated with the trees $l$ and $r$.