Identifier
Values
[.,.] => [1,0] => [2,1] => ([],2) => 0
[.,[.,.]] => [1,0,1,0] => [3,1,2] => ([(1,2)],3) => 1
[[.,.],.] => [1,1,0,0] => [2,3,1] => ([(1,2)],3) => 1
[.,[.,[.,.]]] => [1,0,1,0,1,0] => [4,1,2,3] => ([(1,2),(2,3)],4) => 2
[.,[[.,.],.]] => [1,0,1,1,0,0] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4) => 2
[[.,.],[.,.]] => [1,1,0,0,1,0] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4) => 2
[[.,[.,.]],.] => [1,1,0,1,0,0] => [4,3,1,2] => ([(2,3)],4) => 1
[[[.,.],.],.] => [1,1,1,0,0,0] => [2,3,4,1] => ([(1,2),(2,3)],4) => 2
[.,[.,[.,[.,.]]]] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5) => 3
[.,[.,[[.,.],.]]] => [1,0,1,0,1,1,0,0] => [4,1,2,5,3] => ([(0,4),(1,2),(2,3),(2,4)],5) => 3
[.,[[.,.],[.,.]]] => [1,0,1,1,0,0,1,0] => [3,1,5,2,4] => ([(0,3),(0,4),(1,2),(1,3),(2,4)],5) => 3
[.,[[.,[.,.]],.]] => [1,0,1,1,0,1,0,0] => [5,1,4,2,3] => ([(1,3),(1,4),(4,2)],5) => 2
[.,[[[.,.],.],.]] => [1,0,1,1,1,0,0,0] => [3,1,4,5,2] => ([(0,4),(1,2),(1,4),(4,3)],5) => 3
[[.,.],[.,[.,.]]] => [1,1,0,0,1,0,1,0] => [2,5,1,3,4] => ([(0,4),(1,2),(1,4),(4,3)],5) => 3
[[.,.],[[.,.],.]] => [1,1,0,0,1,1,0,0] => [2,4,1,5,3] => ([(0,3),(0,4),(1,2),(1,3),(2,4)],5) => 3
[[.,[.,.]],[.,.]] => [1,1,0,1,0,0,1,0] => [5,3,1,2,4] => ([(1,4),(2,3),(3,4)],5) => 2
[[[.,.],.],[.,.]] => [1,1,1,0,0,0,1,0] => [2,3,5,1,4] => ([(0,4),(1,2),(2,3),(2,4)],5) => 3
[[.,[.,[.,.]]],.] => [1,1,0,1,0,1,0,0] => [5,4,1,2,3] => ([(2,3),(3,4)],5) => 2
[[.,[[.,.],.]],.] => [1,1,0,1,1,0,0,0] => [4,3,1,5,2] => ([(0,4),(1,4),(2,3),(2,4)],5) => 2
[[[.,.],[.,.]],.] => [1,1,1,0,0,1,0,0] => [2,5,4,1,3] => ([(0,4),(1,2),(1,3),(1,4)],5) => 2
[[[.,[.,.]],.],.] => [1,1,1,0,1,0,0,0] => [5,3,4,1,2] => ([(1,4),(2,3)],5) => 2
[[[[.,.],.],.],.] => [1,1,1,1,0,0,0,0] => [2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5) => 3
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Description
The number of points of the poset minus the width of the poset.
Map
Ringel
Description
The Ringel permutation of the LNakayama algebra corresponding to a Dyck path.
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 $r$.
Map
permutation poset
Description
Sends a permutation to its permutation poset.
For a permutation $\pi$ of length $n$, this poset has vertices
$$\{ (i,\pi(i))\ :\ 1 \leq i \leq n \}$$
and the cover relation is given by $(w, x) \leq (y, z)$ if $w \leq y$ and $x \leq z$.
For example, the permutation $[3,1,5,4,2]$ is mapped to the poset with cover relations
$$\{ (2, 1) \prec (5, 2),\ (2, 1) \prec (4, 4),\ (2, 1) \prec (3, 5),\ (1, 3) \prec (4, 4),\ (1, 3) \prec (3, 5) \}.$$