Identifier
Values
[.,.] => [1,0] => [2,1] => 1
[.,[.,.]] => [1,0,1,0] => [3,1,2] => 1
[[.,.],.] => [1,1,0,0] => [2,3,1] => 1
[.,[.,[.,.]]] => [1,0,1,0,1,0] => [4,1,2,3] => 1
[.,[[.,.],.]] => [1,0,1,1,0,0] => [3,1,4,2] => 2
[[.,.],[.,.]] => [1,1,0,0,1,0] => [2,4,1,3] => 2
[[.,[.,.]],.] => [1,1,0,1,0,0] => [4,3,1,2] => 1
[[[.,.],.],.] => [1,1,1,0,0,0] => [2,3,4,1] => 1
[.,[.,[.,[.,.]]]] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => 1
[.,[.,[[.,.],.]]] => [1,0,1,0,1,1,0,0] => [4,1,2,5,3] => 2
[.,[[.,.],[.,.]]] => [1,0,1,1,0,0,1,0] => [3,1,5,2,4] => 2
[.,[[.,[.,.]],.]] => [1,0,1,1,0,1,0,0] => [5,1,4,2,3] => 2
[.,[[[.,.],.],.]] => [1,0,1,1,1,0,0,0] => [3,1,4,5,2] => 2
[[.,.],[.,[.,.]]] => [1,1,0,0,1,0,1,0] => [2,5,1,3,4] => 2
[[.,.],[[.,.],.]] => [1,1,0,0,1,1,0,0] => [2,4,1,5,3] => 2
[[.,[.,.]],[.,.]] => [1,1,0,1,0,0,1,0] => [5,3,1,2,4] => 1
[[[.,.],.],[.,.]] => [1,1,1,0,0,0,1,0] => [2,3,5,1,4] => 2
[[.,[.,[.,.]]],.] => [1,1,0,1,0,1,0,0] => [5,4,1,2,3] => 1
[[.,[[.,.],.]],.] => [1,1,0,1,1,0,0,0] => [4,3,1,5,2] => 2
[[[.,.],[.,.]],.] => [1,1,1,0,0,1,0,0] => [2,5,4,1,3] => 2
[[[.,[.,.]],.],.] => [1,1,1,0,1,0,0,0] => [5,3,4,1,2] => 2
[[[[.,.],.],.],.] => [1,1,1,1,0,0,0,0] => [2,3,4,5,1] => 1
[.,[.,[.,[.,[.,.]]]]] => [1,0,1,0,1,0,1,0,1,0] => [6,1,2,3,4,5] => 1
[.,[.,[.,[[.,.],.]]]] => [1,0,1,0,1,0,1,1,0,0] => [5,1,2,3,6,4] => 2
[.,[.,[[.,.],[.,.]]]] => [1,0,1,0,1,1,0,0,1,0] => [4,1,2,6,3,5] => 2
[.,[.,[[.,[.,.]],.]]] => [1,0,1,0,1,1,0,1,0,0] => [6,1,2,5,3,4] => 2
[.,[.,[[[.,.],.],.]]] => [1,0,1,0,1,1,1,0,0,0] => [4,1,2,5,6,3] => 2
[.,[[.,.],[.,[.,.]]]] => [1,0,1,1,0,0,1,0,1,0] => [3,1,6,2,4,5] => 2
[.,[[.,.],[[.,.],.]]] => [1,0,1,1,0,0,1,1,0,0] => [3,1,5,2,6,4] => 2
[.,[[.,[.,.]],[.,.]]] => [1,0,1,1,0,1,0,0,1,0] => [6,1,4,2,3,5] => 2
[.,[[[.,.],.],[.,.]]] => [1,0,1,1,1,0,0,0,1,0] => [3,1,4,6,2,5] => 2
[.,[[.,[.,[.,.]]],.]] => [1,0,1,1,0,1,0,1,0,0] => [6,1,5,2,3,4] => 2
[.,[[.,[[.,.],.]],.]] => [1,0,1,1,0,1,1,0,0,0] => [5,1,4,2,6,3] => 3
[.,[[[.,.],[.,.]],.]] => [1,0,1,1,1,0,0,1,0,0] => [3,1,6,5,2,4] => 2
[.,[[[.,[.,.]],.],.]] => [1,0,1,1,1,0,1,0,0,0] => [6,1,4,5,2,3] => 2
[.,[[[[.,.],.],.],.]] => [1,0,1,1,1,1,0,0,0,0] => [3,1,4,5,6,2] => 2
[[.,.],[.,[.,[.,.]]]] => [1,1,0,0,1,0,1,0,1,0] => [2,6,1,3,4,5] => 2
[[.,.],[.,[[.,.],.]]] => [1,1,0,0,1,0,1,1,0,0] => [2,5,1,3,6,4] => 2
[[.,.],[[.,.],[.,.]]] => [1,1,0,0,1,1,0,0,1,0] => [2,4,1,6,3,5] => 3
[[.,.],[[.,[.,.]],.]] => [1,1,0,0,1,1,0,1,0,0] => [2,6,1,5,3,4] => 3
[[.,.],[[[.,.],.],.]] => [1,1,0,0,1,1,1,0,0,0] => [2,4,1,5,6,3] => 2
[[.,[.,.]],[.,[.,.]]] => [1,1,0,1,0,0,1,0,1,0] => [6,3,1,2,4,5] => 1
[[.,[.,.]],[[.,.],.]] => [1,1,0,1,0,0,1,1,0,0] => [5,3,1,2,6,4] => 2
[[[.,.],.],[.,[.,.]]] => [1,1,1,0,0,0,1,0,1,0] => [2,3,6,1,4,5] => 2
[[[.,.],.],[[.,.],.]] => [1,1,1,0,0,0,1,1,0,0] => [2,3,5,1,6,4] => 2
[[.,[.,[.,.]]],[.,.]] => [1,1,0,1,0,1,0,0,1,0] => [6,4,1,2,3,5] => 1
[[.,[[.,.],.]],[.,.]] => [1,1,0,1,1,0,0,0,1,0] => [4,3,1,6,2,5] => 2
[[[.,.],[.,.]],[.,.]] => [1,1,1,0,0,1,0,0,1,0] => [2,6,4,1,3,5] => 2
[[[.,[.,.]],.],[.,.]] => [1,1,1,0,1,0,0,0,1,0] => [6,3,4,1,2,5] => 2
[[[[.,.],.],.],[.,.]] => [1,1,1,1,0,0,0,0,1,0] => [2,3,4,6,1,5] => 2
[[.,[.,[.,[.,.]]]],.] => [1,1,0,1,0,1,0,1,0,0] => [5,6,1,2,3,4] => 1
[[.,[.,[[.,.],.]]],.] => [1,1,0,1,0,1,1,0,0,0] => [5,4,1,2,6,3] => 2
[[.,[[.,.],[.,.]]],.] => [1,1,0,1,1,0,0,1,0,0] => [6,3,1,5,2,4] => 2
[[.,[[.,[.,.]],.]],.] => [1,1,0,1,1,0,1,0,0,0] => [6,4,1,5,2,3] => 2
[[.,[[[.,.],.],.]],.] => [1,1,0,1,1,1,0,0,0,0] => [4,3,1,5,6,2] => 2
[[[.,.],[.,[.,.]]],.] => [1,1,1,0,0,1,0,1,0,0] => [2,6,5,1,3,4] => 2
[[[.,.],[[.,.],.]],.] => [1,1,1,0,0,1,1,0,0,0] => [2,5,4,1,6,3] => 2
[[[.,[.,.]],[.,.]],.] => [1,1,1,0,1,0,0,1,0,0] => [6,3,5,1,2,4] => 2
[[[[.,.],.],[.,.]],.] => [1,1,1,1,0,0,0,1,0,0] => [2,3,6,5,1,4] => 2
[[[.,[.,[.,.]]],.],.] => [1,1,1,0,1,0,1,0,0,0] => [6,5,4,1,2,3] => 1
[[[.,[[.,.],.]],.],.] => [1,1,1,0,1,1,0,0,0,0] => [5,3,4,1,6,2] => 3
[[[[.,.],[.,.]],.],.] => [1,1,1,1,0,0,1,0,0,0] => [2,6,4,5,1,3] => 3
[[[[.,[.,.]],.],.],.] => [1,1,1,1,0,1,0,0,0,0] => [6,3,4,5,1,2] => 2
[[[[[.,.],.],.],.],.] => [1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => 1
[[.,[.,[.,[.,[.,.]]]]],.] => [1,1,0,1,0,1,0,1,0,1,0,0] => [7,6,1,2,3,4,5] => 1
[[[[.,[.,[.,.]]],.],.],.] => [1,1,1,1,0,1,0,1,0,0,0,0] => [7,6,4,5,1,2,3] => 2
[[[.,[.,[.,[.,[.,.]]]]],.],.] => [1,1,1,0,1,0,1,0,1,0,1,0,0,0] => [6,7,8,1,2,3,4,5] => 1
[[[[[.,[.,[.,.]]],.],.],.],.] => [1,1,1,1,1,0,1,0,1,0,0,0,0,0] => [8,7,4,5,6,1,2,3] => 2
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Description
The number of cyclic valleys and cyclic peaks of a permutation.
This is given by the number of indices $i$ such that $\pi_{i-1} > \pi_i < \pi_{i+1}$ with indices considered cyclically. Equivalently, this is the number of indices $i$ such that $\pi_{i-1} < \pi_i > \pi_{i+1}$ with indices considered cyclically.
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 $r$.