Identifier
Values
[.,.] => [1,0] => [1,0] => [2,1] => 0
[.,[.,.]] => [1,0,1,0] => [1,0,1,0] => [3,1,2] => 0
[[.,.],.] => [1,1,0,0] => [1,1,0,0] => [2,3,1] => 0
[.,[.,[.,.]]] => [1,0,1,0,1,0] => [1,0,1,0,1,0] => [4,1,2,3] => 0
[.,[[.,.],.]] => [1,0,1,1,0,0] => [1,0,1,1,0,0] => [3,1,4,2] => 1
[[.,.],[.,.]] => [1,1,0,0,1,0] => [1,1,0,0,1,0] => [2,4,1,3] => 0
[[.,[.,.]],.] => [1,1,0,1,0,0] => [1,0,1,1,0,0] => [3,1,4,2] => 1
[[[.,.],.],.] => [1,1,1,0,0,0] => [1,1,1,0,0,0] => [2,3,4,1] => 0
[.,[.,[.,[.,.]]]] => [1,0,1,0,1,0,1,0] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => 0
[.,[.,[[.,.],.]]] => [1,0,1,0,1,1,0,0] => [1,0,1,0,1,1,0,0] => [4,1,2,5,3] => 1
[.,[[.,.],[.,.]]] => [1,0,1,1,0,0,1,0] => [1,0,1,1,0,0,1,0] => [3,1,5,2,4] => 1
[.,[[.,[.,.]],.]] => [1,0,1,1,0,1,0,0] => [1,0,1,0,1,1,0,0] => [4,1,2,5,3] => 1
[.,[[[.,.],.],.]] => [1,0,1,1,1,0,0,0] => [1,0,1,1,1,0,0,0] => [3,1,4,5,2] => 1
[[.,.],[.,[.,.]]] => [1,1,0,0,1,0,1,0] => [1,1,0,0,1,0,1,0] => [2,5,1,3,4] => 0
[[.,.],[[.,.],.]] => [1,1,0,0,1,1,0,0] => [1,1,0,0,1,1,0,0] => [2,4,1,5,3] => 1
[[.,[.,.]],[.,.]] => [1,1,0,1,0,0,1,0] => [1,0,1,1,0,0,1,0] => [3,1,5,2,4] => 1
[[[.,.],.],[.,.]] => [1,1,1,0,0,0,1,0] => [1,1,1,0,0,0,1,0] => [2,3,5,1,4] => 0
[[.,[.,[.,.]]],.] => [1,1,0,1,0,1,0,0] => [1,1,0,0,1,1,0,0] => [2,4,1,5,3] => 1
[[.,[[.,.],.]],.] => [1,1,0,1,1,0,0,0] => [1,0,1,1,1,0,0,0] => [3,1,4,5,2] => 1
[[[.,.],[.,.]],.] => [1,1,1,0,0,1,0,0] => [1,1,0,0,1,1,0,0] => [2,4,1,5,3] => 1
[[[.,[.,.]],.],.] => [1,1,1,0,1,0,0,0] => [1,0,1,1,1,0,0,0] => [3,1,4,5,2] => 1
[[[[.,.],.],.],.] => [1,1,1,1,0,0,0,0] => [1,1,1,1,0,0,0,0] => [2,3,4,5,1] => 0
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Description
The number of descents of a permutation minus one if its first entry is not one.
This statistic appears in [1, Theorem 2.3] in a gamma-positivity result, see also [2].
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
bounce path
Description
Sends a Dyck path $D$ of length $2n$ to its bounce path.
This path is formed by starting at the endpoint $(n,n)$ of $D$ and travelling west until encountering the first vertical step of $D$, then south until hitting the diagonal, then west again to hit $D$, etc. until the point $(0,0)$ is reached.
This map is the first part of the zeta map Mp00030zeta map.