Identifier
-
Mp00020:
Binary trees
—to Tamari-corresponding Dyck path⟶
Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000232: Set partitions ⟶ ℤ
Values
[.,.] => [1,0] => [2,1] => {{1,2}} => 0
[.,[.,.]] => [1,1,0,0] => [2,3,1] => {{1,2,3}} => 0
[[.,.],.] => [1,0,1,0] => [3,1,2] => {{1,3},{2}} => 0
[.,[.,[.,.]]] => [1,1,1,0,0,0] => [2,3,4,1] => {{1,2,3,4}} => 0
[.,[[.,.],.]] => [1,1,0,1,0,0] => [4,3,1,2] => {{1,4},{2,3}} => 0
[[.,.],[.,.]] => [1,0,1,1,0,0] => [3,1,4,2] => {{1,3,4},{2}} => 0
[[.,[.,.]],.] => [1,1,0,0,1,0] => [2,4,1,3] => {{1,2,4},{3}} => 0
[[[.,.],.],.] => [1,0,1,0,1,0] => [4,1,2,3] => {{1,4},{2},{3}} => 0
[.,[.,[.,[.,.]]]] => [1,1,1,1,0,0,0,0] => [2,3,4,5,1] => {{1,2,3,4,5}} => 0
[.,[.,[[.,.],.]]] => [1,1,1,0,1,0,0,0] => [5,3,4,1,2] => {{1,5},{2,3,4}} => 0
[.,[[.,.],[.,.]]] => [1,1,0,1,1,0,0,0] => [4,3,1,5,2] => {{1,4,5},{2,3}} => 0
[.,[[.,[.,.]],.]] => [1,1,1,0,0,1,0,0] => [2,5,4,1,3] => {{1,2,5},{3,4}} => 0
[.,[[[.,.],.],.]] => [1,1,0,1,0,1,0,0] => [5,4,1,2,3] => {{1,5},{2,4},{3}} => 0
[[.,.],[.,[.,.]]] => [1,0,1,1,1,0,0,0] => [3,1,4,5,2] => {{1,3,4,5},{2}} => 0
[[.,.],[[.,.],.]] => [1,0,1,1,0,1,0,0] => [5,1,4,2,3] => {{1,5},{2},{3,4}} => 0
[[.,[.,.]],[.,.]] => [1,1,0,0,1,1,0,0] => [2,4,1,5,3] => {{1,2,4,5},{3}} => 0
[[[.,.],.],[.,.]] => [1,0,1,0,1,1,0,0] => [4,1,2,5,3] => {{1,4,5},{2},{3}} => 0
[[.,[.,[.,.]]],.] => [1,1,1,0,0,0,1,0] => [2,3,5,1,4] => {{1,2,3,5},{4}} => 0
[[.,[[.,.],.]],.] => [1,1,0,1,0,0,1,0] => [5,3,1,2,4] => {{1,5},{2,3},{4}} => 0
[[[.,.],[.,.]],.] => [1,0,1,1,0,0,1,0] => [3,1,5,2,4] => {{1,3,5},{2},{4}} => 0
[[[.,[.,.]],.],.] => [1,1,0,0,1,0,1,0] => [2,5,1,3,4] => {{1,2,5},{3},{4}} => 0
[[[[.,.],.],.],.] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => {{1,5},{2},{3},{4}} => 0
[.,[.,[.,[.,[.,.]]]]] => [1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => {{1,2,3,4,5,6}} => 0
[.,[.,[.,[[.,.],.]]]] => [1,1,1,1,0,1,0,0,0,0] => [6,3,4,5,1,2] => {{1,6},{2,3,4,5}} => 0
[.,[.,[[.,.],[.,.]]]] => [1,1,1,0,1,1,0,0,0,0] => [5,3,4,1,6,2] => {{1,5,6},{2,3,4}} => 0
[.,[.,[[.,[.,.]],.]]] => [1,1,1,1,0,0,1,0,0,0] => [2,6,4,5,1,3] => {{1,2,6},{3,4,5}} => 0
[.,[.,[[[.,.],.],.]]] => [1,1,1,0,1,0,1,0,0,0] => [6,5,4,1,2,3] => {{1,6},{2,5},{3,4}} => 0
[.,[[.,.],[.,[.,.]]]] => [1,1,0,1,1,1,0,0,0,0] => [4,3,1,5,6,2] => {{1,4,5,6},{2,3}} => 0
[.,[[.,.],[[.,.],.]]] => [1,1,0,1,1,0,1,0,0,0] => [6,4,1,5,2,3] => {{1,6},{2,4,5},{3}} => 0
[.,[[.,[.,.]],[.,.]]] => [1,1,1,0,0,1,1,0,0,0] => [2,5,4,1,6,3] => {{1,2,5,6},{3,4}} => 0
[.,[[[.,.],.],[.,.]]] => [1,1,0,1,0,1,1,0,0,0] => [5,4,1,2,6,3] => {{1,5,6},{2,4},{3}} => 0
[.,[[.,[.,[.,.]]],.]] => [1,1,1,1,0,0,0,1,0,0] => [2,3,6,5,1,4] => {{1,2,3,6},{4,5}} => 0
[.,[[.,[[.,.],.]],.]] => [1,1,1,0,1,0,0,1,0,0] => [6,3,5,1,2,4] => {{1,6},{2,3,5},{4}} => 0
[.,[[[.,.],[.,.]],.]] => [1,1,0,1,1,0,0,1,0,0] => [6,3,1,5,2,4] => {{1,6},{2,3},{4,5}} => 0
[.,[[[.,[.,.]],.],.]] => [1,1,1,0,0,1,0,1,0,0] => [2,6,5,1,3,4] => {{1,2,6},{3,5},{4}} => 0
[.,[[[[.,.],.],.],.]] => [1,1,0,1,0,1,0,1,0,0] => [5,6,1,2,3,4] => {{1,5},{2,6},{3},{4}} => 1
[[.,.],[.,[.,[.,.]]]] => [1,0,1,1,1,1,0,0,0,0] => [3,1,4,5,6,2] => {{1,3,4,5,6},{2}} => 0
[[.,.],[.,[[.,.],.]]] => [1,0,1,1,1,0,1,0,0,0] => [6,1,4,5,2,3] => {{1,6},{2},{3,4,5}} => 0
[[.,.],[[.,.],[.,.]]] => [1,0,1,1,0,1,1,0,0,0] => [5,1,4,2,6,3] => {{1,5,6},{2},{3,4}} => 0
[[.,.],[[.,[.,.]],.]] => [1,0,1,1,1,0,0,1,0,0] => [3,1,6,5,2,4] => {{1,3,6},{2},{4,5}} => 0
[[.,.],[[[.,.],.],.]] => [1,0,1,1,0,1,0,1,0,0] => [6,1,5,2,3,4] => {{1,6},{2},{3,5},{4}} => 0
[[.,[.,.]],[.,[.,.]]] => [1,1,0,0,1,1,1,0,0,0] => [2,4,1,5,6,3] => {{1,2,4,5,6},{3}} => 0
[[.,[.,.]],[[.,.],.]] => [1,1,0,0,1,1,0,1,0,0] => [2,6,1,5,3,4] => {{1,2,6},{3},{4,5}} => 0
[[[.,.],.],[.,[.,.]]] => [1,0,1,0,1,1,1,0,0,0] => [4,1,2,5,6,3] => {{1,4,5,6},{2},{3}} => 0
[[[.,.],.],[[.,.],.]] => [1,0,1,0,1,1,0,1,0,0] => [6,1,2,5,3,4] => {{1,6},{2},{3},{4,5}} => 0
[[.,[.,[.,.]]],[.,.]] => [1,1,1,0,0,0,1,1,0,0] => [2,3,5,1,6,4] => {{1,2,3,5,6},{4}} => 0
[[.,[[.,.],.]],[.,.]] => [1,1,0,1,0,0,1,1,0,0] => [5,3,1,2,6,4] => {{1,5,6},{2,3},{4}} => 0
[[[.,.],[.,.]],[.,.]] => [1,0,1,1,0,0,1,1,0,0] => [3,1,5,2,6,4] => {{1,3,5,6},{2},{4}} => 0
[[[.,[.,.]],.],[.,.]] => [1,1,0,0,1,0,1,1,0,0] => [2,5,1,3,6,4] => {{1,2,5,6},{3},{4}} => 0
[[[[.,.],.],.],[.,.]] => [1,0,1,0,1,0,1,1,0,0] => [5,1,2,3,6,4] => {{1,5,6},{2},{3},{4}} => 0
[[.,[.,[.,[.,.]]]],.] => [1,1,1,1,0,0,0,0,1,0] => [2,3,4,6,1,5] => {{1,2,3,4,6},{5}} => 0
[[.,[.,[[.,.],.]]],.] => [1,1,1,0,1,0,0,0,1,0] => [6,3,4,1,2,5] => {{1,6},{2,3,4},{5}} => 0
[[.,[[.,.],[.,.]]],.] => [1,1,0,1,1,0,0,0,1,0] => [4,3,1,6,2,5] => {{1,4,6},{2,3},{5}} => 0
[[.,[[.,[.,.]],.]],.] => [1,1,1,0,0,1,0,0,1,0] => [2,6,4,1,3,5] => {{1,2,6},{3,4},{5}} => 0
[[.,[[[.,.],.],.]],.] => [1,1,0,1,0,1,0,0,1,0] => [6,4,1,2,3,5] => {{1,6},{2,4},{3},{5}} => 0
[[[.,.],[.,[.,.]]],.] => [1,0,1,1,1,0,0,0,1,0] => [3,1,4,6,2,5] => {{1,3,4,6},{2},{5}} => 0
[[[.,.],[[.,.],.]],.] => [1,0,1,1,0,1,0,0,1,0] => [6,1,4,2,3,5] => {{1,6},{2},{3,4},{5}} => 0
[[[.,[.,.]],[.,.]],.] => [1,1,0,0,1,1,0,0,1,0] => [2,4,1,6,3,5] => {{1,2,4,6},{3},{5}} => 0
[[[[.,.],.],[.,.]],.] => [1,0,1,0,1,1,0,0,1,0] => [4,1,2,6,3,5] => {{1,4,6},{2},{3},{5}} => 0
[[[.,[.,[.,.]]],.],.] => [1,1,1,0,0,0,1,0,1,0] => [2,3,6,1,4,5] => {{1,2,3,6},{4},{5}} => 0
[[[.,[[.,.],.]],.],.] => [1,1,0,1,0,0,1,0,1,0] => [6,3,1,2,4,5] => {{1,6},{2,3},{4},{5}} => 0
[[[[.,.],[.,.]],.],.] => [1,0,1,1,0,0,1,0,1,0] => [3,1,6,2,4,5] => {{1,3,6},{2},{4},{5}} => 0
[[[[.,[.,.]],.],.],.] => [1,1,0,0,1,0,1,0,1,0] => [2,6,1,3,4,5] => {{1,2,6},{3},{4},{5}} => 0
[[[[[.,.],.],.],.],.] => [1,0,1,0,1,0,1,0,1,0] => [6,1,2,3,4,5] => {{1,6},{2},{3},{4},{5}} => 0
[.,[.,[.,[.,[.,[.,.]]]]]] => [1,1,1,1,1,1,0,0,0,0,0,0] => [2,3,4,5,6,7,1] => {{1,2,3,4,5,6,7}} => 0
[.,[.,[.,[.,[[.,.],.]]]]] => [1,1,1,1,1,0,1,0,0,0,0,0] => [7,3,4,5,6,1,2] => {{1,7},{2,3,4,5,6}} => 0
[.,[.,[.,[[.,.],[.,.]]]]] => [1,1,1,1,0,1,1,0,0,0,0,0] => [6,3,4,5,1,7,2] => {{1,6,7},{2,3,4,5}} => 0
[.,[.,[.,[[.,[.,.]],.]]]] => [1,1,1,1,1,0,0,1,0,0,0,0] => [2,7,4,5,6,1,3] => {{1,2,7},{3,4,5,6}} => 0
[.,[.,[.,[[[.,.],.],.]]]] => [1,1,1,1,0,1,0,1,0,0,0,0] => [7,6,4,5,1,2,3] => {{1,7},{2,6},{3,4,5}} => 0
[.,[.,[[.,.],[.,[.,.]]]]] => [1,1,1,0,1,1,1,0,0,0,0,0] => [5,3,4,1,6,7,2] => {{1,5,6,7},{2,3,4}} => 0
[.,[.,[[.,.],[[.,.],.]]]] => [1,1,1,0,1,1,0,1,0,0,0,0] => [7,5,4,1,6,2,3] => {{1,7},{2,5,6},{3,4}} => 0
[.,[.,[[.,[.,.]],[.,.]]]] => [1,1,1,1,0,0,1,1,0,0,0,0] => [2,6,4,5,1,7,3] => {{1,2,6,7},{3,4,5}} => 0
[.,[.,[[[.,.],.],[.,.]]]] => [1,1,1,0,1,0,1,1,0,0,0,0] => [6,5,4,1,2,7,3] => {{1,6,7},{2,5},{3,4}} => 0
[.,[.,[[.,[.,[.,.]]],.]]] => [1,1,1,1,1,0,0,0,1,0,0,0] => [2,3,7,5,6,1,4] => {{1,2,3,7},{4,5,6}} => 0
[.,[.,[[.,[[.,.],.]],.]]] => [1,1,1,1,0,1,0,0,1,0,0,0] => [7,3,6,5,1,2,4] => {{1,7},{2,3,6},{4,5}} => 0
[.,[.,[[[.,.],[.,.]],.]]] => [1,1,1,0,1,1,0,0,1,0,0,0] => [7,3,5,1,6,2,4] => {{1,7},{2,3,5,6},{4}} => 0
[.,[.,[[[.,[.,.]],.],.]]] => [1,1,1,1,0,0,1,0,1,0,0,0] => [2,7,6,5,1,3,4] => {{1,2,7},{3,6},{4,5}} => 0
[.,[.,[[[[.,.],.],.],.]]] => [1,1,1,0,1,0,1,0,1,0,0,0] => [6,7,5,1,2,3,4] => {{1,6},{2,7},{3,5},{4}} => 1
[.,[[.,.],[.,[.,[.,.]]]]] => [1,1,0,1,1,1,1,0,0,0,0,0] => [4,3,1,5,6,7,2] => {{1,4,5,6,7},{2,3}} => 0
[.,[[.,.],[.,[[.,.],.]]]] => [1,1,0,1,1,1,0,1,0,0,0,0] => [7,4,1,5,6,2,3] => {{1,7},{2,4,5,6},{3}} => 0
[.,[[.,.],[[.,.],[.,.]]]] => [1,1,0,1,1,0,1,1,0,0,0,0] => [6,4,1,5,2,7,3] => {{1,6,7},{2,4,5},{3}} => 0
[.,[[.,.],[[.,[.,.]],.]]] => [1,1,0,1,1,1,0,0,1,0,0,0] => [7,3,1,5,6,2,4] => {{1,7},{2,3},{4,5,6}} => 0
[.,[[.,.],[[[.,.],.],.]]] => [1,1,0,1,1,0,1,0,1,0,0,0] => [6,7,1,5,2,3,4] => {{1,6},{2,7},{3},{4,5}} => 1
[.,[[.,[.,.]],[.,[.,.]]]] => [1,1,1,0,0,1,1,1,0,0,0,0] => [2,5,4,1,6,7,3] => {{1,2,5,6,7},{3,4}} => 0
[.,[[.,[.,.]],[[.,.],.]]] => [1,1,1,0,0,1,1,0,1,0,0,0] => [2,7,5,1,6,3,4] => {{1,2,7},{3,5,6},{4}} => 0
[.,[[[.,.],.],[.,[.,.]]]] => [1,1,0,1,0,1,1,1,0,0,0,0] => [5,4,1,2,6,7,3] => {{1,5,6,7},{2,4},{3}} => 0
[.,[[[.,.],.],[[.,.],.]]] => [1,1,0,1,0,1,1,0,1,0,0,0] => [5,7,1,2,6,3,4] => {{1,5,6},{2,7},{3},{4}} => 1
[.,[[.,[.,[.,.]]],[.,.]]] => [1,1,1,1,0,0,0,1,1,0,0,0] => [2,3,6,5,1,7,4] => {{1,2,3,6,7},{4,5}} => 0
[.,[[.,[[.,.],.]],[.,.]]] => [1,1,1,0,1,0,0,1,1,0,0,0] => [6,3,5,1,2,7,4] => {{1,6,7},{2,3,5},{4}} => 0
[.,[[[.,.],[.,.]],[.,.]]] => [1,1,0,1,1,0,0,1,1,0,0,0] => [6,3,1,5,2,7,4] => {{1,6,7},{2,3},{4,5}} => 0
[.,[[[.,[.,.]],.],[.,.]]] => [1,1,1,0,0,1,0,1,1,0,0,0] => [2,6,5,1,3,7,4] => {{1,2,6,7},{3,5},{4}} => 0
[.,[[[[.,.],.],.],[.,.]]] => [1,1,0,1,0,1,0,1,1,0,0,0] => [5,6,1,2,3,7,4] => {{1,5},{2,6,7},{3},{4}} => 1
[.,[[.,[.,[.,[.,.]]]],.]] => [1,1,1,1,1,0,0,0,0,1,0,0] => [2,3,4,7,6,1,5] => {{1,2,3,4,7},{5,6}} => 0
[.,[[.,[.,[[.,.],.]]],.]] => [1,1,1,1,0,1,0,0,0,1,0,0] => [7,3,4,6,1,2,5] => {{1,7},{2,3,4,6},{5}} => 0
[.,[[.,[[.,.],[.,.]]],.]] => [1,1,1,0,1,1,0,0,0,1,0,0] => [7,3,4,1,6,2,5] => {{1,7},{2,3,4},{5,6}} => 0
[.,[[.,[[.,[.,.]],.]],.]] => [1,1,1,1,0,0,1,0,0,1,0,0] => [2,7,4,6,1,3,5] => {{1,2,7},{3,4,6},{5}} => 0
[.,[[.,[[[.,.],.],.]],.]] => [1,1,1,0,1,0,1,0,0,1,0,0] => [6,7,4,1,2,3,5] => {{1,6},{2,7},{3,4},{5}} => 1
[.,[[[.,.],[.,[.,.]]],.]] => [1,1,0,1,1,1,0,0,0,1,0,0] => [4,3,1,7,6,2,5] => {{1,4,7},{2,3},{5,6}} => 0
[.,[[[.,.],[[.,.],.]],.]] => [1,1,0,1,1,0,1,0,0,1,0,0] => [7,4,1,6,2,3,5] => {{1,7},{2,4,6},{3},{5}} => 0
[.,[[[.,[.,.]],[.,.]],.]] => [1,1,1,0,0,1,1,0,0,1,0,0] => [2,7,4,1,6,3,5] => {{1,2,7},{3,4},{5,6}} => 0
[.,[[[[.,.],.],[.,.]],.]] => [1,1,0,1,0,1,1,0,0,1,0,0] => [7,4,1,2,6,3,5] => {{1,7},{2,4},{3},{5,6}} => 0
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Description
The number of crossings of a set partition.
This is given by the number of $i < i' < j < j'$ such that $i,j$ are two consecutive entries on one block, and $i',j'$ are consecutive entries in another block.
This is given by the number of $i < i' < j < j'$ such that $i,j$ are two consecutive entries on one block, and $i',j'$ are consecutive entries in another block.
Map
Ringel
Description
The Ringel permutation of the LNakayama algebra corresponding to a Dyck path.
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:
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 $r$.
Map
weak exceedance partition
Description
The set partition induced by the weak exceedances of a permutation.
This is the coarsest set partition that contains all arcs $(i, \pi(i))$ with $i\leq\pi(i)$.
This is the coarsest set partition that contains all arcs $(i, \pi(i))$ with $i\leq\pi(i)$.
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