Identifier
-
Mp00020:
Binary trees
—to Tamari-corresponding Dyck path⟶
Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000516: Permutations ⟶ 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] => [1,4,3,2] => 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] => [1,5,3,2,4] => 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] => [1,5,3,4,2] => 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] => [1,4,5,3,2] => 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] => [1,5,4,2,3] => 0
[[[.,.],[.,.]],.] => [1,0,1,1,0,0,1,0] => [3,1,5,2,4] => [1,3,5,4,2] => 0
[[[.,[.,.]],.],.] => [1,1,0,0,1,0,1,0] => [2,5,1,3,4] => [1,2,5,4,3] => 0
[[[[.,.],.],.],.] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => [1,5,4,3,2] => 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] => [1,6,3,4,2,5] => 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] => [1,6,3,2,4,5] => 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] => [1,5,6,3,2,4] => 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] => [1,6,4,2,3,5] => 0
[.,[[[.,.],[.,.]],.]] => [1,1,0,1,1,0,0,1,0,0] => [6,3,1,5,2,4] => [1,6,4,5,2,3] => 0
[.,[[[.,[.,.]],.],.]] => [1,1,1,0,0,1,0,1,0,0] => [2,6,5,1,3,4] => [1,2,6,4,3,5] => 0
[.,[[[[.,.],.],.],.]] => [1,1,0,1,0,1,0,1,0,0] => [5,6,1,2,3,4] => [1,5,3,2,6,4] => 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] => [1,6,3,4,5,2] => 0
[[.,.],[[.,.],[.,.]]] => [1,0,1,1,0,1,1,0,0,0] => [5,1,4,2,6,3] => [1,5,6,3,4,2] => 0
[[.,.],[[.,[.,.]],.]] => [1,0,1,1,1,0,0,1,0,0] => [3,1,6,5,2,4] => [1,3,6,4,5,2] => 0
[[.,.],[[[.,.],.],.]] => [1,0,1,1,0,1,0,1,0,0] => [6,1,5,2,3,4] => [1,6,4,2,3,5] => 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] => [1,2,6,4,5,3] => 0
[[[.,.],.],[.,[.,.]]] => [1,0,1,0,1,1,1,0,0,0] => [4,1,2,5,6,3] => [1,4,5,6,3,2] => 0
[[[.,.],.],[[.,.],.]] => [1,0,1,0,1,1,0,1,0,0] => [6,1,2,5,3,4] => [1,6,4,5,3,2] => 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] => [1,5,6,4,2,3] => 0
[[[.,.],[.,.]],[.,.]] => [1,0,1,1,0,0,1,1,0,0] => [3,1,5,2,6,4] => [1,3,5,6,4,2] => 0
[[[.,[.,.]],.],[.,.]] => [1,1,0,0,1,0,1,1,0,0] => [2,5,1,3,6,4] => [1,2,5,6,4,3] => 0
[[[[.,.],.],.],[.,.]] => [1,0,1,0,1,0,1,1,0,0] => [5,1,2,3,6,4] => [1,5,6,4,3,2] => 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] => [1,6,5,2,3,4] => 0
[[.,[[.,.],[.,.]]],.] => [1,1,0,1,1,0,0,0,1,0] => [4,3,1,6,2,5] => [1,4,6,5,2,3] => 0
[[.,[[.,[.,.]],.]],.] => [1,1,1,0,0,1,0,0,1,0] => [2,6,4,1,3,5] => [1,2,6,5,3,4] => 0
[[.,[[[.,.],.],.]],.] => [1,1,0,1,0,1,0,0,1,0] => [6,4,1,2,3,5] => [1,6,5,3,2,4] => 0
[[[.,.],[.,[.,.]]],.] => [1,0,1,1,1,0,0,0,1,0] => [3,1,4,6,2,5] => [1,3,4,6,5,2] => 0
[[[.,.],[[.,.],.]],.] => [1,0,1,1,0,1,0,0,1,0] => [6,1,4,2,3,5] => [1,6,5,3,4,2] => 0
[[[.,[.,.]],[.,.]],.] => [1,1,0,0,1,1,0,0,1,0] => [2,4,1,6,3,5] => [1,2,4,6,5,3] => 0
[[[[.,.],.],[.,.]],.] => [1,0,1,0,1,1,0,0,1,0] => [4,1,2,6,3,5] => [1,4,6,5,3,2] => 0
[[[.,[.,[.,.]]],.],.] => [1,1,1,0,0,0,1,0,1,0] => [2,3,6,1,4,5] => [1,2,3,6,5,4] => 0
[[[.,[[.,.],.]],.],.] => [1,1,0,1,0,0,1,0,1,0] => [6,3,1,2,4,5] => [1,6,5,4,2,3] => 0
[[[[.,.],[.,.]],.],.] => [1,0,1,1,0,0,1,0,1,0] => [3,1,6,2,4,5] => [1,3,6,5,4,2] => 0
[[[[.,[.,.]],.],.],.] => [1,1,0,0,1,0,1,0,1,0] => [2,6,1,3,4,5] => [1,2,6,5,4,3] => 0
[[[[[.,.],.],.],.],.] => [1,0,1,0,1,0,1,0,1,0] => [6,1,2,3,4,5] => [1,6,5,4,3,2] => 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
[.,[.,[.,[[.,[.,.]],.]]]] => [1,1,1,1,1,0,0,1,0,0,0,0] => [2,7,4,5,6,1,3] => [1,2,7,3,4,5,6] => 0
[.,[.,[[.,[.,.]],[.,.]]]] => [1,1,1,1,0,0,1,1,0,0,0,0] => [2,6,4,5,1,7,3] => [1,2,6,7,3,4,5] => 0
[.,[.,[[.,[.,[.,.]]],.]]] => [1,1,1,1,1,0,0,0,1,0,0,0] => [2,3,7,5,6,1,4] => [1,2,3,7,4,5,6] => 0
[.,[.,[[[.,[.,.]],.],.]]] => [1,1,1,1,0,0,1,0,1,0,0,0] => [2,7,6,5,1,3,4] => [1,2,7,4,5,3,6] => 0
[.,[[.,.],[.,[.,[.,.]]]]] => [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
[.,[[.,[.,.]],[.,[.,.]]]] => [1,1,1,0,0,1,1,1,0,0,0,0] => [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] => [1,2,7,4,3,5,6] => 0
[.,[[.,[.,[.,.]]],[.,.]]] => [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
[.,[[[.,[.,.]],.],[.,.]]] => [1,1,1,0,0,1,0,1,1,0,0,0] => [2,6,5,1,3,7,4] => [1,2,6,7,4,3,5] => 0
[.,[[.,[.,[.,[.,.]]]],.]] => [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
[.,[[.,[[.,[.,.]],.]],.]] => [1,1,1,1,0,0,1,0,0,1,0,0] => [2,7,4,6,1,3,5] => [1,2,7,5,3,4,6] => 0
[.,[[[.,[.,.]],[.,.]],.]] => [1,1,1,0,0,1,1,0,0,1,0,0] => [2,7,4,1,6,3,5] => [1,2,7,5,6,3,4] => 0
[.,[[[.,[.,[.,.]]],.],.]] => [1,1,1,1,0,0,0,1,0,1,0,0] => [2,3,7,6,1,4,5] => [1,2,3,7,5,4,6] => 0
[.,[[[[.,[.,.]],.],.],.]] => [1,1,1,0,0,1,0,1,0,1,0,0] => [2,6,7,1,3,4,5] => [1,2,6,4,3,7,5] => 1
[[.,.],[.,[.,[.,[.,.]]]]] => [1,0,1,1,1,1,1,0,0,0,0,0] => [3,1,4,5,6,7,2] => [1,3,4,5,6,7,2] => 0
[[.,.],[.,[[.,[.,.]],.]]] => [1,0,1,1,1,1,0,0,1,0,0,0] => [3,1,7,5,6,2,4] => [1,3,7,4,5,6,2] => 0
[[.,.],[[.,[.,.]],[.,.]]] => [1,0,1,1,1,0,0,1,1,0,0,0] => [3,1,6,5,2,7,4] => [1,3,6,7,4,5,2] => 0
[[.,.],[[.,[.,[.,.]]],.]] => [1,0,1,1,1,1,0,0,0,1,0,0] => [3,1,4,7,6,2,5] => [1,3,4,7,5,6,2] => 0
[[.,.],[[[.,[.,.]],.],.]] => [1,0,1,1,1,0,0,1,0,1,0,0] => [3,1,7,6,2,4,5] => [1,3,7,5,2,4,6] => 0
[[.,[.,.]],[.,[.,[.,.]]]] => [1,1,0,0,1,1,1,1,0,0,0,0] => [2,4,1,5,6,7,3] => [1,2,4,5,6,7,3] => 0
[[.,[.,.]],[.,[[.,.],.]]] => [1,1,0,0,1,1,1,0,1,0,0,0] => [2,7,1,5,6,3,4] => [1,2,7,4,5,6,3] => 0
[[.,[.,.]],[[.,.],[.,.]]] => [1,1,0,0,1,1,0,1,1,0,0,0] => [2,6,1,5,3,7,4] => [1,2,6,7,4,5,3] => 0
[[.,[.,.]],[[.,[.,.]],.]] => [1,1,0,0,1,1,1,0,0,1,0,0] => [2,4,1,7,6,3,5] => [1,2,4,7,5,6,3] => 0
[[.,[.,.]],[[[.,.],.],.]] => [1,1,0,0,1,1,0,1,0,1,0,0] => [2,7,1,6,3,4,5] => [1,2,7,5,3,4,6] => 0
[[[.,.],.],[.,[.,[.,.]]]] => [1,0,1,0,1,1,1,1,0,0,0,0] => [4,1,2,5,6,7,3] => [1,4,5,6,7,3,2] => 0
[[.,[.,[.,.]]],[.,[.,.]]] => [1,1,1,0,0,0,1,1,1,0,0,0] => [2,3,5,1,6,7,4] => [1,2,3,5,6,7,4] => 0
[[.,[.,[.,.]]],[[.,.],.]] => [1,1,1,0,0,0,1,1,0,1,0,0] => [2,3,7,1,6,4,5] => [1,2,3,7,5,6,4] => 0
[[[.,.],[.,.]],[.,[.,.]]] => [1,0,1,1,0,0,1,1,1,0,0,0] => [3,1,5,2,6,7,4] => [1,3,5,6,7,4,2] => 0
[[[.,.],[.,.]],[[.,.],.]] => [1,0,1,1,0,0,1,1,0,1,0,0] => [3,1,7,2,6,4,5] => [1,3,7,5,6,4,2] => 0
[[[.,[.,.]],.],[.,[.,.]]] => [1,1,0,0,1,0,1,1,1,0,0,0] => [2,5,1,3,6,7,4] => [1,2,5,6,7,4,3] => 0
[[[.,[.,.]],.],[[.,.],.]] => [1,1,0,0,1,0,1,1,0,1,0,0] => [2,7,1,3,6,4,5] => [1,2,7,5,6,4,3] => 0
[[.,[.,[.,[.,.]]]],[.,.]] => [1,1,1,1,0,0,0,0,1,1,0,0] => [2,3,4,6,1,7,5] => [1,2,3,4,6,7,5] => 0
[[.,[[.,[.,.]],.]],[.,.]] => [1,1,1,0,0,1,0,0,1,1,0,0] => [2,6,4,1,3,7,5] => [1,2,6,7,5,3,4] => 0
[[[.,.],[.,[.,.]]],[.,.]] => [1,0,1,1,1,0,0,0,1,1,0,0] => [3,1,4,6,2,7,5] => [1,3,4,6,7,5,2] => 0
[[[.,[.,.]],[.,.]],[.,.]] => [1,1,0,0,1,1,0,0,1,1,0,0] => [2,4,1,6,3,7,5] => [1,2,4,6,7,5,3] => 0
[[[.,[.,[.,.]]],.],[.,.]] => [1,1,1,0,0,0,1,0,1,1,0,0] => [2,3,6,1,4,7,5] => [1,2,3,6,7,5,4] => 0
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Description
The number of stretching pairs of a permutation.
This is the number of pairs $(i,j)$ with $\pi(i) < i < j < \pi(j)$.
This is the number of pairs $(i,j)$ with $\pi(i) < i < j < \pi(j)$.
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
cycle-as-one-line notation
Description
Return the permutation obtained by concatenating the cycles of a permutation, each written with minimal element first, sorted by minimal element.
Map
Ringel
Description
The Ringel permutation of the LNakayama algebra corresponding to a Dyck path.
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