Identifier
-
Mp00020:
Binary trees
—to Tamari-corresponding Dyck path⟶
Dyck paths
Mp00327: Dyck paths —inverse Kreweras complement⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000354: Permutations ⟶ ℤ
Values
[.,[.,.]] => [1,1,0,0] => [1,0,1,0] => [2,1] => 1
[[.,.],.] => [1,0,1,0] => [1,1,0,0] => [1,2] => 0
[.,[.,[.,.]]] => [1,1,1,0,0,0] => [1,0,1,0,1,0] => [2,3,1] => 1
[.,[[.,.],.]] => [1,1,0,1,0,0] => [1,1,0,0,1,0] => [1,3,2] => 1
[[.,.],[.,.]] => [1,0,1,1,0,0] => [1,1,0,1,0,0] => [3,1,2] => 1
[[.,[.,.]],.] => [1,1,0,0,1,0] => [1,0,1,1,0,0] => [2,1,3] => 1
[[[.,.],.],.] => [1,0,1,0,1,0] => [1,1,1,0,0,0] => [1,2,3] => 0
[.,[.,[.,[.,.]]]] => [1,1,1,1,0,0,0,0] => [1,0,1,0,1,0,1,0] => [2,3,4,1] => 1
[.,[.,[[.,.],.]]] => [1,1,1,0,1,0,0,0] => [1,0,1,1,0,0,1,0] => [2,1,4,3] => 2
[.,[[.,.],[.,.]]] => [1,1,0,1,1,0,0,0] => [1,1,0,0,1,0,1,0] => [1,3,4,2] => 1
[.,[[.,[.,.]],.]] => [1,1,1,0,0,1,0,0] => [1,1,0,1,0,0,1,0] => [3,1,4,2] => 1
[.,[[[.,.],.],.]] => [1,1,0,1,0,1,0,0] => [1,1,1,0,0,0,1,0] => [1,2,4,3] => 1
[[.,.],[.,[.,.]]] => [1,0,1,1,1,0,0,0] => [1,1,0,1,0,1,0,0] => [3,4,1,2] => 1
[[.,.],[[.,.],.]] => [1,0,1,1,0,1,0,0] => [1,1,1,0,0,1,0,0] => [1,4,2,3] => 1
[[.,[.,.]],[.,.]] => [1,1,0,0,1,1,0,0] => [1,0,1,1,0,1,0,0] => [2,4,1,3] => 2
[[[.,.],.],[.,.]] => [1,0,1,0,1,1,0,0] => [1,1,1,0,1,0,0,0] => [4,1,2,3] => 1
[[.,[.,[.,.]]],.] => [1,1,1,0,0,0,1,0] => [1,0,1,0,1,1,0,0] => [2,3,1,4] => 1
[[.,[[.,.],.]],.] => [1,1,0,1,0,0,1,0] => [1,1,0,0,1,1,0,0] => [1,3,2,4] => 1
[[[.,.],[.,.]],.] => [1,0,1,1,0,0,1,0] => [1,1,0,1,1,0,0,0] => [3,1,2,4] => 1
[[[.,[.,.]],.],.] => [1,1,0,0,1,0,1,0] => [1,0,1,1,1,0,0,0] => [2,1,3,4] => 1
[[[[.,.],.],.],.] => [1,0,1,0,1,0,1,0] => [1,1,1,1,0,0,0,0] => [1,2,3,4] => 0
[.,[.,[.,[.,[.,.]]]]] => [1,1,1,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0] => [2,3,4,5,1] => 1
[.,[.,[.,[[.,.],.]]]] => [1,1,1,1,0,1,0,0,0,0] => [1,0,1,0,1,1,0,0,1,0] => [2,3,1,5,4] => 2
[.,[.,[[.,.],[.,.]]]] => [1,1,1,0,1,1,0,0,0,0] => [1,0,1,1,0,0,1,0,1,0] => [2,1,4,5,3] => 2
[.,[.,[[.,[.,.]],.]]] => [1,1,1,1,0,0,1,0,0,0] => [1,0,1,1,0,1,0,0,1,0] => [2,4,1,5,3] => 2
[.,[.,[[[.,.],.],.]]] => [1,1,1,0,1,0,1,0,0,0] => [1,0,1,1,1,0,0,0,1,0] => [2,1,3,5,4] => 2
[.,[[.,.],[.,[.,.]]]] => [1,1,0,1,1,1,0,0,0,0] => [1,1,0,0,1,0,1,0,1,0] => [1,3,4,5,2] => 1
[.,[[.,.],[[.,.],.]]] => [1,1,0,1,1,0,1,0,0,0] => [1,1,0,0,1,1,0,0,1,0] => [1,3,2,5,4] => 2
[.,[[.,[.,.]],[.,.]]] => [1,1,1,0,0,1,1,0,0,0] => [1,1,0,1,0,0,1,0,1,0] => [3,1,4,5,2] => 1
[.,[[[.,.],.],[.,.]]] => [1,1,0,1,0,1,1,0,0,0] => [1,1,1,0,0,0,1,0,1,0] => [1,2,4,5,3] => 1
[.,[[.,[.,[.,.]]],.]] => [1,1,1,1,0,0,0,1,0,0] => [1,1,0,1,0,1,0,0,1,0] => [3,4,1,5,2] => 1
[.,[[.,[[.,.],.]],.]] => [1,1,1,0,1,0,0,1,0,0] => [1,1,1,0,0,1,0,0,1,0] => [1,4,2,5,3] => 1
[.,[[[.,.],[.,.]],.]] => [1,1,0,1,1,0,0,1,0,0] => [1,1,1,0,1,0,0,0,1,0] => [4,1,2,5,3] => 1
[.,[[[.,[.,.]],.],.]] => [1,1,1,0,0,1,0,1,0,0] => [1,1,0,1,1,0,0,0,1,0] => [3,1,2,5,4] => 2
[.,[[[[.,.],.],.],.]] => [1,1,0,1,0,1,0,1,0,0] => [1,1,1,1,0,0,0,0,1,0] => [1,2,3,5,4] => 1
[[.,.],[.,[.,[.,.]]]] => [1,0,1,1,1,1,0,0,0,0] => [1,1,0,1,0,1,0,1,0,0] => [3,4,5,1,2] => 1
[[.,.],[.,[[.,.],.]]] => [1,0,1,1,1,0,1,0,0,0] => [1,1,0,1,1,0,0,1,0,0] => [3,1,5,2,4] => 2
[[.,.],[[.,.],[.,.]]] => [1,0,1,1,0,1,1,0,0,0] => [1,1,1,0,0,1,0,1,0,0] => [1,4,5,2,3] => 1
[[.,.],[[.,[.,.]],.]] => [1,0,1,1,1,0,0,1,0,0] => [1,1,1,0,1,0,0,1,0,0] => [4,1,5,2,3] => 1
[[.,.],[[[.,.],.],.]] => [1,0,1,1,0,1,0,1,0,0] => [1,1,1,1,0,0,0,1,0,0] => [1,2,5,3,4] => 1
[[.,[.,.]],[.,[.,.]]] => [1,1,0,0,1,1,1,0,0,0] => [1,0,1,1,0,1,0,1,0,0] => [2,4,5,1,3] => 2
[[.,[.,.]],[[.,.],.]] => [1,1,0,0,1,1,0,1,0,0] => [1,0,1,1,1,0,0,1,0,0] => [2,1,5,3,4] => 2
[[[.,.],.],[.,[.,.]]] => [1,0,1,0,1,1,1,0,0,0] => [1,1,1,0,1,0,1,0,0,0] => [4,5,1,2,3] => 1
[[[.,.],.],[[.,.],.]] => [1,0,1,0,1,1,0,1,0,0] => [1,1,1,1,0,0,1,0,0,0] => [1,5,2,3,4] => 1
[[.,[.,[.,.]]],[.,.]] => [1,1,1,0,0,0,1,1,0,0] => [1,0,1,0,1,1,0,1,0,0] => [2,3,5,1,4] => 2
[[.,[[.,.],.]],[.,.]] => [1,1,0,1,0,0,1,1,0,0] => [1,1,0,0,1,1,0,1,0,0] => [1,3,5,2,4] => 2
[[[.,.],[.,.]],[.,.]] => [1,0,1,1,0,0,1,1,0,0] => [1,1,0,1,1,0,1,0,0,0] => [3,5,1,2,4] => 2
[[[.,[.,.]],.],[.,.]] => [1,1,0,0,1,0,1,1,0,0] => [1,0,1,1,1,0,1,0,0,0] => [2,5,1,3,4] => 2
[[[[.,.],.],.],[.,.]] => [1,0,1,0,1,0,1,1,0,0] => [1,1,1,1,0,1,0,0,0,0] => [5,1,2,3,4] => 1
[[.,[.,[.,[.,.]]]],.] => [1,1,1,1,0,0,0,0,1,0] => [1,0,1,0,1,0,1,1,0,0] => [2,3,4,1,5] => 1
[[.,[.,[[.,.],.]]],.] => [1,1,1,0,1,0,0,0,1,0] => [1,0,1,1,0,0,1,1,0,0] => [2,1,4,3,5] => 2
[[.,[[.,.],[.,.]]],.] => [1,1,0,1,1,0,0,0,1,0] => [1,1,0,0,1,0,1,1,0,0] => [1,3,4,2,5] => 1
[[.,[[.,[.,.]],.]],.] => [1,1,1,0,0,1,0,0,1,0] => [1,1,0,1,0,0,1,1,0,0] => [3,1,4,2,5] => 1
[[.,[[[.,.],.],.]],.] => [1,1,0,1,0,1,0,0,1,0] => [1,1,1,0,0,0,1,1,0,0] => [1,2,4,3,5] => 1
[[[.,.],[.,[.,.]]],.] => [1,0,1,1,1,0,0,0,1,0] => [1,1,0,1,0,1,1,0,0,0] => [3,4,1,2,5] => 1
[[[.,.],[[.,.],.]],.] => [1,0,1,1,0,1,0,0,1,0] => [1,1,1,0,0,1,1,0,0,0] => [1,4,2,3,5] => 1
[[[.,[.,.]],[.,.]],.] => [1,1,0,0,1,1,0,0,1,0] => [1,0,1,1,0,1,1,0,0,0] => [2,4,1,3,5] => 2
[[[[.,.],.],[.,.]],.] => [1,0,1,0,1,1,0,0,1,0] => [1,1,1,0,1,1,0,0,0,0] => [4,1,2,3,5] => 1
[[[.,[.,[.,.]]],.],.] => [1,1,1,0,0,0,1,0,1,0] => [1,0,1,0,1,1,1,0,0,0] => [2,3,1,4,5] => 1
[[[.,[[.,.],.]],.],.] => [1,1,0,1,0,0,1,0,1,0] => [1,1,0,0,1,1,1,0,0,0] => [1,3,2,4,5] => 1
[[[[.,.],[.,.]],.],.] => [1,0,1,1,0,0,1,0,1,0] => [1,1,0,1,1,1,0,0,0,0] => [3,1,2,4,5] => 1
[[[[.,[.,.]],.],.],.] => [1,1,0,0,1,0,1,0,1,0] => [1,0,1,1,1,1,0,0,0,0] => [2,1,3,4,5] => 1
[[[[[.,.],.],.],.],.] => [1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => 0
[.,[.,[.,[.,[.,[.,.]]]]]] => [1,1,1,1,1,1,0,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0,1,0] => [2,3,4,5,6,1] => 1
[.,[.,[.,[.,[[.,.],.]]]]] => [1,1,1,1,1,0,1,0,0,0,0,0] => [1,0,1,0,1,0,1,1,0,0,1,0] => [2,3,4,1,6,5] => 2
[.,[.,[.,[[.,.],[.,.]]]]] => [1,1,1,1,0,1,1,0,0,0,0,0] => [1,0,1,0,1,1,0,0,1,0,1,0] => [2,3,1,5,6,4] => 2
[.,[.,[.,[[.,[.,.]],.]]]] => [1,1,1,1,1,0,0,1,0,0,0,0] => [1,0,1,0,1,1,0,1,0,0,1,0] => [2,3,5,1,6,4] => 2
[.,[.,[.,[[[.,.],.],.]]]] => [1,1,1,1,0,1,0,1,0,0,0,0] => [1,0,1,0,1,1,1,0,0,0,1,0] => [2,3,1,4,6,5] => 2
[.,[.,[[.,.],[.,[.,.]]]]] => [1,1,1,0,1,1,1,0,0,0,0,0] => [1,0,1,1,0,0,1,0,1,0,1,0] => [2,1,4,5,6,3] => 2
[.,[.,[[.,.],[[.,.],.]]]] => [1,1,1,0,1,1,0,1,0,0,0,0] => [1,0,1,1,0,0,1,1,0,0,1,0] => [2,1,4,3,6,5] => 3
[.,[.,[[.,[.,.]],[.,.]]]] => [1,1,1,1,0,0,1,1,0,0,0,0] => [1,0,1,1,0,1,0,0,1,0,1,0] => [2,4,1,5,6,3] => 2
[.,[.,[[[.,.],.],[.,.]]]] => [1,1,1,0,1,0,1,1,0,0,0,0] => [1,0,1,1,1,0,0,0,1,0,1,0] => [2,1,3,5,6,4] => 2
[.,[.,[[.,[.,[.,.]]],.]]] => [1,1,1,1,1,0,0,0,1,0,0,0] => [1,0,1,1,0,1,0,1,0,0,1,0] => [2,4,5,1,6,3] => 2
[.,[.,[[.,[[.,.],.]],.]]] => [1,1,1,1,0,1,0,0,1,0,0,0] => [1,0,1,1,1,0,0,1,0,0,1,0] => [2,1,5,3,6,4] => 2
[.,[.,[[[.,.],[.,.]],.]]] => [1,1,1,0,1,1,0,0,1,0,0,0] => [1,0,1,1,1,0,1,0,0,0,1,0] => [2,5,1,3,6,4] => 2
[.,[.,[[[.,[.,.]],.],.]]] => [1,1,1,1,0,0,1,0,1,0,0,0] => [1,0,1,1,0,1,1,0,0,0,1,0] => [2,4,1,3,6,5] => 3
[.,[.,[[[[.,.],.],.],.]]] => [1,1,1,0,1,0,1,0,1,0,0,0] => [1,0,1,1,1,1,0,0,0,0,1,0] => [2,1,3,4,6,5] => 2
[.,[[.,.],[.,[.,[.,.]]]]] => [1,1,0,1,1,1,1,0,0,0,0,0] => [1,1,0,0,1,0,1,0,1,0,1,0] => [1,3,4,5,6,2] => 1
[.,[[.,.],[.,[[.,.],.]]]] => [1,1,0,1,1,1,0,1,0,0,0,0] => [1,1,0,0,1,0,1,1,0,0,1,0] => [1,3,4,2,6,5] => 2
[.,[[.,.],[[.,.],[.,.]]]] => [1,1,0,1,1,0,1,1,0,0,0,0] => [1,1,0,0,1,1,0,0,1,0,1,0] => [1,3,2,5,6,4] => 2
[.,[[.,.],[[.,[.,.]],.]]] => [1,1,0,1,1,1,0,0,1,0,0,0] => [1,1,0,0,1,1,0,1,0,0,1,0] => [1,3,5,2,6,4] => 2
[.,[[.,.],[[[.,.],.],.]]] => [1,1,0,1,1,0,1,0,1,0,0,0] => [1,1,0,0,1,1,1,0,0,0,1,0] => [1,3,2,4,6,5] => 2
[.,[[.,[.,.]],[.,[.,.]]]] => [1,1,1,0,0,1,1,1,0,0,0,0] => [1,1,0,1,0,0,1,0,1,0,1,0] => [3,1,4,5,6,2] => 1
[.,[[.,[.,.]],[[.,.],.]]] => [1,1,1,0,0,1,1,0,1,0,0,0] => [1,1,0,1,0,0,1,1,0,0,1,0] => [3,1,4,2,6,5] => 2
[.,[[[.,.],.],[.,[.,.]]]] => [1,1,0,1,0,1,1,1,0,0,0,0] => [1,1,1,0,0,0,1,0,1,0,1,0] => [1,2,4,5,6,3] => 1
[.,[[[.,.],.],[[.,.],.]]] => [1,1,0,1,0,1,1,0,1,0,0,0] => [1,1,1,0,0,0,1,1,0,0,1,0] => [1,2,4,3,6,5] => 2
[.,[[.,[.,[.,.]]],[.,.]]] => [1,1,1,1,0,0,0,1,1,0,0,0] => [1,1,0,1,0,1,0,0,1,0,1,0] => [3,4,1,5,6,2] => 1
[.,[[.,[[.,.],.]],[.,.]]] => [1,1,1,0,1,0,0,1,1,0,0,0] => [1,1,1,0,0,1,0,0,1,0,1,0] => [1,4,2,5,6,3] => 1
[.,[[[.,.],[.,.]],[.,.]]] => [1,1,0,1,1,0,0,1,1,0,0,0] => [1,1,1,0,1,0,0,0,1,0,1,0] => [4,1,2,5,6,3] => 1
[.,[[[.,[.,.]],.],[.,.]]] => [1,1,1,0,0,1,0,1,1,0,0,0] => [1,1,0,1,1,0,0,0,1,0,1,0] => [3,1,2,5,6,4] => 2
[.,[[[[.,.],.],.],[.,.]]] => [1,1,0,1,0,1,0,1,1,0,0,0] => [1,1,1,1,0,0,0,0,1,0,1,0] => [1,2,3,5,6,4] => 1
[.,[[.,[.,[.,[.,.]]]],.]] => [1,1,1,1,1,0,0,0,0,1,0,0] => [1,1,0,1,0,1,0,1,0,0,1,0] => [3,4,5,1,6,2] => 1
[.,[[.,[.,[[.,.],.]]],.]] => [1,1,1,1,0,1,0,0,0,1,0,0] => [1,1,0,1,1,0,0,1,0,0,1,0] => [3,1,5,2,6,4] => 2
[.,[[.,[[.,.],[.,.]]],.]] => [1,1,1,0,1,1,0,0,0,1,0,0] => [1,1,1,0,0,1,0,1,0,0,1,0] => [1,4,5,2,6,3] => 1
[.,[[.,[[.,[.,.]],.]],.]] => [1,1,1,1,0,0,1,0,0,1,0,0] => [1,1,1,0,1,0,0,1,0,0,1,0] => [4,1,5,2,6,3] => 1
[.,[[.,[[[.,.],.],.]],.]] => [1,1,1,0,1,0,1,0,0,1,0,0] => [1,1,1,1,0,0,0,1,0,0,1,0] => [1,2,5,3,6,4] => 1
[.,[[[.,.],[.,[.,.]]],.]] => [1,1,0,1,1,1,0,0,0,1,0,0] => [1,1,1,0,1,0,1,0,0,0,1,0] => [4,5,1,2,6,3] => 1
[.,[[[.,.],[[.,.],.]],.]] => [1,1,0,1,1,0,1,0,0,1,0,0] => [1,1,1,1,0,0,1,0,0,0,1,0] => [1,5,2,3,6,4] => 1
[.,[[[.,[.,.]],[.,.]],.]] => [1,1,1,0,0,1,1,0,0,1,0,0] => [1,1,0,1,1,0,1,0,0,0,1,0] => [3,5,1,2,6,4] => 2
[.,[[[[.,.],.],[.,.]],.]] => [1,1,0,1,0,1,1,0,0,1,0,0] => [1,1,1,1,0,1,0,0,0,0,1,0] => [5,1,2,3,6,4] => 1
[.,[[[.,[.,[.,.]]],.],.]] => [1,1,1,1,0,0,0,1,0,1,0,0] => [1,1,0,1,0,1,1,0,0,0,1,0] => [3,4,1,2,6,5] => 2
>>> Load all 305 entries. <<<
search for individual values
searching the database for the individual values of this statistic
/
search for generating function
searching the database for statistics with the same generating function
Description
The number of recoils of a permutation.
A recoil, or inverse descent of a permutation $\pi$ is a value $i$ such that $i+1$ appears to the left of $i$ in $\pi_1,\pi_2,\dots,\pi_n$.
In other words, this is the number of descents of the inverse permutation. It can be also be described as the number of occurrences of the mesh pattern $([2,1], {(0,1),(1,1),(2,1)})$, i.e., the middle row is shaded.
A recoil, or inverse descent of a permutation $\pi$ is a value $i$ such that $i+1$ appears to the left of $i$ in $\pi_1,\pi_2,\dots,\pi_n$.
In other words, this is the number of descents of the inverse permutation. It can be also be described as the number of occurrences of the mesh pattern $([2,1], {(0,1),(1,1),(2,1)})$, i.e., the middle row is shaded.
Map
to 321-avoiding permutation (Billey-Jockusch-Stanley)
Description
The Billey-Jockusch-Stanley bijection to 321-avoiding permutations.
Map
inverse Kreweras complement
Description
Return the inverse of the Kreweras complement of a Dyck path, regarded as a noncrossing set partition.
To identify Dyck paths and noncrossing set partitions, this maps uses the following classical bijection. The number of down steps after the $i$-th up step of the Dyck path is the size of the block of the set partition whose maximal element is $i$. If $i$ is not a maximal element of a block, the $(i+1)$-st step is also an up step.
To identify Dyck paths and noncrossing set partitions, this maps uses the following classical bijection. The number of down steps after the $i$-th up step of the Dyck path is the size of the block of the set partition whose maximal element is $i$. If $i$ is not a maximal element of a block, the $(i+1)$-st step is also an up step.
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 $r$.
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