Identifier
-
Mp00025:
Dyck paths
—to 132-avoiding permutation⟶
Permutations
Mp00239: Permutations —Corteel⟶ Permutations
St000541: Permutations ⟶ ℤ (values match St000542The number of left-to-right-minima of a permutation.)
Values
[1,0,1,0] => [2,1] => [2,1] => 1
[1,1,0,0] => [1,2] => [1,2] => 0
[1,0,1,0,1,0] => [3,2,1] => [2,3,1] => 1
[1,0,1,1,0,0] => [2,3,1] => [3,2,1] => 2
[1,1,0,0,1,0] => [3,1,2] => [3,1,2] => 1
[1,1,0,1,0,0] => [2,1,3] => [2,1,3] => 1
[1,1,1,0,0,0] => [1,2,3] => [1,2,3] => 0
[1,0,1,0,1,0,1,0] => [4,3,2,1] => [3,4,1,2] => 1
[1,0,1,0,1,1,0,0] => [3,4,2,1] => [4,3,1,2] => 2
[1,0,1,1,0,0,1,0] => [4,2,3,1] => [2,3,4,1] => 1
[1,0,1,1,0,1,0,0] => [3,2,4,1] => [2,4,3,1] => 1
[1,0,1,1,1,0,0,0] => [2,3,4,1] => [4,2,3,1] => 2
[1,1,0,0,1,0,1,0] => [4,3,1,2] => [3,4,2,1] => 2
[1,1,0,0,1,1,0,0] => [3,4,1,2] => [4,3,2,1] => 3
[1,1,0,1,0,0,1,0] => [4,2,1,3] => [2,4,1,3] => 1
[1,1,0,1,0,1,0,0] => [3,2,1,4] => [2,3,1,4] => 1
[1,1,0,1,1,0,0,0] => [2,3,1,4] => [3,2,1,4] => 2
[1,1,1,0,0,0,1,0] => [4,1,2,3] => [4,1,2,3] => 1
[1,1,1,0,0,1,0,0] => [3,1,2,4] => [3,1,2,4] => 1
[1,1,1,0,1,0,0,0] => [2,1,3,4] => [2,1,3,4] => 1
[1,1,1,1,0,0,0,0] => [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,0,1,0,1,0] => [5,4,3,2,1] => [3,4,5,1,2] => 1
[1,0,1,0,1,0,1,1,0,0] => [4,5,3,2,1] => [4,3,5,1,2] => 2
[1,0,1,0,1,1,0,0,1,0] => [5,3,4,2,1] => [3,5,4,1,2] => 1
[1,0,1,0,1,1,0,1,0,0] => [4,3,5,2,1] => [4,5,3,1,2] => 2
[1,0,1,0,1,1,1,0,0,0] => [3,4,5,2,1] => [5,4,3,1,2] => 3
[1,0,1,1,0,0,1,0,1,0] => [5,4,2,3,1] => [4,5,1,2,3] => 1
[1,0,1,1,0,0,1,1,0,0] => [4,5,2,3,1] => [5,4,1,2,3] => 2
[1,0,1,1,0,1,0,0,1,0] => [5,3,2,4,1] => [3,4,1,5,2] => 1
[1,0,1,1,0,1,0,1,0,0] => [4,3,2,5,1] => [3,5,1,4,2] => 1
[1,0,1,1,0,1,1,0,0,0] => [3,4,2,5,1] => [5,3,1,4,2] => 2
[1,0,1,1,1,0,0,0,1,0] => [5,2,3,4,1] => [2,3,4,5,1] => 1
[1,0,1,1,1,0,0,1,0,0] => [4,2,3,5,1] => [2,3,5,4,1] => 1
[1,0,1,1,1,0,1,0,0,0] => [3,2,4,5,1] => [2,5,3,4,1] => 1
[1,0,1,1,1,1,0,0,0,0] => [2,3,4,5,1] => [5,2,3,4,1] => 2
[1,1,0,0,1,0,1,0,1,0] => [5,4,3,1,2] => [3,4,5,2,1] => 2
[1,1,0,0,1,0,1,1,0,0] => [4,5,3,1,2] => [4,3,5,2,1] => 3
[1,1,0,0,1,1,0,0,1,0] => [5,3,4,1,2] => [3,5,4,2,1] => 2
[1,1,0,0,1,1,0,1,0,0] => [4,3,5,1,2] => [4,5,3,2,1] => 3
[1,1,0,0,1,1,1,0,0,0] => [3,4,5,1,2] => [5,4,3,2,1] => 4
[1,1,0,1,0,0,1,0,1,0] => [5,4,2,1,3] => [4,5,1,3,2] => 1
[1,1,0,1,0,0,1,1,0,0] => [4,5,2,1,3] => [5,4,1,3,2] => 2
[1,1,0,1,0,1,0,0,1,0] => [5,3,2,1,4] => [3,5,1,2,4] => 1
[1,1,0,1,0,1,0,1,0,0] => [4,3,2,1,5] => [3,4,1,2,5] => 1
[1,1,0,1,0,1,1,0,0,0] => [3,4,2,1,5] => [4,3,1,2,5] => 2
[1,1,0,1,1,0,0,0,1,0] => [5,2,3,1,4] => [2,3,5,1,4] => 1
[1,1,0,1,1,0,0,1,0,0] => [4,2,3,1,5] => [2,3,4,1,5] => 1
[1,1,0,1,1,0,1,0,0,0] => [3,2,4,1,5] => [2,4,3,1,5] => 1
[1,1,0,1,1,1,0,0,0,0] => [2,3,4,1,5] => [4,2,3,1,5] => 2
[1,1,1,0,0,0,1,0,1,0] => [5,4,1,2,3] => [4,5,2,3,1] => 2
[1,1,1,0,0,0,1,1,0,0] => [4,5,1,2,3] => [5,4,2,3,1] => 3
[1,1,1,0,0,1,0,0,1,0] => [5,3,1,2,4] => [3,5,2,1,4] => 2
[1,1,1,0,0,1,0,1,0,0] => [4,3,1,2,5] => [3,4,2,1,5] => 2
[1,1,1,0,0,1,1,0,0,0] => [3,4,1,2,5] => [4,3,2,1,5] => 3
[1,1,1,0,1,0,0,0,1,0] => [5,2,1,3,4] => [2,5,1,3,4] => 1
[1,1,1,0,1,0,0,1,0,0] => [4,2,1,3,5] => [2,4,1,3,5] => 1
[1,1,1,0,1,0,1,0,0,0] => [3,2,1,4,5] => [2,3,1,4,5] => 1
[1,1,1,0,1,1,0,0,0,0] => [2,3,1,4,5] => [3,2,1,4,5] => 2
[1,1,1,1,0,0,0,0,1,0] => [5,1,2,3,4] => [5,1,2,3,4] => 1
[1,1,1,1,0,0,0,1,0,0] => [4,1,2,3,5] => [4,1,2,3,5] => 1
[1,1,1,1,0,0,1,0,0,0] => [3,1,2,4,5] => [3,1,2,4,5] => 1
[1,1,1,1,0,1,0,0,0,0] => [2,1,3,4,5] => [2,1,3,4,5] => 1
[1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,0,1,0,1,0] => [6,5,4,3,2,1] => [4,5,6,1,2,3] => 1
[1,0,1,0,1,0,1,0,1,1,0,0] => [5,6,4,3,2,1] => [5,4,6,1,2,3] => 2
[1,0,1,0,1,0,1,1,0,0,1,0] => [6,4,5,3,2,1] => [4,6,5,1,2,3] => 1
[1,0,1,0,1,0,1,1,0,1,0,0] => [5,4,6,3,2,1] => [5,6,4,1,2,3] => 2
[1,0,1,0,1,0,1,1,1,0,0,0] => [4,5,6,3,2,1] => [6,5,4,1,2,3] => 3
[1,0,1,0,1,1,0,0,1,0,1,0] => [6,5,3,4,2,1] => [3,4,5,6,1,2] => 1
[1,0,1,0,1,1,0,0,1,1,0,0] => [5,6,3,4,2,1] => [4,3,5,6,1,2] => 2
[1,0,1,0,1,1,0,1,0,0,1,0] => [6,4,3,5,2,1] => [3,4,6,5,1,2] => 1
[1,0,1,0,1,1,0,1,0,1,0,0] => [5,4,3,6,2,1] => [3,5,6,4,1,2] => 1
[1,0,1,0,1,1,0,1,1,0,0,0] => [4,5,3,6,2,1] => [5,3,6,4,1,2] => 2
[1,0,1,0,1,1,1,0,0,0,1,0] => [6,3,4,5,2,1] => [3,6,4,5,1,2] => 1
[1,0,1,0,1,1,1,0,0,1,0,0] => [5,3,4,6,2,1] => [3,6,5,4,1,2] => 1
[1,0,1,0,1,1,1,0,1,0,0,0] => [4,3,5,6,2,1] => [5,6,3,4,1,2] => 2
[1,0,1,0,1,1,1,1,0,0,0,0] => [3,4,5,6,2,1] => [6,5,3,4,1,2] => 3
[1,0,1,1,0,0,1,0,1,0,1,0] => [6,5,4,2,3,1] => [4,5,6,2,1,3] => 2
[1,0,1,1,0,0,1,0,1,1,0,0] => [5,6,4,2,3,1] => [5,4,6,2,1,3] => 3
[1,0,1,1,0,0,1,1,0,0,1,0] => [6,4,5,2,3,1] => [4,6,5,2,1,3] => 2
[1,0,1,1,0,0,1,1,0,1,0,0] => [5,4,6,2,3,1] => [5,6,4,2,1,3] => 3
[1,0,1,1,0,0,1,1,1,0,0,0] => [4,5,6,2,3,1] => [6,5,4,2,1,3] => 4
[1,0,1,1,0,1,0,0,1,0,1,0] => [6,5,3,2,4,1] => [3,5,6,1,2,4] => 1
[1,0,1,1,0,1,0,0,1,1,0,0] => [5,6,3,2,4,1] => [5,3,6,1,2,4] => 2
[1,0,1,1,0,1,0,1,0,0,1,0] => [6,4,3,2,5,1] => [3,4,5,1,6,2] => 1
[1,0,1,1,0,1,0,1,0,1,0,0] => [5,4,3,2,6,1] => [3,4,6,1,5,2] => 1
[1,0,1,1,0,1,0,1,1,0,0,0] => [4,5,3,2,6,1] => [4,3,6,1,5,2] => 2
[1,0,1,1,0,1,1,0,0,0,1,0] => [6,3,4,2,5,1] => [3,5,4,1,6,2] => 1
[1,0,1,1,0,1,1,0,0,1,0,0] => [5,3,4,2,6,1] => [3,6,4,1,5,2] => 1
[1,0,1,1,0,1,1,0,1,0,0,0] => [4,3,5,2,6,1] => [4,6,3,1,5,2] => 2
[1,0,1,1,0,1,1,1,0,0,0,0] => [3,4,5,2,6,1] => [6,4,3,1,5,2] => 3
[1,0,1,1,1,0,0,0,1,0,1,0] => [6,5,2,3,4,1] => [5,6,1,2,3,4] => 1
[1,0,1,1,1,0,0,0,1,1,0,0] => [5,6,2,3,4,1] => [6,5,1,2,3,4] => 2
[1,0,1,1,1,0,0,1,0,0,1,0] => [6,4,2,3,5,1] => [4,5,1,2,6,3] => 1
[1,0,1,1,1,0,0,1,0,1,0,0] => [5,4,2,3,6,1] => [4,6,1,2,5,3] => 1
[1,0,1,1,1,0,0,1,1,0,0,0] => [4,5,2,3,6,1] => [6,4,1,2,5,3] => 2
[1,0,1,1,1,0,1,0,0,0,1,0] => [6,3,2,4,5,1] => [3,4,1,5,6,2] => 1
[1,0,1,1,1,0,1,0,0,1,0,0] => [5,3,2,4,6,1] => [3,4,1,6,5,2] => 1
[1,0,1,1,1,0,1,0,1,0,0,0] => [4,3,2,5,6,1] => [3,6,1,4,5,2] => 1
[1,0,1,1,1,0,1,1,0,0,0,0] => [3,4,2,5,6,1] => [6,3,1,4,5,2] => 2
[1,0,1,1,1,1,0,0,0,0,1,0] => [6,2,3,4,5,1] => [2,3,4,5,6,1] => 1
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Description
The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right.
For a permutation $\pi$ of length $n$, this is the number of indices $2 \leq j \leq n$ such that for all $1 \leq i < j$, the pair $(i,j)$ is an inversion of $\pi$.
For a permutation $\pi$ of length $n$, this is the number of indices $2 \leq j \leq n$ such that for all $1 \leq i < j$, the pair $(i,j)$ is an inversion of $\pi$.
Map
Corteel
Description
Corteel's map interchanging the number of crossings and the number of nestings of a permutation.
This involution creates a labelled bicoloured Motzkin path, using the Foata-Zeilberger map. In the corresponding bump diagram, each label records the number of arcs nesting the given arc. Then each label is replaced by its complement, and the inverse of the Foata-Zeilberger map is applied.
This involution creates a labelled bicoloured Motzkin path, using the Foata-Zeilberger map. In the corresponding bump diagram, each label records the number of arcs nesting the given arc. Then each label is replaced by its complement, and the inverse of the Foata-Zeilberger map is applied.
Map
to 132-avoiding permutation
Description
Sends a Dyck path to a 132-avoiding permutation.
This bijection is defined in [1, Section 2].
This bijection is defined in [1, Section 2].
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