Identifier
-
Mp00025:
Dyck paths
—to 132-avoiding permutation⟶
Permutations
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000007: Permutations ⟶ ℤ
Values
[1,0] => [1] => [1] => [1] => 1
[1,0,1,0] => [2,1] => [2,1] => [1,2] => 1
[1,1,0,0] => [1,2] => [1,2] => [2,1] => 2
[1,0,1,0,1,0] => [3,2,1] => [3,2,1] => [1,2,3] => 1
[1,0,1,1,0,0] => [2,3,1] => [3,1,2] => [1,3,2] => 2
[1,1,0,0,1,0] => [3,1,2] => [2,3,1] => [2,1,3] => 1
[1,1,0,1,0,0] => [2,1,3] => [2,1,3] => [2,3,1] => 2
[1,1,1,0,0,0] => [1,2,3] => [1,2,3] => [3,2,1] => 3
[1,0,1,0,1,0,1,0] => [4,3,2,1] => [4,3,2,1] => [1,2,3,4] => 1
[1,0,1,0,1,1,0,0] => [3,4,2,1] => [4,2,1,3] => [1,3,4,2] => 2
[1,0,1,1,0,0,1,0] => [4,2,3,1] => [3,1,4,2] => [2,4,1,3] => 2
[1,0,1,1,0,1,0,0] => [3,2,4,1] => [4,1,3,2] => [1,4,2,3] => 2
[1,0,1,1,1,0,0,0] => [2,3,4,1] => [4,1,2,3] => [1,4,3,2] => 3
[1,1,0,0,1,0,1,0] => [4,3,1,2] => [2,4,3,1] => [3,1,2,4] => 1
[1,1,0,0,1,1,0,0] => [3,4,1,2] => [2,4,1,3] => [3,1,4,2] => 2
[1,1,0,1,0,0,1,0] => [4,2,1,3] => [3,4,2,1] => [2,1,3,4] => 1
[1,1,0,1,0,1,0,0] => [3,2,1,4] => [3,2,1,4] => [2,3,4,1] => 2
[1,1,0,1,1,0,0,0] => [2,3,1,4] => [3,1,2,4] => [2,4,3,1] => 3
[1,1,1,0,0,0,1,0] => [4,1,2,3] => [2,3,4,1] => [3,2,1,4] => 1
[1,1,1,0,0,1,0,0] => [3,1,2,4] => [2,3,1,4] => [3,2,4,1] => 2
[1,1,1,0,1,0,0,0] => [2,1,3,4] => [2,1,3,4] => [3,4,2,1] => 3
[1,1,1,1,0,0,0,0] => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 4
[1,0,1,0,1,0,1,0,1,0] => [5,4,3,2,1] => [5,4,3,2,1] => [1,2,3,4,5] => 1
[1,0,1,0,1,0,1,1,0,0] => [4,5,3,2,1] => [5,3,2,1,4] => [1,3,4,5,2] => 2
[1,0,1,0,1,1,0,0,1,0] => [5,3,4,2,1] => [4,2,1,5,3] => [2,4,5,1,3] => 2
[1,0,1,0,1,1,0,1,0,0] => [4,3,5,2,1] => [5,2,1,4,3] => [1,4,5,2,3] => 2
[1,0,1,0,1,1,1,0,0,0] => [3,4,5,2,1] => [5,2,1,3,4] => [1,4,5,3,2] => 3
[1,0,1,1,0,0,1,0,1,0] => [5,4,2,3,1] => [3,1,5,4,2] => [3,5,1,2,4] => 2
[1,0,1,1,0,0,1,1,0,0] => [4,5,2,3,1] => [3,1,5,2,4] => [3,5,1,4,2] => 3
[1,0,1,1,0,1,0,0,1,0] => [5,3,2,4,1] => [4,1,5,3,2] => [2,5,1,3,4] => 2
[1,0,1,1,0,1,0,1,0,0] => [4,3,2,5,1] => [5,1,4,3,2] => [1,5,2,3,4] => 2
[1,0,1,1,0,1,1,0,0,0] => [3,4,2,5,1] => [5,1,4,2,3] => [1,5,2,4,3] => 3
[1,0,1,1,1,0,0,0,1,0] => [5,2,3,4,1] => [4,1,3,5,2] => [2,5,3,1,4] => 2
[1,0,1,1,1,0,0,1,0,0] => [4,2,3,5,1] => [5,1,3,4,2] => [1,5,3,2,4] => 2
[1,0,1,1,1,0,1,0,0,0] => [3,2,4,5,1] => [5,1,3,2,4] => [1,5,3,4,2] => 3
[1,0,1,1,1,1,0,0,0,0] => [2,3,4,5,1] => [5,1,2,3,4] => [1,5,4,3,2] => 4
[1,1,0,0,1,0,1,0,1,0] => [5,4,3,1,2] => [2,5,4,3,1] => [4,1,2,3,5] => 1
[1,1,0,0,1,0,1,1,0,0] => [4,5,3,1,2] => [2,5,3,1,4] => [4,1,3,5,2] => 2
[1,1,0,0,1,1,0,0,1,0] => [5,3,4,1,2] => [2,4,1,5,3] => [4,2,5,1,3] => 2
[1,1,0,0,1,1,0,1,0,0] => [4,3,5,1,2] => [2,5,1,4,3] => [4,1,5,2,3] => 2
[1,1,0,0,1,1,1,0,0,0] => [3,4,5,1,2] => [2,5,1,3,4] => [4,1,5,3,2] => 3
[1,1,0,1,0,0,1,0,1,0] => [5,4,2,1,3] => [3,5,4,2,1] => [3,1,2,4,5] => 1
[1,1,0,1,0,0,1,1,0,0] => [4,5,2,1,3] => [3,5,2,1,4] => [3,1,4,5,2] => 2
[1,1,0,1,0,1,0,0,1,0] => [5,3,2,1,4] => [4,5,3,2,1] => [2,1,3,4,5] => 1
[1,1,0,1,0,1,0,1,0,0] => [4,3,2,1,5] => [4,3,2,1,5] => [2,3,4,5,1] => 2
[1,1,0,1,0,1,1,0,0,0] => [3,4,2,1,5] => [4,2,1,3,5] => [2,4,5,3,1] => 3
[1,1,0,1,1,0,0,0,1,0] => [5,2,3,1,4] => [3,1,4,5,2] => [3,5,2,1,4] => 2
[1,1,0,1,1,0,0,1,0,0] => [4,2,3,1,5] => [3,1,4,2,5] => [3,5,2,4,1] => 3
[1,1,0,1,1,0,1,0,0,0] => [3,2,4,1,5] => [4,1,3,2,5] => [2,5,3,4,1] => 3
[1,1,0,1,1,1,0,0,0,0] => [2,3,4,1,5] => [4,1,2,3,5] => [2,5,4,3,1] => 4
[1,1,1,0,0,0,1,0,1,0] => [5,4,1,2,3] => [2,3,5,4,1] => [4,3,1,2,5] => 1
[1,1,1,0,0,0,1,1,0,0] => [4,5,1,2,3] => [2,3,5,1,4] => [4,3,1,5,2] => 2
[1,1,1,0,0,1,0,0,1,0] => [5,3,1,2,4] => [2,4,5,3,1] => [4,2,1,3,5] => 1
[1,1,1,0,0,1,0,1,0,0] => [4,3,1,2,5] => [2,4,3,1,5] => [4,2,3,5,1] => 2
[1,1,1,0,0,1,1,0,0,0] => [3,4,1,2,5] => [2,4,1,3,5] => [4,2,5,3,1] => 3
[1,1,1,0,1,0,0,0,1,0] => [5,2,1,3,4] => [3,4,5,2,1] => [3,2,1,4,5] => 1
[1,1,1,0,1,0,0,1,0,0] => [4,2,1,3,5] => [3,4,2,1,5] => [3,2,4,5,1] => 2
[1,1,1,0,1,0,1,0,0,0] => [3,2,1,4,5] => [3,2,1,4,5] => [3,4,5,2,1] => 3
[1,1,1,0,1,1,0,0,0,0] => [2,3,1,4,5] => [3,1,2,4,5] => [3,5,4,2,1] => 4
[1,1,1,1,0,0,0,0,1,0] => [5,1,2,3,4] => [2,3,4,5,1] => [4,3,2,1,5] => 1
[1,1,1,1,0,0,0,1,0,0] => [4,1,2,3,5] => [2,3,4,1,5] => [4,3,2,5,1] => 2
[1,1,1,1,0,0,1,0,0,0] => [3,1,2,4,5] => [2,3,1,4,5] => [4,3,5,2,1] => 3
[1,1,1,1,0,1,0,0,0,0] => [2,1,3,4,5] => [2,1,3,4,5] => [4,5,3,2,1] => 4
[1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 5
[1,0,1,0,1,0,1,0,1,0,1,0] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => [1,2,3,4,5,6] => 1
[1,0,1,0,1,0,1,0,1,1,0,0] => [5,6,4,3,2,1] => [6,4,3,2,1,5] => [1,3,4,5,6,2] => 2
[1,0,1,0,1,0,1,1,0,0,1,0] => [6,4,5,3,2,1] => [5,3,2,1,6,4] => [2,4,5,6,1,3] => 2
[1,0,1,0,1,0,1,1,0,1,0,0] => [5,4,6,3,2,1] => [6,3,2,1,5,4] => [1,4,5,6,2,3] => 2
[1,0,1,0,1,0,1,1,1,0,0,0] => [4,5,6,3,2,1] => [6,3,2,1,4,5] => [1,4,5,6,3,2] => 3
[1,0,1,0,1,1,0,0,1,0,1,0] => [6,5,3,4,2,1] => [4,2,1,6,5,3] => [3,5,6,1,2,4] => 2
[1,0,1,0,1,1,0,0,1,1,0,0] => [5,6,3,4,2,1] => [4,2,1,6,3,5] => [3,5,6,1,4,2] => 3
[1,0,1,0,1,1,0,1,0,0,1,0] => [6,4,3,5,2,1] => [5,2,1,6,4,3] => [2,5,6,1,3,4] => 2
[1,0,1,0,1,1,0,1,0,1,0,0] => [5,4,3,6,2,1] => [6,2,1,5,4,3] => [1,5,6,2,3,4] => 2
[1,0,1,0,1,1,0,1,1,0,0,0] => [4,5,3,6,2,1] => [6,2,1,5,3,4] => [1,5,6,2,4,3] => 3
[1,0,1,0,1,1,1,0,0,0,1,0] => [6,3,4,5,2,1] => [5,2,1,4,6,3] => [2,5,6,3,1,4] => 2
[1,0,1,0,1,1,1,0,0,1,0,0] => [5,3,4,6,2,1] => [6,2,1,4,5,3] => [1,5,6,3,2,4] => 2
[1,0,1,0,1,1,1,0,1,0,0,0] => [4,3,5,6,2,1] => [6,2,1,4,3,5] => [1,5,6,3,4,2] => 3
[1,0,1,0,1,1,1,1,0,0,0,0] => [3,4,5,6,2,1] => [6,2,1,3,4,5] => [1,5,6,4,3,2] => 4
[1,0,1,1,0,0,1,0,1,0,1,0] => [6,5,4,2,3,1] => [3,1,6,5,4,2] => [4,6,1,2,3,5] => 2
[1,0,1,1,0,0,1,0,1,1,0,0] => [5,6,4,2,3,1] => [3,1,6,4,2,5] => [4,6,1,3,5,2] => 3
[1,0,1,1,0,0,1,1,0,0,1,0] => [6,4,5,2,3,1] => [3,1,5,2,6,4] => [4,6,2,5,1,3] => 3
[1,0,1,1,0,0,1,1,0,1,0,0] => [5,4,6,2,3,1] => [3,1,6,2,5,4] => [4,6,1,5,2,3] => 3
[1,0,1,1,0,0,1,1,1,0,0,0] => [4,5,6,2,3,1] => [3,1,6,2,4,5] => [4,6,1,5,3,2] => 4
[1,0,1,1,0,1,0,0,1,0,1,0] => [6,5,3,2,4,1] => [4,1,6,5,3,2] => [3,6,1,2,4,5] => 2
[1,0,1,1,0,1,0,0,1,1,0,0] => [5,6,3,2,4,1] => [4,1,6,3,2,5] => [3,6,1,4,5,2] => 3
[1,0,1,1,0,1,0,1,0,0,1,0] => [6,4,3,2,5,1] => [5,1,6,4,3,2] => [2,6,1,3,4,5] => 2
[1,0,1,1,0,1,0,1,0,1,0,0] => [5,4,3,2,6,1] => [6,1,5,4,3,2] => [1,6,2,3,4,5] => 2
[1,0,1,1,0,1,0,1,1,0,0,0] => [4,5,3,2,6,1] => [6,1,5,3,2,4] => [1,6,2,4,5,3] => 3
[1,0,1,1,0,1,1,0,0,0,1,0] => [6,3,4,2,5,1] => [5,1,4,2,6,3] => [2,6,3,5,1,4] => 3
[1,0,1,1,0,1,1,0,0,1,0,0] => [5,3,4,2,6,1] => [6,1,4,2,5,3] => [1,6,3,5,2,4] => 3
[1,0,1,1,0,1,1,0,1,0,0,0] => [4,3,5,2,6,1] => [6,1,5,2,4,3] => [1,6,2,5,3,4] => 3
[1,0,1,1,0,1,1,1,0,0,0,0] => [3,4,5,2,6,1] => [6,1,5,2,3,4] => [1,6,2,5,4,3] => 4
[1,0,1,1,1,0,0,0,1,0,1,0] => [6,5,2,3,4,1] => [4,1,3,6,5,2] => [3,6,4,1,2,5] => 2
[1,0,1,1,1,0,0,0,1,1,0,0] => [5,6,2,3,4,1] => [4,1,3,6,2,5] => [3,6,4,1,5,2] => 3
[1,0,1,1,1,0,0,1,0,0,1,0] => [6,4,2,3,5,1] => [5,1,3,6,4,2] => [2,6,4,1,3,5] => 2
[1,0,1,1,1,0,0,1,0,1,0,0] => [5,4,2,3,6,1] => [6,1,3,5,4,2] => [1,6,4,2,3,5] => 2
[1,0,1,1,1,0,0,1,1,0,0,0] => [4,5,2,3,6,1] => [6,1,3,5,2,4] => [1,6,4,2,5,3] => 3
[1,0,1,1,1,0,1,0,0,0,1,0] => [6,3,2,4,5,1] => [5,1,4,6,3,2] => [2,6,3,1,4,5] => 2
[1,0,1,1,1,0,1,0,0,1,0,0] => [5,3,2,4,6,1] => [6,1,4,5,3,2] => [1,6,3,2,4,5] => 2
[1,0,1,1,1,0,1,0,1,0,0,0] => [4,3,2,5,6,1] => [6,1,4,3,2,5] => [1,6,3,4,5,2] => 3
[1,0,1,1,1,0,1,1,0,0,0,0] => [3,4,2,5,6,1] => [6,1,4,2,3,5] => [1,6,3,5,4,2] => 4
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Description
The number of saliances of the permutation.
A saliance is a right-to-left maximum. This can be described as an occurrence of the mesh pattern $([1], {(1,1)})$, i.e., the upper right quadrant is shaded, see [1].
A saliance is a right-to-left maximum. This can be described as an occurrence of the mesh pattern $([1], {(1,1)})$, i.e., the upper right quadrant is shaded, see [1].
Map
invert Laguerre heap
Description
The permutation obtained by inverting the corresponding Laguerre heap, according to Viennot.
Let $\pi$ be a permutation. Following Viennot [1], we associate to $\pi$ a heap of pieces, by considering each decreasing run $(\pi_i, \pi_{i+1}, \dots, \pi_j)$ of $\pi$ as one piece, beginning with the left most run. Two pieces commute if and only if the minimal element of one piece is larger than the maximal element of the other piece.
This map yields the permutation corresponding to the heap obtained by reversing the reading direction of the heap.
Equivalently, this is the permutation obtained by flipping the noncrossing arc diagram of Reading [2] vertically.
By definition, this map preserves the set of decreasing runs.
Let $\pi$ be a permutation. Following Viennot [1], we associate to $\pi$ a heap of pieces, by considering each decreasing run $(\pi_i, \pi_{i+1}, \dots, \pi_j)$ of $\pi$ as one piece, beginning with the left most run. Two pieces commute if and only if the minimal element of one piece is larger than the maximal element of the other piece.
This map yields the permutation corresponding to the heap obtained by reversing the reading direction of the heap.
Equivalently, this is the permutation obtained by flipping the noncrossing arc diagram of Reading [2] vertically.
By definition, this map preserves the set of decreasing runs.
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].
Map
complement
Description
Sents a permutation to its complement.
The complement of a permutation $\sigma$ of length $n$ is the permutation $\tau$ with $\tau(i) = n+1-\sigma(i)$
The complement of a permutation $\sigma$ of length $n$ is the permutation $\tau$ with $\tau(i) = n+1-\sigma(i)$
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