Identifier
-
Mp00025:
Dyck paths
—to 132-avoiding permutation⟶
Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00241: Permutations —invert Laguerre heap⟶ 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] => [2,1] => 1
[1,1,0,0] => [1,2] => [1,2] => [1,2] => 0
[1,0,1,0,1,0] => [3,2,1] => [3,1,2] => [2,3,1] => 1
[1,0,1,1,0,0] => [2,3,1] => [3,2,1] => [3,2,1] => 2
[1,1,0,0,1,0] => [3,1,2] => [2,3,1] => [3,1,2] => 1
[1,1,0,1,0,0] => [2,1,3] => [2,1,3] => [2,1,3] => 1
[1,1,1,0,0,0] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,0,1,0,1,0,1,0] => [4,3,2,1] => [4,1,2,3] => [2,3,4,1] => 1
[1,0,1,0,1,1,0,0] => [3,4,2,1] => [4,1,3,2] => [3,2,4,1] => 2
[1,0,1,1,0,0,1,0] => [4,2,3,1] => [4,3,1,2] => [2,4,3,1] => 1
[1,0,1,1,0,1,0,0] => [3,2,4,1] => [4,3,2,1] => [4,3,2,1] => 3
[1,0,1,1,1,0,0,0] => [2,3,4,1] => [4,2,3,1] => [3,1,4,2] => 1
[1,1,0,0,1,0,1,0] => [4,3,1,2] => [2,4,1,3] => [3,4,1,2] => 1
[1,1,0,0,1,1,0,0] => [3,4,1,2] => [2,4,3,1] => [4,3,1,2] => 2
[1,1,0,1,0,0,1,0] => [4,2,1,3] => [3,1,4,2] => [4,2,3,1] => 2
[1,1,0,1,0,1,0,0] => [3,2,1,4] => [3,1,2,4] => [2,3,1,4] => 1
[1,1,0,1,1,0,0,0] => [2,3,1,4] => [3,2,1,4] => [3,2,1,4] => 2
[1,1,1,0,0,0,1,0] => [4,1,2,3] => [2,3,4,1] => [4,1,2,3] => 1
[1,1,1,0,0,1,0,0] => [3,1,2,4] => [2,3,1,4] => [3,1,2,4] => 1
[1,1,1,0,1,0,0,0] => [2,1,3,4] => [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] => [1,2,3,4] => 0
[1,0,1,0,1,0,1,0,1,0] => [5,4,3,2,1] => [5,1,2,3,4] => [2,3,4,5,1] => 1
[1,0,1,0,1,0,1,1,0,0] => [4,5,3,2,1] => [5,1,2,4,3] => [2,4,3,5,1] => 1
[1,0,1,0,1,1,0,0,1,0] => [5,3,4,2,1] => [5,1,4,2,3] => [3,4,2,5,1] => 2
[1,0,1,0,1,1,0,1,0,0] => [4,3,5,2,1] => [5,1,4,3,2] => [4,3,2,5,1] => 3
[1,0,1,0,1,1,1,0,0,0] => [3,4,5,2,1] => [5,1,3,4,2] => [4,2,3,5,1] => 2
[1,0,1,1,0,0,1,0,1,0] => [5,4,2,3,1] => [5,3,1,2,4] => [2,4,5,3,1] => 1
[1,0,1,1,0,0,1,1,0,0] => [4,5,2,3,1] => [5,3,1,4,2] => [4,2,5,3,1] => 2
[1,0,1,1,0,1,0,0,1,0] => [5,3,2,4,1] => [5,4,2,1,3] => [3,5,4,2,1] => 2
[1,0,1,1,0,1,0,1,0,0] => [4,3,2,5,1] => [5,4,2,3,1] => [3,1,5,4,2] => 1
[1,0,1,1,0,1,1,0,0,0] => [3,4,2,5,1] => [5,4,3,2,1] => [5,4,3,2,1] => 4
[1,0,1,1,1,0,0,0,1,0] => [5,2,3,4,1] => [5,3,4,1,2] => [2,4,1,5,3] => 1
[1,0,1,1,1,0,0,1,0,0] => [4,2,3,5,1] => [5,3,4,2,1] => [4,2,1,5,3] => 2
[1,0,1,1,1,0,1,0,0,0] => [3,2,4,5,1] => [5,3,2,4,1] => [4,1,5,3,2] => 1
[1,0,1,1,1,1,0,0,0,0] => [2,3,4,5,1] => [5,2,3,4,1] => [4,1,3,5,2] => 1
[1,1,0,0,1,0,1,0,1,0] => [5,4,3,1,2] => [2,5,1,3,4] => [3,4,5,1,2] => 1
[1,1,0,0,1,0,1,1,0,0] => [4,5,3,1,2] => [2,5,1,4,3] => [4,3,5,1,2] => 2
[1,1,0,0,1,1,0,0,1,0] => [5,3,4,1,2] => [2,5,4,1,3] => [3,5,4,1,2] => 1
[1,1,0,0,1,1,0,1,0,0] => [4,3,5,1,2] => [2,5,4,3,1] => [5,4,3,1,2] => 3
[1,1,0,0,1,1,1,0,0,0] => [3,4,5,1,2] => [2,5,3,4,1] => [4,1,2,5,3] => 1
[1,1,0,1,0,0,1,0,1,0] => [5,4,2,1,3] => [3,1,5,2,4] => [4,5,2,3,1] => 2
[1,1,0,1,0,0,1,1,0,0] => [4,5,2,1,3] => [3,1,5,4,2] => [5,4,2,3,1] => 3
[1,1,0,1,0,1,0,0,1,0] => [5,3,2,1,4] => [4,1,2,5,3] => [2,5,3,4,1] => 1
[1,1,0,1,0,1,0,1,0,0] => [4,3,2,1,5] => [4,1,2,3,5] => [2,3,4,1,5] => 1
[1,1,0,1,0,1,1,0,0,0] => [3,4,2,1,5] => [4,1,3,2,5] => [3,2,4,1,5] => 2
[1,1,0,1,1,0,0,0,1,0] => [5,2,3,1,4] => [4,3,1,5,2] => [5,2,4,3,1] => 2
[1,1,0,1,1,0,0,1,0,0] => [4,2,3,1,5] => [4,3,1,2,5] => [2,4,3,1,5] => 1
[1,1,0,1,1,0,1,0,0,0] => [3,2,4,1,5] => [4,3,2,1,5] => [4,3,2,1,5] => 3
[1,1,0,1,1,1,0,0,0,0] => [2,3,4,1,5] => [4,2,3,1,5] => [3,1,4,2,5] => 1
[1,1,1,0,0,0,1,0,1,0] => [5,4,1,2,3] => [2,3,5,1,4] => [4,5,1,2,3] => 1
[1,1,1,0,0,0,1,1,0,0] => [4,5,1,2,3] => [2,3,5,4,1] => [5,4,1,2,3] => 2
[1,1,1,0,0,1,0,0,1,0] => [5,3,1,2,4] => [2,4,1,5,3] => [5,3,4,1,2] => 2
[1,1,1,0,0,1,0,1,0,0] => [4,3,1,2,5] => [2,4,1,3,5] => [3,4,1,2,5] => 1
[1,1,1,0,0,1,1,0,0,0] => [3,4,1,2,5] => [2,4,3,1,5] => [4,3,1,2,5] => 2
[1,1,1,0,1,0,0,0,1,0] => [5,2,1,3,4] => [3,1,4,5,2] => [5,2,3,1,4] => 2
[1,1,1,0,1,0,0,1,0,0] => [4,2,1,3,5] => [3,1,4,2,5] => [4,2,3,1,5] => 2
[1,1,1,0,1,0,1,0,0,0] => [3,2,1,4,5] => [3,1,2,4,5] => [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] => [3,2,1,4,5] => 2
[1,1,1,1,0,0,0,0,1,0] => [5,1,2,3,4] => [2,3,4,5,1] => [5,1,2,3,4] => 1
[1,1,1,1,0,0,0,1,0,0] => [4,1,2,3,5] => [2,3,4,1,5] => [4,1,2,3,5] => 1
[1,1,1,1,0,0,1,0,0,0] => [3,1,2,4,5] => [2,3,1,4,5] => [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] => [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] => [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,0,1,0,1,0] => [6,5,4,3,2,1] => [6,1,2,3,4,5] => [2,3,4,5,6,1] => 1
[1,0,1,0,1,0,1,0,1,1,0,0] => [5,6,4,3,2,1] => [6,1,2,3,5,4] => [2,3,5,4,6,1] => 1
[1,0,1,0,1,0,1,1,0,0,1,0] => [6,4,5,3,2,1] => [6,1,2,5,3,4] => [2,4,5,3,6,1] => 1
[1,0,1,0,1,0,1,1,0,1,0,0] => [5,4,6,3,2,1] => [6,1,2,5,4,3] => [2,5,4,3,6,1] => 1
[1,0,1,0,1,0,1,1,1,0,0,0] => [4,5,6,3,2,1] => [6,1,2,4,5,3] => [2,5,3,4,6,1] => 1
[1,0,1,0,1,1,0,0,1,0,1,0] => [6,5,3,4,2,1] => [6,1,4,2,3,5] => [3,4,2,5,6,1] => 2
[1,0,1,0,1,1,0,0,1,1,0,0] => [5,6,3,4,2,1] => [6,1,4,2,5,3] => [5,3,4,2,6,1] => 3
[1,0,1,0,1,1,0,1,0,0,1,0] => [6,4,3,5,2,1] => [6,1,5,3,2,4] => [4,5,3,2,6,1] => 3
[1,0,1,0,1,1,0,1,0,1,0,0] => [5,4,3,6,2,1] => [6,1,5,3,4,2] => [4,2,5,3,6,1] => 2
[1,0,1,0,1,1,0,1,1,0,0,0] => [4,5,3,6,2,1] => [6,1,5,4,3,2] => [5,4,3,2,6,1] => 4
[1,0,1,0,1,1,1,0,0,0,1,0] => [6,3,4,5,2,1] => [6,1,4,5,2,3] => [3,5,2,4,6,1] => 2
[1,0,1,0,1,1,1,0,0,1,0,0] => [5,3,4,6,2,1] => [6,1,4,5,3,2] => [5,3,2,4,6,1] => 3
[1,0,1,0,1,1,1,0,1,0,0,0] => [4,3,5,6,2,1] => [6,1,4,3,5,2] => [5,2,4,3,6,1] => 2
[1,0,1,0,1,1,1,1,0,0,0,0] => [3,4,5,6,2,1] => [6,1,3,4,5,2] => [5,2,3,4,6,1] => 2
[1,0,1,1,0,0,1,0,1,0,1,0] => [6,5,4,2,3,1] => [6,3,1,2,4,5] => [2,4,5,6,3,1] => 1
[1,0,1,1,0,0,1,0,1,1,0,0] => [5,6,4,2,3,1] => [6,3,1,2,5,4] => [2,5,4,6,3,1] => 1
[1,0,1,1,0,0,1,1,0,0,1,0] => [6,4,5,2,3,1] => [6,3,1,5,2,4] => [4,5,2,6,3,1] => 2
[1,0,1,1,0,0,1,1,0,1,0,0] => [5,4,6,2,3,1] => [6,3,1,5,4,2] => [5,4,2,6,3,1] => 3
[1,0,1,1,0,0,1,1,1,0,0,0] => [4,5,6,2,3,1] => [6,3,1,4,5,2] => [5,2,4,6,3,1] => 2
[1,0,1,1,0,1,0,0,1,0,1,0] => [6,5,3,2,4,1] => [6,4,2,1,3,5] => [3,5,6,4,2,1] => 2
[1,0,1,1,0,1,0,0,1,1,0,0] => [5,6,3,2,4,1] => [6,4,2,1,5,3] => [5,3,6,4,2,1] => 3
[1,0,1,1,0,1,0,1,0,0,1,0] => [6,4,3,2,5,1] => [6,5,2,3,1,4] => [3,1,4,6,5,2] => 1
[1,0,1,1,0,1,0,1,0,1,0,0] => [5,4,3,2,6,1] => [6,5,2,3,4,1] => [4,1,3,6,5,2] => 1
[1,0,1,1,0,1,0,1,1,0,0,0] => [4,5,3,2,6,1] => [6,5,2,4,3,1] => [4,3,1,6,5,2] => 2
[1,0,1,1,0,1,1,0,0,0,1,0] => [6,3,4,2,5,1] => [6,5,4,2,1,3] => [3,6,5,4,2,1] => 2
[1,0,1,1,0,1,1,0,0,1,0,0] => [5,3,4,2,6,1] => [6,5,4,2,3,1] => [3,1,6,5,4,2] => 1
[1,0,1,1,0,1,1,0,1,0,0,0] => [4,3,5,2,6,1] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => 5
[1,0,1,1,0,1,1,1,0,0,0,0] => [3,4,5,2,6,1] => [6,5,3,4,2,1] => [4,2,1,6,5,3] => 2
[1,0,1,1,1,0,0,0,1,0,1,0] => [6,5,2,3,4,1] => [6,3,4,1,2,5] => [2,4,1,5,6,3] => 1
[1,0,1,1,1,0,0,0,1,1,0,0] => [5,6,2,3,4,1] => [6,3,4,1,5,2] => [5,2,4,1,6,3] => 2
[1,0,1,1,1,0,0,1,0,0,1,0] => [6,4,2,3,5,1] => [6,3,5,2,1,4] => [4,5,2,1,6,3] => 2
[1,0,1,1,1,0,0,1,0,1,0,0] => [5,4,2,3,6,1] => [6,3,5,2,4,1] => [4,1,5,2,6,3] => 1
[1,0,1,1,1,0,0,1,1,0,0,0] => [4,5,2,3,6,1] => [6,3,5,4,2,1] => [5,4,2,1,6,3] => 3
[1,0,1,1,1,0,1,0,0,0,1,0] => [6,3,2,4,5,1] => [6,4,2,5,1,3] => [3,5,1,6,4,2] => 1
[1,0,1,1,1,0,1,0,0,1,0,0] => [5,3,2,4,6,1] => [6,4,2,5,3,1] => [5,3,1,6,4,2] => 2
[1,0,1,1,1,0,1,0,1,0,0,0] => [4,3,2,5,6,1] => [6,4,2,3,5,1] => [5,1,3,6,4,2] => 1
[1,0,1,1,1,0,1,1,0,0,0,0] => [3,4,2,5,6,1] => [6,4,3,2,5,1] => [5,1,6,4,3,2] => 1
[1,0,1,1,1,1,0,0,0,0,1,0] => [6,2,3,4,5,1] => [6,3,4,5,1,2] => [2,5,1,4,6,3] => 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
first fundamental transformation
Description
Return the permutation whose cycles are the subsequences between successive left to right maxima.
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
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.
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