Identifier
-
Mp00119:
Dyck paths
—to 321-avoiding permutation (Krattenthaler)⟶
Permutations
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000542: Permutations ⟶ ℤ (values match St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right.)
Values
[1,0] => [1] => [1] => [1] => 1
[1,0,1,0] => [1,2] => [1,2] => [2,1] => 2
[1,1,0,0] => [2,1] => [2,1] => [1,2] => 1
[1,0,1,0,1,0] => [1,2,3] => [1,2,3] => [3,2,1] => 3
[1,0,1,1,0,0] => [1,3,2] => [1,3,2] => [3,1,2] => 2
[1,1,0,0,1,0] => [2,1,3] => [2,1,3] => [2,3,1] => 2
[1,1,0,1,0,0] => [2,3,1] => [3,2,1] => [1,2,3] => 1
[1,1,1,0,0,0] => [3,1,2] => [3,1,2] => [1,3,2] => 1
[1,0,1,0,1,0,1,0] => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 4
[1,0,1,0,1,1,0,0] => [1,2,4,3] => [1,2,4,3] => [4,3,1,2] => 3
[1,0,1,1,0,0,1,0] => [1,3,2,4] => [1,3,2,4] => [4,2,3,1] => 3
[1,0,1,1,0,1,0,0] => [1,3,4,2] => [1,4,3,2] => [4,1,2,3] => 2
[1,0,1,1,1,0,0,0] => [1,4,2,3] => [1,4,2,3] => [4,1,3,2] => 2
[1,1,0,0,1,0,1,0] => [2,1,3,4] => [2,1,3,4] => [3,4,2,1] => 3
[1,1,0,0,1,1,0,0] => [2,1,4,3] => [2,1,4,3] => [3,4,1,2] => 2
[1,1,0,1,0,0,1,0] => [2,3,1,4] => [3,2,1,4] => [2,3,4,1] => 2
[1,1,0,1,0,1,0,0] => [2,3,4,1] => [4,2,3,1] => [1,3,2,4] => 1
[1,1,0,1,1,0,0,0] => [2,4,1,3] => [4,2,1,3] => [1,3,4,2] => 1
[1,1,1,0,0,0,1,0] => [3,1,2,4] => [3,1,2,4] => [2,4,3,1] => 2
[1,1,1,0,0,1,0,0] => [3,1,4,2] => [3,4,1,2] => [2,1,4,3] => 2
[1,1,1,0,1,0,0,0] => [3,4,1,2] => [4,1,3,2] => [1,4,2,3] => 1
[1,1,1,1,0,0,0,0] => [4,1,2,3] => [4,1,2,3] => [1,4,3,2] => 1
[1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 5
[1,0,1,0,1,0,1,1,0,0] => [1,2,3,5,4] => [1,2,3,5,4] => [5,4,3,1,2] => 4
[1,0,1,0,1,1,0,0,1,0] => [1,2,4,3,5] => [1,2,4,3,5] => [5,4,2,3,1] => 4
[1,0,1,0,1,1,0,1,0,0] => [1,2,4,5,3] => [1,2,5,4,3] => [5,4,1,2,3] => 3
[1,0,1,0,1,1,1,0,0,0] => [1,2,5,3,4] => [1,2,5,3,4] => [5,4,1,3,2] => 3
[1,0,1,1,0,0,1,0,1,0] => [1,3,2,4,5] => [1,3,2,4,5] => [5,3,4,2,1] => 4
[1,0,1,1,0,0,1,1,0,0] => [1,3,2,5,4] => [1,3,2,5,4] => [5,3,4,1,2] => 3
[1,0,1,1,0,1,0,0,1,0] => [1,3,4,2,5] => [1,4,3,2,5] => [5,2,3,4,1] => 3
[1,0,1,1,0,1,0,1,0,0] => [1,3,4,5,2] => [1,5,3,4,2] => [5,1,3,2,4] => 2
[1,0,1,1,0,1,1,0,0,0] => [1,3,5,2,4] => [1,5,3,2,4] => [5,1,3,4,2] => 2
[1,0,1,1,1,0,0,0,1,0] => [1,4,2,3,5] => [1,4,2,3,5] => [5,2,4,3,1] => 3
[1,0,1,1,1,0,0,1,0,0] => [1,4,2,5,3] => [1,4,5,2,3] => [5,2,1,4,3] => 3
[1,0,1,1,1,0,1,0,0,0] => [1,4,5,2,3] => [1,5,2,4,3] => [5,1,4,2,3] => 2
[1,0,1,1,1,1,0,0,0,0] => [1,5,2,3,4] => [1,5,2,3,4] => [5,1,4,3,2] => 2
[1,1,0,0,1,0,1,0,1,0] => [2,1,3,4,5] => [2,1,3,4,5] => [4,5,3,2,1] => 4
[1,1,0,0,1,0,1,1,0,0] => [2,1,3,5,4] => [2,1,3,5,4] => [4,5,3,1,2] => 3
[1,1,0,0,1,1,0,0,1,0] => [2,1,4,3,5] => [2,1,4,3,5] => [4,5,2,3,1] => 3
[1,1,0,0,1,1,0,1,0,0] => [2,1,4,5,3] => [2,1,5,4,3] => [4,5,1,2,3] => 2
[1,1,0,0,1,1,1,0,0,0] => [2,1,5,3,4] => [2,1,5,3,4] => [4,5,1,3,2] => 2
[1,1,0,1,0,0,1,0,1,0] => [2,3,1,4,5] => [3,2,1,4,5] => [3,4,5,2,1] => 3
[1,1,0,1,0,0,1,1,0,0] => [2,3,1,5,4] => [3,2,1,5,4] => [3,4,5,1,2] => 2
[1,1,0,1,0,1,0,0,1,0] => [2,3,4,1,5] => [4,2,3,1,5] => [2,4,3,5,1] => 2
[1,1,0,1,0,1,0,1,0,0] => [2,3,4,5,1] => [5,2,3,4,1] => [1,4,3,2,5] => 1
[1,1,0,1,0,1,1,0,0,0] => [2,3,5,1,4] => [5,2,3,1,4] => [1,4,3,5,2] => 1
[1,1,0,1,1,0,0,0,1,0] => [2,4,1,3,5] => [4,2,1,3,5] => [2,4,5,3,1] => 2
[1,1,0,1,1,0,0,1,0,0] => [2,4,1,5,3] => [4,2,5,1,3] => [2,4,1,5,3] => 2
[1,1,0,1,1,0,1,0,0,0] => [2,4,5,1,3] => [5,2,1,4,3] => [1,4,5,2,3] => 1
[1,1,0,1,1,1,0,0,0,0] => [2,5,1,3,4] => [5,2,1,3,4] => [1,4,5,3,2] => 1
[1,1,1,0,0,0,1,0,1,0] => [3,1,2,4,5] => [3,1,2,4,5] => [3,5,4,2,1] => 3
[1,1,1,0,0,0,1,1,0,0] => [3,1,2,5,4] => [3,1,2,5,4] => [3,5,4,1,2] => 2
[1,1,1,0,0,1,0,0,1,0] => [3,1,4,2,5] => [3,4,1,2,5] => [3,2,5,4,1] => 3
[1,1,1,0,0,1,0,1,0,0] => [3,1,4,5,2] => [3,5,1,4,2] => [3,1,5,2,4] => 2
[1,1,1,0,0,1,1,0,0,0] => [3,1,5,2,4] => [3,5,1,2,4] => [3,1,5,4,2] => 2
[1,1,1,0,1,0,0,0,1,0] => [3,4,1,2,5] => [4,1,3,2,5] => [2,5,3,4,1] => 2
[1,1,1,0,1,0,0,1,0,0] => [3,4,1,5,2] => [4,5,3,1,2] => [2,1,3,5,4] => 2
[1,1,1,0,1,0,1,0,0,0] => [3,4,5,1,2] => [5,1,3,4,2] => [1,5,3,2,4] => 1
[1,1,1,0,1,1,0,0,0,0] => [3,5,1,2,4] => [5,1,3,2,4] => [1,5,3,4,2] => 1
[1,1,1,1,0,0,0,0,1,0] => [4,1,2,3,5] => [4,1,2,3,5] => [2,5,4,3,1] => 2
[1,1,1,1,0,0,0,1,0,0] => [4,1,2,5,3] => [4,1,5,2,3] => [2,5,1,4,3] => 2
[1,1,1,1,0,0,1,0,0,0] => [4,1,5,2,3] => [4,5,2,1,3] => [2,1,4,5,3] => 2
[1,1,1,1,0,1,0,0,0,0] => [4,5,1,2,3] => [5,1,2,4,3] => [1,5,4,2,3] => 1
[1,1,1,1,1,0,0,0,0,0] => [5,1,2,3,4] => [5,1,2,3,4] => [1,5,4,3,2] => 1
[1,0,1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => 6
[1,0,1,0,1,0,1,0,1,1,0,0] => [1,2,3,4,6,5] => [1,2,3,4,6,5] => [6,5,4,3,1,2] => 5
[1,0,1,0,1,0,1,1,0,0,1,0] => [1,2,3,5,4,6] => [1,2,3,5,4,6] => [6,5,4,2,3,1] => 5
[1,0,1,0,1,0,1,1,0,1,0,0] => [1,2,3,5,6,4] => [1,2,3,6,5,4] => [6,5,4,1,2,3] => 4
[1,0,1,0,1,0,1,1,1,0,0,0] => [1,2,3,6,4,5] => [1,2,3,6,4,5] => [6,5,4,1,3,2] => 4
[1,0,1,0,1,1,0,0,1,0,1,0] => [1,2,4,3,5,6] => [1,2,4,3,5,6] => [6,5,3,4,2,1] => 5
[1,0,1,0,1,1,0,0,1,1,0,0] => [1,2,4,3,6,5] => [1,2,4,3,6,5] => [6,5,3,4,1,2] => 4
[1,0,1,0,1,1,0,1,0,0,1,0] => [1,2,4,5,3,6] => [1,2,5,4,3,6] => [6,5,2,3,4,1] => 4
[1,0,1,0,1,1,0,1,0,1,0,0] => [1,2,4,5,6,3] => [1,2,6,4,5,3] => [6,5,1,3,2,4] => 3
[1,0,1,0,1,1,0,1,1,0,0,0] => [1,2,4,6,3,5] => [1,2,6,4,3,5] => [6,5,1,3,4,2] => 3
[1,0,1,0,1,1,1,0,0,0,1,0] => [1,2,5,3,4,6] => [1,2,5,3,4,6] => [6,5,2,4,3,1] => 4
[1,0,1,0,1,1,1,0,0,1,0,0] => [1,2,5,3,6,4] => [1,2,5,6,3,4] => [6,5,2,1,4,3] => 4
[1,0,1,0,1,1,1,0,1,0,0,0] => [1,2,5,6,3,4] => [1,2,6,3,5,4] => [6,5,1,4,2,3] => 3
[1,0,1,0,1,1,1,1,0,0,0,0] => [1,2,6,3,4,5] => [1,2,6,3,4,5] => [6,5,1,4,3,2] => 3
[1,0,1,1,0,0,1,0,1,0,1,0] => [1,3,2,4,5,6] => [1,3,2,4,5,6] => [6,4,5,3,2,1] => 5
[1,0,1,1,0,0,1,0,1,1,0,0] => [1,3,2,4,6,5] => [1,3,2,4,6,5] => [6,4,5,3,1,2] => 4
[1,0,1,1,0,0,1,1,0,0,1,0] => [1,3,2,5,4,6] => [1,3,2,5,4,6] => [6,4,5,2,3,1] => 4
[1,0,1,1,0,0,1,1,0,1,0,0] => [1,3,2,5,6,4] => [1,3,2,6,5,4] => [6,4,5,1,2,3] => 3
[1,0,1,1,0,0,1,1,1,0,0,0] => [1,3,2,6,4,5] => [1,3,2,6,4,5] => [6,4,5,1,3,2] => 3
[1,0,1,1,0,1,0,0,1,0,1,0] => [1,3,4,2,5,6] => [1,4,3,2,5,6] => [6,3,4,5,2,1] => 4
[1,0,1,1,0,1,0,0,1,1,0,0] => [1,3,4,2,6,5] => [1,4,3,2,6,5] => [6,3,4,5,1,2] => 3
[1,0,1,1,0,1,0,1,0,0,1,0] => [1,3,4,5,2,6] => [1,5,3,4,2,6] => [6,2,4,3,5,1] => 3
[1,0,1,1,0,1,0,1,0,1,0,0] => [1,3,4,5,6,2] => [1,6,3,4,5,2] => [6,1,4,3,2,5] => 2
[1,0,1,1,0,1,0,1,1,0,0,0] => [1,3,4,6,2,5] => [1,6,3,4,2,5] => [6,1,4,3,5,2] => 2
[1,0,1,1,0,1,1,0,0,0,1,0] => [1,3,5,2,4,6] => [1,5,3,2,4,6] => [6,2,4,5,3,1] => 3
[1,0,1,1,0,1,1,0,0,1,0,0] => [1,3,5,2,6,4] => [1,5,3,6,2,4] => [6,2,4,1,5,3] => 3
[1,0,1,1,0,1,1,0,1,0,0,0] => [1,3,5,6,2,4] => [1,6,3,2,5,4] => [6,1,4,5,2,3] => 2
[1,0,1,1,0,1,1,1,0,0,0,0] => [1,3,6,2,4,5] => [1,6,3,2,4,5] => [6,1,4,5,3,2] => 2
[1,0,1,1,1,0,0,0,1,0,1,0] => [1,4,2,3,5,6] => [1,4,2,3,5,6] => [6,3,5,4,2,1] => 4
[1,0,1,1,1,0,0,0,1,1,0,0] => [1,4,2,3,6,5] => [1,4,2,3,6,5] => [6,3,5,4,1,2] => 3
[1,0,1,1,1,0,0,1,0,0,1,0] => [1,4,2,5,3,6] => [1,4,5,2,3,6] => [6,3,2,5,4,1] => 4
[1,0,1,1,1,0,0,1,0,1,0,0] => [1,4,2,5,6,3] => [1,4,6,2,5,3] => [6,3,1,5,2,4] => 3
[1,0,1,1,1,0,0,1,1,0,0,0] => [1,4,2,6,3,5] => [1,4,6,2,3,5] => [6,3,1,5,4,2] => 3
[1,0,1,1,1,0,1,0,0,0,1,0] => [1,4,5,2,3,6] => [1,5,2,4,3,6] => [6,2,5,3,4,1] => 3
[1,0,1,1,1,0,1,0,0,1,0,0] => [1,4,5,2,6,3] => [1,5,6,4,2,3] => [6,2,1,3,5,4] => 3
[1,0,1,1,1,0,1,0,1,0,0,0] => [1,4,5,6,2,3] => [1,6,2,4,5,3] => [6,1,5,3,2,4] => 2
[1,0,1,1,1,0,1,1,0,0,0,0] => [1,4,6,2,3,5] => [1,6,2,4,3,5] => [6,1,5,3,4,2] => 2
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Description
The number of left-to-right-minima of a permutation.
An integer $\sigma_i$ in the one-line notation of a permutation $\sigma$ is a left-to-right-minimum if there does not exist a j < i such that $\sigma_j < \sigma_i$.
An integer $\sigma_i$ in the one-line notation of a permutation $\sigma$ is a left-to-right-minimum if there does not exist a j < i such that $\sigma_j < \sigma_i$.
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 $\tau(i) = n+1-\sigma(i)$
Map
to 321-avoiding permutation (Krattenthaler)
Description
Krattenthaler's bijection to 321-avoiding permutations.
Draw the path of semilength $n$ in an $n\times n$ square matrix, starting at the upper left corner, with right and down steps, and staying below the diagonal. Then the permutation matrix is obtained by placing ones into the cells corresponding to the peaks of the path and placing ones into the remaining columns from left to right, such that the row indices of the cells increase.
Draw the path of semilength $n$ in an $n\times n$ square matrix, starting at the upper left corner, with right and down steps, and staying below the diagonal. Then the permutation matrix is obtained by placing ones into the cells corresponding to the peaks of the path and placing ones into the remaining columns from left to right, such that the row indices of the cells increase.
Map
descent views to invisible inversion bottoms
Description
Return a permutation whose multiset of invisible inversion bottoms is the multiset of descent views of the given permutation.
An invisible inversion of a permutation $\sigma$ is a pair $i < j$ such that $i < \sigma(j) < \sigma(i)$. The element $\sigma(j)$ is then an invisible inversion bottom.
A descent view in a permutation $\pi$ is an element $\pi(j)$ such that $\pi(i+1) < \pi(j) < \pi(i)$, and additionally the smallest element in the decreasing run containing $\pi(i)$ is smaller than the smallest element in the decreasing run containing $\pi(j)$.
This map is a bijection $\chi:\mathfrak S_n \to \mathfrak S_n$, such that
An invisible inversion of a permutation $\sigma$ is a pair $i < j$ such that $i < \sigma(j) < \sigma(i)$. The element $\sigma(j)$ is then an invisible inversion bottom.
A descent view in a permutation $\pi$ is an element $\pi(j)$ such that $\pi(i+1) < \pi(j) < \pi(i)$, and additionally the smallest element in the decreasing run containing $\pi(i)$ is smaller than the smallest element in the decreasing run containing $\pi(j)$.
This map is a bijection $\chi:\mathfrak S_n \to \mathfrak S_n$, such that
- the multiset of descent views in $\pi$ is the multiset of invisible inversion bottoms in $\chi(\pi)$,
- the set of left-to-right maxima of $\pi$ is the set of maximal elements in the cycles of $\chi(\pi)$,
- the set of global ascent of $\pi$ is the set of global ascent of $\chi(\pi)$,
- the set of maximal elements in the decreasing runs of $\pi$ is the set of weak deficiency positions of $\chi(\pi)$, and
- the set of minimal elements in the decreasing runs of $\pi$ is the set of weak deficiency values of $\chi(\pi)$.
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