Identifier
-
Mp00024:
Dyck paths
—to 321-avoiding permutation⟶
Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
St000039: Permutations ⟶ ℤ
Values
[1,0] => [1] => [1] => [1] => 0
[1,0,1,0] => [2,1] => [2,1] => [2,1] => 0
[1,1,0,0] => [1,2] => [1,2] => [1,2] => 0
[1,0,1,0,1,0] => [2,1,3] => [2,1,3] => [2,1,3] => 0
[1,0,1,1,0,0] => [2,3,1] => [3,2,1] => [2,3,1] => 1
[1,1,0,0,1,0] => [3,1,2] => [3,1,2] => [3,1,2] => 0
[1,1,0,1,0,0] => [1,3,2] => [1,3,2] => [1,3,2] => 0
[1,1,1,0,0,0] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,0,1,0,1,0,1,0] => [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 0
[1,0,1,0,1,1,0,0] => [2,4,1,3] => [4,2,1,3] => [2,4,1,3] => 1
[1,0,1,1,0,0,1,0] => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 0
[1,0,1,1,0,1,0,0] => [2,3,1,4] => [3,2,1,4] => [2,3,1,4] => 1
[1,0,1,1,1,0,0,0] => [2,3,4,1] => [4,3,2,1] => [2,3,4,1] => 2
[1,1,0,0,1,0,1,0] => [3,1,4,2] => [4,3,1,2] => [3,1,4,2] => 1
[1,1,0,0,1,1,0,0] => [3,4,1,2] => [4,1,3,2] => [4,3,1,2] => 1
[1,1,0,1,0,0,1,0] => [3,1,2,4] => [3,1,2,4] => [3,1,2,4] => 0
[1,1,0,1,0,1,0,0] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 0
[1,1,0,1,1,0,0,0] => [1,3,4,2] => [1,4,3,2] => [1,3,4,2] => 1
[1,1,1,0,0,0,1,0] => [4,1,2,3] => [4,1,2,3] => [4,1,2,3] => 0
[1,1,1,0,0,1,0,0] => [1,4,2,3] => [1,4,2,3] => [1,4,2,3] => 0
[1,1,1,0,1,0,0,0] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 0
[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] => [2,1,4,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => 0
[1,0,1,0,1,0,1,1,0,0] => [2,4,1,3,5] => [4,2,1,3,5] => [2,4,1,3,5] => 1
[1,0,1,0,1,1,0,0,1,0] => [2,1,4,5,3] => [2,1,5,4,3] => [2,1,4,5,3] => 1
[1,0,1,0,1,1,0,1,0,0] => [2,4,1,5,3] => [5,4,2,1,3] => [2,4,1,5,3] => 2
[1,0,1,0,1,1,1,0,0,0] => [2,4,5,1,3] => [5,2,1,4,3] => [2,5,4,1,3] => 2
[1,0,1,1,0,0,1,0,1,0] => [2,1,5,3,4] => [2,1,5,3,4] => [2,1,5,3,4] => 0
[1,0,1,1,0,0,1,1,0,0] => [2,5,1,3,4] => [5,2,1,3,4] => [2,5,1,3,4] => 1
[1,0,1,1,0,1,0,0,1,0] => [2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => 0
[1,0,1,1,0,1,0,1,0,0] => [2,3,1,5,4] => [3,2,1,5,4] => [2,3,1,5,4] => 1
[1,0,1,1,0,1,1,0,0,0] => [2,3,5,1,4] => [5,3,2,1,4] => [2,3,5,1,4] => 2
[1,0,1,1,1,0,0,0,1,0] => [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => 0
[1,0,1,1,1,0,0,1,0,0] => [2,3,1,4,5] => [3,2,1,4,5] => [2,3,1,4,5] => 1
[1,0,1,1,1,0,1,0,0,0] => [2,3,4,1,5] => [4,3,2,1,5] => [2,3,4,1,5] => 2
[1,0,1,1,1,1,0,0,0,0] => [2,3,4,5,1] => [5,4,3,2,1] => [2,3,4,5,1] => 3
[1,1,0,0,1,0,1,0,1,0] => [3,1,4,2,5] => [4,3,1,2,5] => [3,1,4,2,5] => 1
[1,1,0,0,1,0,1,1,0,0] => [3,4,1,2,5] => [4,1,3,2,5] => [4,3,1,2,5] => 1
[1,1,0,0,1,1,0,0,1,0] => [3,1,4,5,2] => [5,4,3,1,2] => [3,1,4,5,2] => 2
[1,1,0,0,1,1,0,1,0,0] => [3,4,1,5,2] => [5,4,1,3,2] => [4,3,1,5,2] => 2
[1,1,0,0,1,1,1,0,0,0] => [3,4,5,1,2] => [5,1,4,3,2] => [5,3,4,1,2] => 2
[1,1,0,1,0,0,1,0,1,0] => [3,1,5,2,4] => [5,3,1,2,4] => [3,1,5,2,4] => 1
[1,1,0,1,0,0,1,1,0,0] => [3,5,1,2,4] => [5,1,3,2,4] => [5,3,1,2,4] => 1
[1,1,0,1,0,1,0,0,1,0] => [3,1,2,5,4] => [3,1,2,5,4] => [3,1,2,5,4] => 0
[1,1,0,1,0,1,0,1,0,0] => [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 0
[1,1,0,1,0,1,1,0,0,0] => [1,3,5,2,4] => [1,5,3,2,4] => [1,3,5,2,4] => 1
[1,1,0,1,1,0,0,0,1,0] => [3,1,2,4,5] => [3,1,2,4,5] => [3,1,2,4,5] => 0
[1,1,0,1,1,0,0,1,0,0] => [1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 0
[1,1,0,1,1,0,1,0,0,0] => [1,3,4,2,5] => [1,4,3,2,5] => [1,3,4,2,5] => 1
[1,1,0,1,1,1,0,0,0,0] => [1,3,4,5,2] => [1,5,4,3,2] => [1,3,4,5,2] => 2
[1,1,1,0,0,0,1,0,1,0] => [4,1,5,2,3] => [5,1,2,4,3] => [5,1,4,2,3] => 1
[1,1,1,0,0,0,1,1,0,0] => [4,5,1,2,3] => [5,1,4,2,3] => [5,4,2,1,3] => 1
[1,1,1,0,0,1,0,0,1,0] => [4,1,2,5,3] => [5,4,1,2,3] => [4,1,2,5,3] => 1
[1,1,1,0,0,1,0,1,0,0] => [1,4,2,5,3] => [1,5,4,2,3] => [1,4,2,5,3] => 1
[1,1,1,0,0,1,1,0,0,0] => [1,4,5,2,3] => [1,5,2,4,3] => [1,5,4,2,3] => 1
[1,1,1,0,1,0,0,0,1,0] => [4,1,2,3,5] => [4,1,2,3,5] => [4,1,2,3,5] => 0
[1,1,1,0,1,0,0,1,0,0] => [1,4,2,3,5] => [1,4,2,3,5] => [1,4,2,3,5] => 0
[1,1,1,0,1,0,1,0,0,0] => [1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 0
[1,1,1,0,1,1,0,0,0,0] => [1,2,4,5,3] => [1,2,5,4,3] => [1,2,4,5,3] => 1
[1,1,1,1,0,0,0,0,1,0] => [5,1,2,3,4] => [5,1,2,3,4] => [5,1,2,3,4] => 0
[1,1,1,1,0,0,0,1,0,0] => [1,5,2,3,4] => [1,5,2,3,4] => [1,5,2,3,4] => 0
[1,1,1,1,0,0,1,0,0,0] => [1,2,5,3,4] => [1,2,5,3,4] => [1,2,5,3,4] => 0
[1,1,1,1,0,1,0,0,0,0] => [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 0
[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] => [2,1,4,3,6,5] => [2,1,4,3,6,5] => [2,1,4,3,6,5] => 0
[1,0,1,0,1,0,1,0,1,1,0,0] => [2,4,1,3,6,5] => [4,2,1,3,6,5] => [2,4,1,3,6,5] => 1
[1,0,1,0,1,0,1,1,0,0,1,0] => [2,1,4,6,3,5] => [2,1,6,4,3,5] => [2,1,4,6,3,5] => 1
[1,0,1,0,1,0,1,1,0,1,0,0] => [2,4,1,6,3,5] => [6,4,2,1,3,5] => [2,4,1,6,3,5] => 2
[1,0,1,0,1,0,1,1,1,0,0,0] => [2,4,6,1,3,5] => [6,2,1,4,3,5] => [2,6,4,1,3,5] => 2
[1,0,1,0,1,1,0,0,1,0,1,0] => [2,1,4,3,5,6] => [2,1,4,3,5,6] => [2,1,4,3,5,6] => 0
[1,0,1,0,1,1,0,0,1,1,0,0] => [2,4,1,3,5,6] => [4,2,1,3,5,6] => [2,4,1,3,5,6] => 1
[1,0,1,0,1,1,0,1,0,0,1,0] => [2,1,4,5,3,6] => [2,1,5,4,3,6] => [2,1,4,5,3,6] => 1
[1,0,1,0,1,1,0,1,0,1,0,0] => [2,4,1,5,3,6] => [5,4,2,1,3,6] => [2,4,1,5,3,6] => 2
[1,0,1,0,1,1,0,1,1,0,0,0] => [2,4,5,1,3,6] => [5,2,1,4,3,6] => [2,5,4,1,3,6] => 2
[1,0,1,0,1,1,1,0,0,0,1,0] => [2,1,4,5,6,3] => [2,1,6,5,4,3] => [2,1,4,5,6,3] => 2
[1,0,1,0,1,1,1,0,0,1,0,0] => [2,4,1,5,6,3] => [6,5,4,2,1,3] => [2,4,1,5,6,3] => 3
[1,0,1,0,1,1,1,0,1,0,0,0] => [2,4,5,1,6,3] => [6,5,2,1,4,3] => [2,5,4,1,6,3] => 3
[1,0,1,0,1,1,1,1,0,0,0,0] => [2,4,5,6,1,3] => [6,2,1,5,4,3] => [2,6,4,5,1,3] => 3
[1,0,1,1,0,0,1,0,1,0,1,0] => [2,1,5,3,6,4] => [2,1,6,5,3,4] => [2,1,5,3,6,4] => 1
[1,0,1,1,0,0,1,0,1,1,0,0] => [2,5,1,3,6,4] => [6,5,2,1,3,4] => [2,5,1,3,6,4] => 2
[1,0,1,1,0,0,1,1,0,0,1,0] => [2,1,5,6,3,4] => [2,1,6,3,5,4] => [2,1,6,5,3,4] => 1
[1,0,1,1,0,0,1,1,0,1,0,0] => [2,5,1,6,3,4] => [6,2,1,3,5,4] => [2,6,1,5,3,4] => 2
[1,0,1,1,0,0,1,1,1,0,0,0] => [2,5,6,1,3,4] => [6,2,1,5,3,4] => [2,6,5,3,1,4] => 2
[1,0,1,1,0,1,0,0,1,0,1,0] => [2,1,5,3,4,6] => [2,1,5,3,4,6] => [2,1,5,3,4,6] => 0
[1,0,1,1,0,1,0,0,1,1,0,0] => [2,5,1,3,4,6] => [5,2,1,3,4,6] => [2,5,1,3,4,6] => 1
[1,0,1,1,0,1,0,1,0,0,1,0] => [2,1,3,5,4,6] => [2,1,3,5,4,6] => [2,1,3,5,4,6] => 0
[1,0,1,1,0,1,0,1,0,1,0,0] => [2,3,1,5,4,6] => [3,2,1,5,4,6] => [2,3,1,5,4,6] => 1
[1,0,1,1,0,1,0,1,1,0,0,0] => [2,3,5,1,4,6] => [5,3,2,1,4,6] => [2,3,5,1,4,6] => 2
[1,0,1,1,0,1,1,0,0,0,1,0] => [2,1,3,5,6,4] => [2,1,3,6,5,4] => [2,1,3,5,6,4] => 1
[1,0,1,1,0,1,1,0,0,1,0,0] => [2,3,1,5,6,4] => [3,2,1,6,5,4] => [2,3,1,5,6,4] => 2
[1,0,1,1,0,1,1,0,1,0,0,0] => [2,3,5,1,6,4] => [6,5,3,2,1,4] => [2,3,5,1,6,4] => 3
[1,0,1,1,0,1,1,1,0,0,0,0] => [2,3,5,6,1,4] => [6,3,2,1,5,4] => [2,3,6,5,1,4] => 3
[1,0,1,1,1,0,0,0,1,0,1,0] => [2,1,6,3,4,5] => [2,1,6,3,4,5] => [2,1,6,3,4,5] => 0
[1,0,1,1,1,0,0,0,1,1,0,0] => [2,6,1,3,4,5] => [6,2,1,3,4,5] => [2,6,1,3,4,5] => 1
[1,0,1,1,1,0,0,1,0,0,1,0] => [2,1,3,6,4,5] => [2,1,3,6,4,5] => [2,1,3,6,4,5] => 0
[1,0,1,1,1,0,0,1,0,1,0,0] => [2,3,1,6,4,5] => [3,2,1,6,4,5] => [2,3,1,6,4,5] => 1
[1,0,1,1,1,0,0,1,1,0,0,0] => [2,3,6,1,4,5] => [6,3,2,1,4,5] => [2,3,6,1,4,5] => 2
[1,0,1,1,1,0,1,0,0,0,1,0] => [2,1,3,4,6,5] => [2,1,3,4,6,5] => [2,1,3,4,6,5] => 0
[1,0,1,1,1,0,1,0,0,1,0,0] => [2,3,1,4,6,5] => [3,2,1,4,6,5] => [2,3,1,4,6,5] => 1
[1,0,1,1,1,0,1,0,1,0,0,0] => [2,3,4,1,6,5] => [4,3,2,1,6,5] => [2,3,4,1,6,5] => 2
[1,0,1,1,1,0,1,1,0,0,0,0] => [2,3,4,6,1,5] => [6,4,3,2,1,5] => [2,3,4,6,1,5] => 3
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Description
The number of crossings of a permutation.
A crossing of a permutation $\pi$ is given by a pair $(i,j)$ such that either $i < j \leq \pi(i) \leq \pi(j)$ or $\pi(i) < \pi(j) < i < j$.
Pictorially, the diagram of a permutation is obtained by writing the numbers from $1$ to $n$ in this order on a line, and connecting $i$ and $\pi(i)$ with an arc above the line if $i\leq\pi(i)$ and with an arc below the line if $i > \pi(i)$. Then the number of crossings is the number of pairs of arcs above the line that cross or touch, plus the number of arcs below the line that cross.
A crossing of a permutation $\pi$ is given by a pair $(i,j)$ such that either $i < j \leq \pi(i) \leq \pi(j)$ or $\pi(i) < \pi(j) < i < j$.
Pictorially, the diagram of a permutation is obtained by writing the numbers from $1$ to $n$ in this order on a line, and connecting $i$ and $\pi(i)$ with an arc above the line if $i\leq\pi(i)$ and with an arc below the line if $i > \pi(i)$. Then the number of crossings is the number of pairs of arcs above the line that cross or touch, plus the number of arcs below the line that cross.
Map
Clarke-Steingrimsson-Zeng inverse
Description
The inverse of the Clarke-Steingrimsson-Zeng map, sending excedances to descents.
This is the inverse of the map $\Phi$ in [1, sec.3].
This is the inverse of the map $\Phi$ in [1, sec.3].
Map
to 321-avoiding permutation
Description
Sends a Dyck path to a 321-avoiding permutation.
This bijection defined in [3, pp. 60] and in [2, Section 3.1].
It is shown in [1] that it sends the number of centered tunnels to the number of fixed points, the number of right tunnels to the number of exceedences, and the semilength plus the height of the middle point to 2 times the length of the longest increasing subsequence.
This bijection defined in [3, pp. 60] and in [2, Section 3.1].
It is shown in [1] that it sends the number of centered tunnels to the number of fixed points, the number of right tunnels to the number of exceedences, and the semilength plus the height of the middle point to 2 times the length of the longest increasing subsequence.
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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