Identifier
-
Mp00025:
Dyck paths
—to 132-avoiding permutation⟶
Permutations
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
St000325: Permutations ⟶ ℤ (values match St000021The number of descents of a permutation., St000470The number of runs in a permutation.)
Values
[1,0] => [1] => [1] => 1
[1,0,1,0] => [2,1] => [2,1] => 2
[1,1,0,0] => [1,2] => [1,2] => 1
[1,0,1,0,1,0] => [3,2,1] => [2,3,1] => 2
[1,0,1,1,0,0] => [2,3,1] => [3,2,1] => 3
[1,1,0,0,1,0] => [3,1,2] => [3,1,2] => 2
[1,1,0,1,0,0] => [2,1,3] => [2,1,3] => 2
[1,1,1,0,0,0] => [1,2,3] => [1,2,3] => 1
[1,0,1,0,1,0,1,0] => [4,3,2,1] => [2,3,4,1] => 2
[1,0,1,0,1,1,0,0] => [3,4,2,1] => [2,4,3,1] => 3
[1,0,1,1,0,0,1,0] => [4,2,3,1] => [3,4,2,1] => 3
[1,0,1,1,0,1,0,0] => [3,2,4,1] => [4,3,2,1] => 4
[1,0,1,1,1,0,0,0] => [2,3,4,1] => [4,2,3,1] => 3
[1,1,0,0,1,0,1,0] => [4,3,1,2] => [3,1,4,2] => 3
[1,1,0,0,1,1,0,0] => [3,4,1,2] => [4,1,3,2] => 3
[1,1,0,1,0,0,1,0] => [4,2,1,3] => [2,4,1,3] => 2
[1,1,0,1,0,1,0,0] => [3,2,1,4] => [2,3,1,4] => 2
[1,1,0,1,1,0,0,0] => [2,3,1,4] => [3,2,1,4] => 3
[1,1,1,0,0,0,1,0] => [4,1,2,3] => [4,1,2,3] => 2
[1,1,1,0,0,1,0,0] => [3,1,2,4] => [3,1,2,4] => 2
[1,1,1,0,1,0,0,0] => [2,1,3,4] => [2,1,3,4] => 2
[1,1,1,1,0,0,0,0] => [1,2,3,4] => [1,2,3,4] => 1
[1,0,1,0,1,0,1,0,1,0] => [5,4,3,2,1] => [2,3,4,5,1] => 2
[1,0,1,0,1,0,1,1,0,0] => [4,5,3,2,1] => [2,3,5,4,1] => 3
[1,0,1,0,1,1,0,0,1,0] => [5,3,4,2,1] => [2,4,5,3,1] => 3
[1,0,1,0,1,1,0,1,0,0] => [4,3,5,2,1] => [2,5,4,3,1] => 4
[1,0,1,0,1,1,1,0,0,0] => [3,4,5,2,1] => [2,5,3,4,1] => 3
[1,0,1,1,0,0,1,0,1,0] => [5,4,2,3,1] => [3,4,2,5,1] => 3
[1,0,1,1,0,0,1,1,0,0] => [4,5,2,3,1] => [3,5,2,4,1] => 3
[1,0,1,1,0,1,0,0,1,0] => [5,3,2,4,1] => [4,3,5,2,1] => 4
[1,0,1,1,0,1,0,1,0,0] => [4,3,2,5,1] => [5,3,4,2,1] => 4
[1,0,1,1,0,1,1,0,0,0] => [3,4,2,5,1] => [5,4,3,2,1] => 5
[1,0,1,1,1,0,0,0,1,0] => [5,2,3,4,1] => [4,5,2,3,1] => 3
[1,0,1,1,1,0,0,1,0,0] => [4,2,3,5,1] => [5,4,2,3,1] => 4
[1,0,1,1,1,0,1,0,0,0] => [3,2,4,5,1] => [5,3,2,4,1] => 4
[1,0,1,1,1,1,0,0,0,0] => [2,3,4,5,1] => [5,2,3,4,1] => 3
[1,1,0,0,1,0,1,0,1,0] => [5,4,3,1,2] => [3,1,4,5,2] => 3
[1,1,0,0,1,0,1,1,0,0] => [4,5,3,1,2] => [3,1,5,4,2] => 4
[1,1,0,0,1,1,0,0,1,0] => [5,3,4,1,2] => [4,1,5,3,2] => 4
[1,1,0,0,1,1,0,1,0,0] => [4,3,5,1,2] => [5,1,4,3,2] => 4
[1,1,0,0,1,1,1,0,0,0] => [3,4,5,1,2] => [5,1,3,4,2] => 3
[1,1,0,1,0,0,1,0,1,0] => [5,4,2,1,3] => [2,4,1,5,3] => 3
[1,1,0,1,0,0,1,1,0,0] => [4,5,2,1,3] => [2,5,1,4,3] => 3
[1,1,0,1,0,1,0,0,1,0] => [5,3,2,1,4] => [2,3,5,1,4] => 2
[1,1,0,1,0,1,0,1,0,0] => [4,3,2,1,5] => [2,3,4,1,5] => 2
[1,1,0,1,0,1,1,0,0,0] => [3,4,2,1,5] => [2,4,3,1,5] => 3
[1,1,0,1,1,0,0,0,1,0] => [5,2,3,1,4] => [3,5,2,1,4] => 3
[1,1,0,1,1,0,0,1,0,0] => [4,2,3,1,5] => [3,4,2,1,5] => 3
[1,1,0,1,1,0,1,0,0,0] => [3,2,4,1,5] => [4,3,2,1,5] => 4
[1,1,0,1,1,1,0,0,0,0] => [2,3,4,1,5] => [4,2,3,1,5] => 3
[1,1,1,0,0,0,1,0,1,0] => [5,4,1,2,3] => [4,1,2,5,3] => 3
[1,1,1,0,0,0,1,1,0,0] => [4,5,1,2,3] => [5,1,2,4,3] => 3
[1,1,1,0,0,1,0,0,1,0] => [5,3,1,2,4] => [3,1,5,2,4] => 3
[1,1,1,0,0,1,0,1,0,0] => [4,3,1,2,5] => [3,1,4,2,5] => 3
[1,1,1,0,0,1,1,0,0,0] => [3,4,1,2,5] => [4,1,3,2,5] => 3
[1,1,1,0,1,0,0,0,1,0] => [5,2,1,3,4] => [2,5,1,3,4] => 2
[1,1,1,0,1,0,0,1,0,0] => [4,2,1,3,5] => [2,4,1,3,5] => 2
[1,1,1,0,1,0,1,0,0,0] => [3,2,1,4,5] => [2,3,1,4,5] => 2
[1,1,1,0,1,1,0,0,0,0] => [2,3,1,4,5] => [3,2,1,4,5] => 3
[1,1,1,1,0,0,0,0,1,0] => [5,1,2,3,4] => [5,1,2,3,4] => 2
[1,1,1,1,0,0,0,1,0,0] => [4,1,2,3,5] => [4,1,2,3,5] => 2
[1,1,1,1,0,0,1,0,0,0] => [3,1,2,4,5] => [3,1,2,4,5] => 2
[1,1,1,1,0,1,0,0,0,0] => [2,1,3,4,5] => [2,1,3,4,5] => 2
[1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => [1,2,3,4,5] => 1
[1,0,1,0,1,0,1,0,1,0,1,0] => [6,5,4,3,2,1] => [2,3,4,5,6,1] => 2
[1,0,1,0,1,0,1,0,1,1,0,0] => [5,6,4,3,2,1] => [2,3,4,6,5,1] => 3
[1,0,1,0,1,0,1,1,0,0,1,0] => [6,4,5,3,2,1] => [2,3,5,6,4,1] => 3
[1,0,1,0,1,0,1,1,0,1,0,0] => [5,4,6,3,2,1] => [2,3,6,5,4,1] => 4
[1,0,1,0,1,0,1,1,1,0,0,0] => [4,5,6,3,2,1] => [2,3,6,4,5,1] => 3
[1,0,1,0,1,1,0,0,1,0,1,0] => [6,5,3,4,2,1] => [2,4,5,3,6,1] => 3
[1,0,1,0,1,1,0,0,1,1,0,0] => [5,6,3,4,2,1] => [2,4,6,3,5,1] => 3
[1,0,1,0,1,1,0,1,0,0,1,0] => [6,4,3,5,2,1] => [2,5,4,6,3,1] => 4
[1,0,1,0,1,1,0,1,0,1,0,0] => [5,4,3,6,2,1] => [2,6,4,5,3,1] => 4
[1,0,1,0,1,1,0,1,1,0,0,0] => [4,5,3,6,2,1] => [2,6,5,4,3,1] => 5
[1,0,1,0,1,1,1,0,0,0,1,0] => [6,3,4,5,2,1] => [2,5,6,3,4,1] => 3
[1,0,1,0,1,1,1,0,0,1,0,0] => [5,3,4,6,2,1] => [2,6,5,3,4,1] => 4
[1,0,1,0,1,1,1,0,1,0,0,0] => [4,3,5,6,2,1] => [2,6,4,3,5,1] => 4
[1,0,1,0,1,1,1,1,0,0,0,0] => [3,4,5,6,2,1] => [2,6,3,4,5,1] => 3
[1,0,1,1,0,0,1,0,1,0,1,0] => [6,5,4,2,3,1] => [3,4,2,5,6,1] => 3
[1,0,1,1,0,0,1,0,1,1,0,0] => [5,6,4,2,3,1] => [3,4,2,6,5,1] => 4
[1,0,1,1,0,0,1,1,0,0,1,0] => [6,4,5,2,3,1] => [3,5,2,6,4,1] => 4
[1,0,1,1,0,0,1,1,0,1,0,0] => [5,4,6,2,3,1] => [3,6,2,5,4,1] => 4
[1,0,1,1,0,0,1,1,1,0,0,0] => [4,5,6,2,3,1] => [3,6,2,4,5,1] => 3
[1,0,1,1,0,1,0,0,1,0,1,0] => [6,5,3,2,4,1] => [4,3,5,2,6,1] => 4
[1,0,1,1,0,1,0,0,1,1,0,0] => [5,6,3,2,4,1] => [4,3,6,2,5,1] => 4
[1,0,1,1,0,1,0,1,0,0,1,0] => [6,4,3,2,5,1] => [5,3,4,6,2,1] => 4
[1,0,1,1,0,1,0,1,0,1,0,0] => [5,4,3,2,6,1] => [6,3,4,5,2,1] => 4
[1,0,1,1,0,1,0,1,1,0,0,0] => [4,5,3,2,6,1] => [6,3,5,4,2,1] => 5
[1,0,1,1,0,1,1,0,0,0,1,0] => [6,3,4,2,5,1] => [5,4,6,3,2,1] => 5
[1,0,1,1,0,1,1,0,0,1,0,0] => [5,3,4,2,6,1] => [6,4,5,3,2,1] => 5
[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
[1,0,1,1,0,1,1,1,0,0,0,0] => [3,4,5,2,6,1] => [6,5,3,4,2,1] => 5
[1,0,1,1,1,0,0,0,1,0,1,0] => [6,5,2,3,4,1] => [4,5,2,3,6,1] => 3
[1,0,1,1,1,0,0,0,1,1,0,0] => [5,6,2,3,4,1] => [4,6,2,3,5,1] => 3
[1,0,1,1,1,0,0,1,0,0,1,0] => [6,4,2,3,5,1] => [5,4,2,6,3,1] => 5
[1,0,1,1,1,0,0,1,0,1,0,0] => [5,4,2,3,6,1] => [6,4,2,5,3,1] => 5
[1,0,1,1,1,0,0,1,1,0,0,0] => [4,5,2,3,6,1] => [6,5,2,4,3,1] => 5
[1,0,1,1,1,0,1,0,0,0,1,0] => [6,3,2,4,5,1] => [5,3,6,2,4,1] => 4
[1,0,1,1,1,0,1,0,0,1,0,0] => [5,3,2,4,6,1] => [6,3,5,2,4,1] => 4
[1,0,1,1,1,0,1,0,1,0,0,0] => [4,3,2,5,6,1] => [6,3,4,2,5,1] => 4
[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
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Description
The width of the tree associated to a permutation.
A permutation can be mapped to a rooted tree with vertices $\{0,1,2,\ldots,n\}$ and root $0$ in the following way. Entries of the permutations are inserted one after the other, each child is larger than its parent and the children are in strict order from left to right. Details of the construction are found in [1].
The width of the tree is given by the number of leaves of this tree.
Note that, due to the construction of this tree, the width of the tree is always one more than the number of descents St000021The number of descents of a permutation.. This also matches the number of runs in a permutation St000470The number of runs in a permutation..
See also St000308The height of the tree associated to a permutation. for the height of this tree.
A permutation can be mapped to a rooted tree with vertices $\{0,1,2,\ldots,n\}$ and root $0$ in the following way. Entries of the permutations are inserted one after the other, each child is larger than its parent and the children are in strict order from left to right. Details of the construction are found in [1].
The width of the tree is given by the number of leaves of this tree.
Note that, due to the construction of this tree, the width of the tree is always one more than the number of descents St000021The number of descents of a permutation.. This also matches the number of runs in a permutation St000470The number of runs in a permutation..
See also St000308The height of the tree associated to a permutation. for the height of this tree.
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 $\chi(\pi)$.
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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