Identifier
Mp00231:
Integer compositions
—bounce path⟶
Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
Images
[1] => [1,0] => [2,1] => [2,1]
[1,1] => [1,0,1,0] => [3,1,2] => [3,1,2]
[2] => [1,1,0,0] => [2,3,1] => [3,2,1]
[1,1,1] => [1,0,1,0,1,0] => [4,1,2,3] => [4,1,2,3]
[1,2] => [1,0,1,1,0,0] => [3,1,4,2] => [3,4,1,2]
[2,1] => [1,1,0,0,1,0] => [2,4,1,3] => [4,2,1,3]
[3] => [1,1,1,0,0,0] => [2,3,4,1] => [4,2,3,1]
[1,1,1,1] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => [5,1,2,3,4]
[1,1,2] => [1,0,1,0,1,1,0,0] => [4,1,2,5,3] => [4,1,5,2,3]
[1,2,1] => [1,0,1,1,0,0,1,0] => [3,1,5,2,4] => [3,5,1,2,4]
[1,3] => [1,0,1,1,1,0,0,0] => [3,1,4,5,2] => [3,5,1,4,2]
[2,1,1] => [1,1,0,0,1,0,1,0] => [2,5,1,3,4] => [5,2,1,3,4]
[2,2] => [1,1,0,0,1,1,0,0] => [2,4,1,5,3] => [4,2,5,1,3]
[3,1] => [1,1,1,0,0,0,1,0] => [2,3,5,1,4] => [5,2,3,1,4]
[4] => [1,1,1,1,0,0,0,0] => [2,3,4,5,1] => [5,2,3,4,1]
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0] => [6,1,2,3,4,5] => [6,1,2,3,4,5]
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0] => [5,1,2,3,6,4] => [5,1,2,6,3,4]
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => [4,1,2,6,3,5] => [4,1,6,2,3,5]
[1,1,3] => [1,0,1,0,1,1,1,0,0,0] => [4,1,2,5,6,3] => [4,1,6,2,5,3]
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => [3,1,6,2,4,5] => [3,6,1,2,4,5]
[1,2,2] => [1,0,1,1,0,0,1,1,0,0] => [3,1,5,2,6,4] => [3,5,1,6,2,4]
[1,3,1] => [1,0,1,1,1,0,0,0,1,0] => [3,1,4,6,2,5] => [3,6,1,4,2,5]
[1,4] => [1,0,1,1,1,1,0,0,0,0] => [3,1,4,5,6,2] => [3,6,1,4,5,2]
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => [2,6,1,3,4,5] => [6,2,1,3,4,5]
[2,1,2] => [1,1,0,0,1,0,1,1,0,0] => [2,5,1,3,6,4] => [5,2,1,6,3,4]
[2,2,1] => [1,1,0,0,1,1,0,0,1,0] => [2,4,1,6,3,5] => [4,2,6,1,3,5]
[2,3] => [1,1,0,0,1,1,1,0,0,0] => [2,4,1,5,6,3] => [4,2,6,1,5,3]
[3,1,1] => [1,1,1,0,0,0,1,0,1,0] => [2,3,6,1,4,5] => [6,2,3,1,4,5]
[3,2] => [1,1,1,0,0,0,1,1,0,0] => [2,3,5,1,6,4] => [5,2,3,6,1,4]
[4,1] => [1,1,1,1,0,0,0,0,1,0] => [2,3,4,6,1,5] => [6,2,3,4,1,5]
[5] => [1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => [6,2,3,4,5,1]
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => [7,1,2,3,4,5,6] => [7,1,2,3,4,5,6]
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0] => [6,1,2,3,4,7,5] => [6,1,2,3,7,4,5]
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0] => [5,1,2,3,7,4,6] => [5,1,2,7,3,4,6]
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0] => [5,1,2,3,6,7,4] => [5,1,2,7,3,6,4]
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0] => [4,1,2,7,3,5,6] => [4,1,7,2,3,5,6]
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0] => [4,1,2,6,3,7,5] => [4,1,6,2,7,3,5]
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0] => [4,1,2,5,7,3,6] => [4,1,7,2,5,3,6]
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0] => [4,1,2,5,6,7,3] => [4,1,7,2,5,6,3]
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0] => [3,1,7,2,4,5,6] => [3,7,1,2,4,5,6]
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0] => [3,1,6,2,4,7,5] => [3,6,1,2,7,4,5]
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0] => [3,1,5,2,7,4,6] => [3,5,1,7,2,4,6]
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0] => [3,1,5,2,6,7,4] => [3,5,1,7,2,6,4]
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0] => [3,1,4,7,2,5,6] => [3,7,1,4,2,5,6]
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0] => [3,1,4,6,2,7,5] => [3,6,1,4,7,2,5]
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0] => [3,1,4,5,7,2,6] => [3,7,1,4,5,2,6]
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0] => [3,1,4,5,6,7,2] => [3,7,1,4,5,6,2]
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0] => [2,7,1,3,4,5,6] => [7,2,1,3,4,5,6]
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0] => [2,6,1,3,4,7,5] => [6,2,1,3,7,4,5]
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0] => [2,5,1,3,7,4,6] => [5,2,1,7,3,4,6]
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0] => [2,5,1,3,6,7,4] => [5,2,1,7,3,6,4]
[2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0] => [2,4,1,7,3,5,6] => [4,2,7,1,3,5,6]
[2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0] => [2,4,1,6,3,7,5] => [4,2,6,1,7,3,5]
[2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0] => [2,4,1,5,7,3,6] => [4,2,7,1,5,3,6]
[2,4] => [1,1,0,0,1,1,1,1,0,0,0,0] => [2,4,1,5,6,7,3] => [4,2,7,1,5,6,3]
[3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0] => [2,3,7,1,4,5,6] => [7,2,3,1,4,5,6]
[3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0] => [2,3,6,1,4,7,5] => [6,2,3,1,7,4,5]
[3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0] => [2,3,5,1,7,4,6] => [5,2,3,7,1,4,6]
[3,3] => [1,1,1,0,0,0,1,1,1,0,0,0] => [2,3,5,1,6,7,4] => [5,2,3,7,1,6,4]
[4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0] => [2,3,4,7,1,5,6] => [7,2,3,4,1,5,6]
[4,2] => [1,1,1,1,0,0,0,0,1,1,0,0] => [2,3,4,6,1,7,5] => [6,2,3,4,7,1,5]
[5,1] => [1,1,1,1,1,0,0,0,0,0,1,0] => [2,3,4,5,7,1,6] => [7,2,3,4,5,1,6]
[6] => [1,1,1,1,1,1,0,0,0,0,0,0] => [2,3,4,5,6,7,1] => [7,2,3,4,5,6,1]
[1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [8,1,2,3,4,5,6,7] => [8,1,2,3,4,5,6,7]
[1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0] => [7,1,2,3,4,5,8,6] => [7,1,2,3,4,8,5,6]
[1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0] => [6,1,2,3,4,8,5,7] => [6,1,2,3,8,4,5,7]
[1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0] => [6,1,2,3,4,7,8,5] => [6,1,2,3,8,4,7,5]
[1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0] => [5,1,2,3,8,4,6,7] => [5,1,2,8,3,4,6,7]
[1,1,1,2,2] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0] => [5,1,2,3,7,4,8,6] => [5,1,2,7,3,8,4,6]
[1,1,1,3,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0] => [5,1,2,3,6,8,4,7] => [5,1,2,8,3,6,4,7]
[1,1,1,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0] => [5,1,2,3,6,7,8,4] => [5,1,2,8,3,6,7,4]
[1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0] => [4,1,2,8,3,5,6,7] => [4,1,8,2,3,5,6,7]
[1,1,2,1,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0] => [4,1,2,7,3,5,8,6] => [4,1,7,2,3,8,5,6]
[1,1,2,2,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0] => [4,1,2,6,3,8,5,7] => [4,1,6,2,8,3,5,7]
[1,1,2,3] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0] => [4,1,2,6,3,7,8,5] => [4,1,6,2,8,3,7,5]
[1,1,3,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0] => [4,1,2,5,8,3,6,7] => [4,1,8,2,5,3,6,7]
[1,1,3,2] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0] => [4,1,2,5,7,3,8,6] => [4,1,7,2,5,8,3,6]
[1,1,4,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0] => [4,1,2,5,6,8,3,7] => [4,1,8,2,5,6,3,7]
[1,1,5] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0] => [4,1,2,5,6,7,8,3] => [4,1,8,2,5,6,7,3]
[1,2,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0] => [3,1,8,2,4,5,6,7] => [3,8,1,2,4,5,6,7]
[1,2,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0] => [3,1,7,2,4,5,8,6] => [3,7,1,2,4,8,5,6]
[1,2,1,2,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0] => [3,1,6,2,4,8,5,7] => [3,6,1,2,8,4,5,7]
[1,2,1,3] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0] => [3,1,6,2,4,7,8,5] => [3,6,1,2,8,4,7,5]
[1,2,2,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0] => [3,1,5,2,8,4,6,7] => [3,5,1,8,2,4,6,7]
[1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0] => [3,1,5,2,7,4,8,6] => [3,5,1,7,2,8,4,6]
[1,2,3,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0] => [3,1,5,2,6,8,4,7] => [3,5,1,8,2,6,4,7]
[1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0] => [3,1,5,2,6,7,8,4] => [3,5,1,8,2,6,7,4]
[1,3,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0] => [3,1,4,8,2,5,6,7] => [3,8,1,4,2,5,6,7]
[1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0] => [3,1,4,7,2,5,8,6] => [3,7,1,4,2,8,5,6]
[1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0] => [3,1,4,6,2,8,5,7] => [3,6,1,4,8,2,5,7]
[1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0] => [3,1,4,6,2,7,8,5] => [3,6,1,4,8,2,7,5]
[1,4,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0] => [3,1,4,5,8,2,6,7] => [3,8,1,4,5,2,6,7]
[1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0] => [3,1,4,5,7,2,8,6] => [3,7,1,4,5,8,2,6]
[1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0] => [3,1,4,5,6,8,2,7] => [3,8,1,4,5,6,2,7]
[1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0] => [3,1,4,5,6,7,8,2] => [3,8,1,4,5,6,7,2]
[2,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0] => [2,8,1,3,4,5,6,7] => [8,2,1,3,4,5,6,7]
[2,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0] => [2,7,1,3,4,5,8,6] => [7,2,1,3,4,8,5,6]
[2,1,1,2,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0] => [2,6,1,3,4,8,5,7] => [6,2,1,3,8,4,5,7]
[2,1,1,3] => [1,1,0,0,1,0,1,0,1,1,1,0,0,0] => [2,6,1,3,4,7,8,5] => [6,2,1,3,8,4,7,5]
[2,1,2,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0] => [2,5,1,3,8,4,6,7] => [5,2,1,8,3,4,6,7]
[2,1,2,2] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0] => [2,5,1,3,7,4,8,6] => [5,2,1,7,3,8,4,6]
>>> Load all 267 entries. <<<Map
bounce path
Description
The bounce path determined by an integer composition.
Map
Ringel
Description
The Ringel permutation of the LNakayama algebra corresponding to a Dyck path.
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 $\chi(\pi)$.
searching the database
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