Identifier
Mp00283:
Perfect matchings
—non-nesting-exceedence permutation⟶
Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00237: Permutations —descent views to invisible inversion bottoms⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00237: Permutations —descent views to invisible inversion bottoms⟶ Permutations
Images
[(1,2)] => [2,1] => [1,2] => [1,2]
[(1,2),(3,4)] => [2,1,4,3] => [1,2,3,4] => [1,2,3,4]
[(1,3),(2,4)] => [3,4,1,2] => [1,3,2,4] => [1,3,2,4]
[(1,4),(2,3)] => [3,4,2,1] => [1,3,2,4] => [1,3,2,4]
[(1,2),(3,4),(5,6)] => [2,1,4,3,6,5] => [1,2,3,4,5,6] => [1,2,3,4,5,6]
[(1,3),(2,4),(5,6)] => [3,4,1,2,6,5] => [1,3,2,4,5,6] => [1,3,2,4,5,6]
[(1,4),(2,3),(5,6)] => [3,4,2,1,6,5] => [1,3,2,4,5,6] => [1,3,2,4,5,6]
[(1,5),(2,3),(4,6)] => [3,5,2,6,1,4] => [1,3,2,5,4,6] => [1,3,2,5,4,6]
[(1,6),(2,3),(4,5)] => [3,5,2,6,4,1] => [1,3,2,5,4,6] => [1,3,2,5,4,6]
[(1,6),(2,4),(3,5)] => [4,5,6,2,3,1] => [1,4,2,5,3,6] => [1,4,5,2,3,6]
[(1,5),(2,4),(3,6)] => [4,5,6,2,1,3] => [1,4,2,5,3,6] => [1,4,5,2,3,6]
[(1,4),(2,5),(3,6)] => [4,5,6,1,2,3] => [1,4,2,5,3,6] => [1,4,5,2,3,6]
[(1,3),(2,5),(4,6)] => [3,5,1,6,2,4] => [1,3,2,5,4,6] => [1,3,2,5,4,6]
[(1,2),(3,5),(4,6)] => [2,1,5,6,3,4] => [1,2,3,5,4,6] => [1,2,3,5,4,6]
[(1,2),(3,6),(4,5)] => [2,1,5,6,4,3] => [1,2,3,5,4,6] => [1,2,3,5,4,6]
[(1,3),(2,6),(4,5)] => [3,5,1,6,4,2] => [1,3,2,5,4,6] => [1,3,2,5,4,6]
[(1,4),(2,6),(3,5)] => [4,5,6,1,3,2] => [1,4,2,5,3,6] => [1,4,5,2,3,6]
[(1,5),(2,6),(3,4)] => [4,5,6,3,1,2] => [1,4,3,6,2,5] => [1,6,4,3,2,5]
[(1,6),(2,5),(3,4)] => [4,5,6,3,2,1] => [1,4,3,6,2,5] => [1,6,4,3,2,5]
[(1,2),(3,4),(5,6),(7,8)] => [2,1,4,3,6,5,8,7] => [1,2,3,4,5,6,7,8] => [1,2,3,4,5,6,7,8]
[(1,3),(2,4),(5,6),(7,8)] => [3,4,1,2,6,5,8,7] => [1,3,2,4,5,6,7,8] => [1,3,2,4,5,6,7,8]
[(1,4),(2,3),(5,6),(7,8)] => [3,4,2,1,6,5,8,7] => [1,3,2,4,5,6,7,8] => [1,3,2,4,5,6,7,8]
[(1,5),(2,3),(4,6),(7,8)] => [3,5,2,6,1,4,8,7] => [1,3,2,5,4,6,7,8] => [1,3,2,5,4,6,7,8]
[(1,6),(2,3),(4,5),(7,8)] => [3,5,2,6,4,1,8,7] => [1,3,2,5,4,6,7,8] => [1,3,2,5,4,6,7,8]
[(1,7),(2,3),(4,5),(6,8)] => [3,5,2,7,4,8,1,6] => [1,3,2,5,4,7,6,8] => [1,3,2,5,4,7,6,8]
[(1,8),(2,3),(4,5),(6,7)] => [3,5,2,7,4,8,6,1] => [1,3,2,5,4,7,6,8] => [1,3,2,5,4,7,6,8]
[(1,8),(2,4),(3,5),(6,7)] => [4,5,7,2,3,8,6,1] => [1,4,2,5,3,7,6,8] => [1,4,5,2,3,7,6,8]
[(1,7),(2,4),(3,5),(6,8)] => [4,5,7,2,3,8,1,6] => [1,4,2,5,3,7,6,8] => [1,4,5,2,3,7,6,8]
[(1,6),(2,4),(3,5),(7,8)] => [4,5,6,2,3,1,8,7] => [1,4,2,5,3,6,7,8] => [1,4,5,2,3,6,7,8]
[(1,5),(2,4),(3,6),(7,8)] => [4,5,6,2,1,3,8,7] => [1,4,2,5,3,6,7,8] => [1,4,5,2,3,6,7,8]
[(1,4),(2,5),(3,6),(7,8)] => [4,5,6,1,2,3,8,7] => [1,4,2,5,3,6,7,8] => [1,4,5,2,3,6,7,8]
[(1,3),(2,5),(4,6),(7,8)] => [3,5,1,6,2,4,8,7] => [1,3,2,5,4,6,7,8] => [1,3,2,5,4,6,7,8]
[(1,2),(3,5),(4,6),(7,8)] => [2,1,5,6,3,4,8,7] => [1,2,3,5,4,6,7,8] => [1,2,3,5,4,6,7,8]
[(1,2),(3,6),(4,5),(7,8)] => [2,1,5,6,4,3,8,7] => [1,2,3,5,4,6,7,8] => [1,2,3,5,4,6,7,8]
[(1,3),(2,6),(4,5),(7,8)] => [3,5,1,6,4,2,8,7] => [1,3,2,5,4,6,7,8] => [1,3,2,5,4,6,7,8]
[(1,4),(2,6),(3,5),(7,8)] => [4,5,6,1,3,2,8,7] => [1,4,2,5,3,6,7,8] => [1,4,5,2,3,6,7,8]
[(1,5),(2,6),(3,4),(7,8)] => [4,5,6,3,1,2,8,7] => [1,4,3,6,2,5,7,8] => [1,6,4,3,2,5,7,8]
[(1,6),(2,5),(3,4),(7,8)] => [4,5,6,3,2,1,8,7] => [1,4,3,6,2,5,7,8] => [1,6,4,3,2,5,7,8]
[(1,7),(2,5),(3,4),(6,8)] => [4,5,7,3,2,8,1,6] => [1,4,3,7,2,5,6,8] => [1,7,4,3,2,5,6,8]
[(1,8),(2,5),(3,4),(6,7)] => [4,5,7,3,2,8,6,1] => [1,4,3,7,6,8,2,5] => [1,8,4,3,2,7,6,5]
[(1,8),(2,6),(3,4),(5,7)] => [4,6,7,3,8,2,5,1] => [1,4,3,7,5,8,2,6] => [1,8,4,3,7,2,5,6]
[(1,7),(2,6),(3,4),(5,8)] => [4,6,7,3,8,2,1,5] => [1,4,3,7,2,6,5,8] => [1,7,4,3,6,2,5,8]
[(1,6),(2,7),(3,4),(5,8)] => [4,6,7,3,8,1,2,5] => [1,4,3,7,2,6,5,8] => [1,7,4,3,6,2,5,8]
[(1,5),(2,7),(3,4),(6,8)] => [4,5,7,3,1,8,2,6] => [1,4,3,7,2,5,6,8] => [1,7,4,3,2,5,6,8]
[(1,4),(2,7),(3,5),(6,8)] => [4,5,7,1,3,8,2,6] => [1,4,2,5,3,7,6,8] => [1,4,5,2,3,7,6,8]
[(1,3),(2,7),(4,5),(6,8)] => [3,5,1,7,4,8,2,6] => [1,3,2,5,4,7,6,8] => [1,3,2,5,4,7,6,8]
[(1,2),(3,7),(4,5),(6,8)] => [2,1,5,7,4,8,3,6] => [1,2,3,5,4,7,6,8] => [1,2,3,5,4,7,6,8]
[(1,2),(3,8),(4,5),(6,7)] => [2,1,5,7,4,8,6,3] => [1,2,3,5,4,7,6,8] => [1,2,3,5,4,7,6,8]
[(1,3),(2,8),(4,5),(6,7)] => [3,5,1,7,4,8,6,2] => [1,3,2,5,4,7,6,8] => [1,3,2,5,4,7,6,8]
[(1,4),(2,8),(3,5),(6,7)] => [4,5,7,1,3,8,6,2] => [1,4,2,5,3,7,6,8] => [1,4,5,2,3,7,6,8]
[(1,5),(2,8),(3,4),(6,7)] => [4,5,7,3,1,8,6,2] => [1,4,3,7,6,8,2,5] => [1,8,4,3,2,7,6,5]
[(1,6),(2,8),(3,4),(5,7)] => [4,6,7,3,8,1,5,2] => [1,4,3,7,5,8,2,6] => [1,8,4,3,7,2,5,6]
[(1,7),(2,8),(3,4),(5,6)] => [4,6,7,3,8,5,1,2] => [1,4,3,7,2,6,5,8] => [1,7,4,3,6,2,5,8]
[(1,8),(2,7),(3,4),(5,6)] => [4,6,7,3,8,5,2,1] => [1,4,3,7,2,6,5,8] => [1,7,4,3,6,2,5,8]
[(1,8),(2,7),(3,5),(4,6)] => [5,6,7,8,3,4,2,1] => [1,5,3,7,2,6,4,8] => [1,7,5,6,3,2,4,8]
[(1,7),(2,8),(3,5),(4,6)] => [5,6,7,8,3,4,1,2] => [1,5,3,7,2,6,4,8] => [1,7,5,6,3,2,4,8]
[(1,6),(2,8),(3,5),(4,7)] => [5,6,7,8,3,1,4,2] => [1,5,3,7,4,8,2,6] => [1,8,5,7,3,2,4,6]
[(1,5),(2,8),(3,6),(4,7)] => [5,6,7,8,1,3,4,2] => [1,5,2,6,3,7,4,8] => [1,5,6,7,2,3,4,8]
[(1,4),(2,8),(3,6),(5,7)] => [4,6,7,1,8,3,5,2] => [1,4,2,6,3,7,5,8] => [1,4,6,2,7,3,5,8]
[(1,3),(2,8),(4,6),(5,7)] => [3,6,1,7,8,4,5,2] => [1,3,2,6,4,7,5,8] => [1,3,2,6,7,4,5,8]
[(1,2),(3,8),(4,6),(5,7)] => [2,1,6,7,8,4,5,3] => [1,2,3,6,4,7,5,8] => [1,2,3,6,7,4,5,8]
[(1,2),(3,7),(4,6),(5,8)] => [2,1,6,7,8,4,3,5] => [1,2,3,6,4,7,5,8] => [1,2,3,6,7,4,5,8]
[(1,3),(2,7),(4,6),(5,8)] => [3,6,1,7,8,4,2,5] => [1,3,2,6,4,7,5,8] => [1,3,2,6,7,4,5,8]
[(1,4),(2,7),(3,6),(5,8)] => [4,6,7,1,8,3,2,5] => [1,4,2,6,3,7,5,8] => [1,4,6,2,7,3,5,8]
[(1,5),(2,7),(3,6),(4,8)] => [5,6,7,8,1,3,2,4] => [1,5,2,6,3,7,4,8] => [1,5,6,7,2,3,4,8]
[(1,6),(2,7),(3,5),(4,8)] => [5,6,7,8,3,1,2,4] => [1,5,3,7,2,6,4,8] => [1,7,5,6,3,2,4,8]
[(1,7),(2,6),(3,5),(4,8)] => [5,6,7,8,3,2,1,4] => [1,5,3,7,2,6,4,8] => [1,7,5,6,3,2,4,8]
[(1,8),(2,6),(3,5),(4,7)] => [5,6,7,8,3,2,4,1] => [1,5,3,7,4,8,2,6] => [1,8,5,7,3,2,4,6]
[(1,8),(2,5),(3,6),(4,7)] => [5,6,7,8,2,3,4,1] => [1,5,2,6,3,7,4,8] => [1,5,6,7,2,3,4,8]
[(1,7),(2,5),(3,6),(4,8)] => [5,6,7,8,2,3,1,4] => [1,5,2,6,3,7,4,8] => [1,5,6,7,2,3,4,8]
[(1,6),(2,5),(3,7),(4,8)] => [5,6,7,8,2,1,3,4] => [1,5,2,6,3,7,4,8] => [1,5,6,7,2,3,4,8]
[(1,5),(2,6),(3,7),(4,8)] => [5,6,7,8,1,2,3,4] => [1,5,2,6,3,7,4,8] => [1,5,6,7,2,3,4,8]
[(1,4),(2,6),(3,7),(5,8)] => [4,6,7,1,8,2,3,5] => [1,4,2,6,3,7,5,8] => [1,4,6,2,7,3,5,8]
[(1,3),(2,6),(4,7),(5,8)] => [3,6,1,7,8,2,4,5] => [1,3,2,6,4,7,5,8] => [1,3,2,6,7,4,5,8]
[(1,2),(3,6),(4,7),(5,8)] => [2,1,6,7,8,3,4,5] => [1,2,3,6,4,7,5,8] => [1,2,3,6,7,4,5,8]
[(1,2),(3,5),(4,7),(6,8)] => [2,1,5,7,3,8,4,6] => [1,2,3,5,4,7,6,8] => [1,2,3,5,4,7,6,8]
[(1,3),(2,5),(4,7),(6,8)] => [3,5,1,7,2,8,4,6] => [1,3,2,5,4,7,6,8] => [1,3,2,5,4,7,6,8]
[(1,4),(2,5),(3,7),(6,8)] => [4,5,7,1,2,8,3,6] => [1,4,2,5,3,7,6,8] => [1,4,5,2,3,7,6,8]
[(1,5),(2,4),(3,7),(6,8)] => [4,5,7,2,1,8,3,6] => [1,4,2,5,3,7,6,8] => [1,4,5,2,3,7,6,8]
[(1,6),(2,4),(3,7),(5,8)] => [4,6,7,2,8,1,3,5] => [1,4,2,6,3,7,5,8] => [1,4,6,2,7,3,5,8]
[(1,7),(2,4),(3,6),(5,8)] => [4,6,7,2,8,3,1,5] => [1,4,2,6,3,7,5,8] => [1,4,6,2,7,3,5,8]
[(1,8),(2,4),(3,6),(5,7)] => [4,6,7,2,8,3,5,1] => [1,4,2,6,3,7,5,8] => [1,4,6,2,7,3,5,8]
[(1,8),(2,3),(4,6),(5,7)] => [3,6,2,7,8,4,5,1] => [1,3,2,6,4,7,5,8] => [1,3,2,6,7,4,5,8]
[(1,7),(2,3),(4,6),(5,8)] => [3,6,2,7,8,4,1,5] => [1,3,2,6,4,7,5,8] => [1,3,2,6,7,4,5,8]
[(1,6),(2,3),(4,7),(5,8)] => [3,6,2,7,8,1,4,5] => [1,3,2,6,4,7,5,8] => [1,3,2,6,7,4,5,8]
[(1,5),(2,3),(4,7),(6,8)] => [3,5,2,7,1,8,4,6] => [1,3,2,5,4,7,6,8] => [1,3,2,5,4,7,6,8]
[(1,4),(2,3),(5,7),(6,8)] => [3,4,2,1,7,8,5,6] => [1,3,2,4,5,7,6,8] => [1,3,2,4,5,7,6,8]
[(1,3),(2,4),(5,7),(6,8)] => [3,4,1,2,7,8,5,6] => [1,3,2,4,5,7,6,8] => [1,3,2,4,5,7,6,8]
[(1,2),(3,4),(5,7),(6,8)] => [2,1,4,3,7,8,5,6] => [1,2,3,4,5,7,6,8] => [1,2,3,4,5,7,6,8]
[(1,2),(3,4),(5,8),(6,7)] => [2,1,4,3,7,8,6,5] => [1,2,3,4,5,7,6,8] => [1,2,3,4,5,7,6,8]
[(1,3),(2,4),(5,8),(6,7)] => [3,4,1,2,7,8,6,5] => [1,3,2,4,5,7,6,8] => [1,3,2,4,5,7,6,8]
[(1,4),(2,3),(5,8),(6,7)] => [3,4,2,1,7,8,6,5] => [1,3,2,4,5,7,6,8] => [1,3,2,4,5,7,6,8]
[(1,5),(2,3),(4,8),(6,7)] => [3,5,2,7,1,8,6,4] => [1,3,2,5,4,7,6,8] => [1,3,2,5,4,7,6,8]
[(1,6),(2,3),(4,8),(5,7)] => [3,6,2,7,8,1,5,4] => [1,3,2,6,4,7,5,8] => [1,3,2,6,7,4,5,8]
[(1,7),(2,3),(4,8),(5,6)] => [3,6,2,7,8,5,1,4] => [1,3,2,6,5,8,4,7] => [1,3,2,8,6,5,4,7]
[(1,8),(2,3),(4,7),(5,6)] => [3,6,2,7,8,5,4,1] => [1,3,2,6,5,8,4,7] => [1,3,2,8,6,5,4,7]
[(1,8),(2,4),(3,7),(5,6)] => [4,6,7,2,8,5,3,1] => [1,4,2,6,5,8,3,7] => [1,4,8,2,6,5,3,7]
[(1,7),(2,4),(3,8),(5,6)] => [4,6,7,2,8,5,1,3] => [1,4,2,6,5,8,3,7] => [1,4,8,2,6,5,3,7]
[(1,6),(2,4),(3,8),(5,7)] => [4,6,7,2,8,1,5,3] => [1,4,2,6,3,7,5,8] => [1,4,6,2,7,3,5,8]
[(1,5),(2,4),(3,8),(6,7)] => [4,5,7,2,1,8,6,3] => [1,4,2,5,3,7,6,8] => [1,4,5,2,3,7,6,8]
[(1,4),(2,5),(3,8),(6,7)] => [4,5,7,1,2,8,6,3] => [1,4,2,5,3,7,6,8] => [1,4,5,2,3,7,6,8]
>>> Load all 135 entries. <<<Map
non-nesting-exceedence permutation
Description
The fixed-point-free permutation with deficiencies given by the perfect matching, no alignments and no inversions between exceedences.
Put differently, the exceedences form the unique non-nesting perfect matching whose openers coincide with those of the given perfect matching.
Put differently, the exceedences form the unique non-nesting perfect matching whose openers coincide with those of the given perfect matching.
Map
cycle-as-one-line notation
Description
Return the permutation obtained by concatenating the cycles of a permutation, each written with minimal element first, sorted by minimal element.
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.
This map is similar to Mp00235descent views to invisible inversion bottoms, but different beginning with permutations of six elements.
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
This map is similar to Mp00235descent views to invisible inversion bottoms, but different beginning with permutations of six elements.
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 maximima 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 deficiency positions of $\chi(\pi)$, and
- the set of minimal elements in the decreasing runs of $\pi$ is the set of deficiency values of $\chi(\pi)$.
searching the database
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