Identifier
Mp00058:
Perfect matchings
—to permutation⟶
Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
Images
=>
Cc0012;cc-rep-0
[(1,2)]=>[2,1]=>[2,1]=>[2,1]
[(1,2),(3,4)]=>[2,1,4,3]=>[3,2,4,1]=>[4,3,2,1]
[(1,3),(2,4)]=>[3,4,1,2]=>[1,4,2,3]=>[1,4,2,3]
[(1,4),(2,3)]=>[4,3,2,1]=>[4,3,2,1]=>[2,3,4,1]
[(1,2),(3,4),(5,6)]=>[2,1,4,3,6,5]=>[4,3,5,2,6,1]=>[6,5,4,3,2,1]
[(1,3),(2,4),(5,6)]=>[3,4,1,2,6,5]=>[2,5,3,4,6,1]=>[6,2,5,3,4,1]
[(1,4),(2,3),(5,6)]=>[4,3,2,1,6,5]=>[5,4,3,2,6,1]=>[6,3,4,5,2,1]
[(1,5),(2,3),(4,6)]=>[5,3,2,6,1,4]=>[1,5,4,6,2,3]=>[1,6,2,5,4,3]
[(1,6),(2,3),(4,5)]=>[6,3,2,5,4,1]=>[6,4,2,5,3,1]=>[3,5,4,6,1,2]
[(1,6),(2,4),(3,5)]=>[6,4,5,2,3,1]=>[6,1,5,2,4,3]=>[6,5,4,2,1,3]
[(1,5),(2,4),(3,6)]=>[5,4,6,2,1,3]=>[5,1,6,3,2,4]=>[5,3,6,2,1,4]
[(1,4),(2,5),(3,6)]=>[4,5,6,1,2,3]=>[1,2,6,3,4,5]=>[1,2,6,3,4,5]
[(1,3),(2,5),(4,6)]=>[3,5,1,6,2,4]=>[4,1,3,6,2,5]=>[4,6,1,3,2,5]
[(1,2),(3,5),(4,6)]=>[2,1,5,6,3,4]=>[3,2,1,6,4,5]=>[2,3,1,6,4,5]
[(1,2),(3,6),(4,5)]=>[2,1,6,5,4,3]=>[5,4,6,3,2,1]=>[2,3,6,5,4,1]
[(1,3),(2,6),(4,5)]=>[3,6,1,5,4,2]=>[3,6,4,5,2,1]=>[2,5,3,6,4,1]
[(1,4),(2,6),(3,5)]=>[4,6,5,1,3,2]=>[2,6,5,3,4,1]=>[4,2,5,3,6,1]
[(1,5),(2,6),(3,4)]=>[5,6,4,3,1,2]=>[1,6,5,4,2,3]=>[1,4,2,5,6,3]
[(1,6),(2,5),(3,4)]=>[6,5,4,3,2,1]=>[6,5,4,3,2,1]=>[2,3,4,5,6,1]
[(1,2),(3,4),(5,6),(7,8)]=>[2,1,4,3,6,5,8,7]=>[5,4,6,3,7,2,8,1]=>[8,7,6,5,4,3,2,1]
[(1,3),(2,4),(5,6),(7,8)]=>[3,4,1,2,6,5,8,7]=>[3,6,4,5,7,2,8,1]=>[8,7,3,6,4,5,2,1]
[(1,4),(2,3),(5,6),(7,8)]=>[4,3,2,1,6,5,8,7]=>[6,5,4,3,7,2,8,1]=>[8,7,4,5,6,3,2,1]
[(1,5),(2,3),(4,6),(7,8)]=>[5,3,2,6,1,4,8,7]=>[2,6,5,7,3,4,8,1]=>[8,2,7,3,6,5,4,1]
[(1,6),(2,3),(4,5),(7,8)]=>[6,3,2,5,4,1,8,7]=>[7,5,3,6,4,2,8,1]=>[8,4,6,5,7,2,3,1]
[(1,7),(2,3),(4,5),(6,8)]=>[7,3,2,5,4,8,1,6]=>[1,6,4,7,5,8,2,3]=>[1,8,2,6,7,4,5,3]
[(1,8),(2,3),(4,5),(6,7)]=>[8,3,2,5,4,7,6,1]=>[8,5,3,6,2,7,4,1]=>[4,7,6,5,8,1,3,2]
[(1,8),(2,4),(3,5),(6,7)]=>[8,4,5,2,3,7,6,1]=>[8,2,6,3,4,7,5,1]=>[5,7,8,1,6,3,4,2]
[(1,7),(2,4),(3,5),(6,8)]=>[7,4,5,2,3,8,1,6]=>[1,3,7,4,6,8,2,5]=>[1,8,3,7,2,4,6,5]
[(1,6),(2,4),(3,5),(7,8)]=>[6,4,5,2,3,1,8,7]=>[7,2,6,3,5,4,8,1]=>[8,7,6,5,3,2,4,1]
[(1,5),(2,4),(3,6),(7,8)]=>[5,4,6,2,1,3,8,7]=>[6,2,7,4,3,5,8,1]=>[8,6,4,7,3,2,5,1]
[(1,4),(2,5),(3,6),(7,8)]=>[4,5,6,1,2,3,8,7]=>[2,3,7,4,5,6,8,1]=>[8,2,3,7,4,5,6,1]
[(1,3),(2,5),(4,6),(7,8)]=>[3,5,1,6,2,4,8,7]=>[5,2,4,7,3,6,8,1]=>[8,5,7,2,4,3,6,1]
[(1,2),(3,5),(4,6),(7,8)]=>[2,1,5,6,3,4,8,7]=>[4,3,2,7,5,6,8,1]=>[8,3,4,2,7,5,6,1]
[(1,2),(3,6),(4,5),(7,8)]=>[2,1,6,5,4,3,8,7]=>[6,5,7,4,3,2,8,1]=>[8,3,4,7,6,5,2,1]
[(1,3),(2,6),(4,5),(7,8)]=>[3,6,1,5,4,2,8,7]=>[4,7,5,6,3,2,8,1]=>[8,3,6,4,7,5,2,1]
[(1,5),(2,6),(3,4),(7,8)]=>[5,6,4,3,1,2,8,7]=>[2,7,6,5,3,4,8,1]=>[8,2,5,3,6,7,4,1]
[(1,6),(2,5),(3,4),(7,8)]=>[6,5,4,3,2,1,8,7]=>[7,6,5,4,3,2,8,1]=>[8,3,4,5,6,7,2,1]
[(1,7),(2,5),(3,4),(6,8)]=>[7,5,4,3,2,8,1,6]=>[1,7,6,5,4,8,2,3]=>[1,8,2,5,6,7,4,3]
[(1,8),(2,5),(3,4),(6,7)]=>[8,5,4,3,2,7,6,1]=>[8,6,5,4,2,7,3,1]=>[3,7,4,5,6,8,1,2]
[(1,8),(2,6),(3,4),(5,7)]=>[8,6,4,3,7,2,5,1]=>[8,1,6,5,7,2,4,3]=>[8,7,4,2,6,1,5,3]
[(1,7),(2,6),(3,4),(5,8)]=>[7,6,4,3,8,2,1,5]=>[7,1,6,5,8,3,2,4]=>[7,3,8,2,6,1,5,4]
[(1,6),(2,7),(3,4),(5,8)]=>[6,7,4,3,8,1,2,5]=>[1,2,7,6,8,3,4,5]=>[1,2,8,3,4,7,6,5]
[(1,4),(2,7),(3,5),(6,8)]=>[4,7,5,1,3,8,2,6]=>[2,1,7,5,6,8,3,4]=>[2,1,8,3,7,5,6,4]
[(1,3),(2,7),(4,5),(6,8)]=>[3,7,1,5,4,8,2,6]=>[3,1,6,7,5,8,2,4]=>[3,8,1,2,7,6,5,4]
[(1,2),(3,8),(4,5),(6,7)]=>[2,1,8,5,4,7,6,3]=>[6,5,8,4,2,7,3,1]=>[3,7,4,8,6,5,1,2]
[(1,3),(2,8),(4,5),(6,7)]=>[3,8,1,5,4,7,6,2]=>[4,8,5,6,2,7,3,1]=>[3,7,6,4,8,5,1,2]
[(1,4),(2,8),(3,5),(6,7)]=>[4,8,5,1,3,7,6,2]=>[3,8,6,4,2,7,5,1]=>[5,4,3,7,6,8,1,2]
[(1,5),(2,8),(3,4),(6,7)]=>[5,8,4,3,1,7,6,2]=>[7,8,5,2,4,6,3,1]=>[3,6,5,1,8,4,7,2]
[(1,7),(2,8),(3,4),(5,6)]=>[7,8,4,3,6,5,1,2]=>[1,8,6,3,7,5,2,4]=>[1,5,7,3,6,8,2,4]
[(1,8),(2,7),(3,4),(5,6)]=>[8,7,4,3,6,5,2,1]=>[8,7,5,2,6,4,3,1]=>[3,4,6,5,7,2,8,1]
[(1,8),(2,7),(3,5),(4,6)]=>[8,7,5,6,3,4,2,1]=>[8,7,1,6,2,5,4,3]=>[7,6,4,5,2,1,8,3]
[(1,7),(2,8),(3,5),(4,6)]=>[7,8,5,6,3,4,1,2]=>[1,8,2,7,3,6,4,5]=>[1,8,7,6,4,3,2,5]
[(1,6),(2,8),(3,5),(4,7)]=>[6,8,5,7,3,1,4,2]=>[7,8,1,6,2,4,5,3]=>[8,6,5,2,4,1,7,3]
[(1,5),(2,8),(3,6),(4,7)]=>[5,8,6,7,1,3,4,2]=>[2,8,1,7,4,3,6,5]=>[8,2,4,7,6,3,1,5]
[(1,4),(2,8),(3,6),(5,7)]=>[4,8,6,1,7,3,5,2]=>[6,8,1,4,7,2,5,3]=>[8,7,5,1,2,6,4,3]
[(1,3),(2,8),(4,6),(5,7)]=>[3,8,1,6,7,4,5,2]=>[3,8,4,1,7,2,6,5]=>[4,8,3,7,6,1,2,5]
[(1,2),(3,8),(4,6),(5,7)]=>[2,1,8,6,7,4,5,3]=>[5,4,8,1,7,2,6,3]=>[8,7,6,5,4,2,1,3]
[(1,3),(2,7),(4,6),(5,8)]=>[3,7,1,6,8,4,2,5]=>[3,7,4,1,8,5,2,6]=>[4,7,3,5,8,1,2,6]
[(1,4),(2,7),(3,6),(5,8)]=>[4,7,6,1,8,3,2,5]=>[6,7,1,4,8,3,2,5]=>[7,3,8,1,2,6,4,5]
[(1,5),(2,7),(3,6),(4,8)]=>[5,7,6,8,1,3,2,4]=>[2,7,1,8,4,5,3,6]=>[7,2,5,8,4,3,1,6]
[(1,6),(2,7),(3,5),(4,8)]=>[6,7,5,8,3,1,2,4]=>[1,7,2,8,5,3,4,6]=>[1,7,5,3,8,4,2,6]
[(1,7),(2,6),(3,5),(4,8)]=>[7,6,5,8,3,2,1,4]=>[7,6,1,8,4,3,2,5]=>[6,3,4,7,1,8,2,5]
[(1,8),(2,6),(3,5),(4,7)]=>[8,6,5,7,3,2,4,1]=>[8,6,1,7,4,2,5,3]=>[6,4,8,5,1,7,3,2]
[(1,8),(2,5),(3,6),(4,7)]=>[8,5,6,7,2,3,4,1]=>[8,1,2,7,3,4,6,5]=>[8,1,7,3,6,4,2,5]
[(1,7),(2,5),(3,6),(4,8)]=>[7,5,6,8,2,3,1,4]=>[7,1,2,8,3,5,4,6]=>[7,1,8,5,3,4,2,6]
[(1,6),(2,5),(3,7),(4,8)]=>[6,5,7,8,2,1,3,4]=>[6,1,2,8,4,3,5,7]=>[6,1,4,8,3,2,5,7]
[(1,5),(2,6),(3,7),(4,8)]=>[5,6,7,8,1,2,3,4]=>[1,2,3,8,4,5,6,7]=>[1,2,3,8,4,5,6,7]
[(1,4),(2,6),(3,7),(5,8)]=>[4,6,7,1,8,2,3,5]=>[5,1,2,4,8,3,6,7]=>[5,1,8,2,4,3,6,7]
[(1,3),(2,6),(4,7),(5,8)]=>[3,6,1,7,8,2,4,5]=>[4,1,3,2,8,5,6,7]=>[4,3,1,2,8,5,6,7]
[(1,2),(3,6),(4,7),(5,8)]=>[2,1,6,7,8,3,4,5]=>[3,2,1,4,8,5,6,7]=>[2,3,1,4,8,5,6,7]
[(1,2),(3,5),(4,7),(6,8)]=>[2,1,5,7,3,8,4,6]=>[4,3,6,1,5,8,2,7]=>[6,8,4,3,1,5,2,7]
[(1,3),(2,5),(4,7),(6,8)]=>[3,5,1,7,2,8,4,6]=>[5,6,2,1,4,8,3,7]=>[2,6,8,1,5,4,3,7]
[(1,4),(2,5),(3,7),(6,8)]=>[4,5,7,1,2,8,3,6]=>[2,6,1,4,5,8,3,7]=>[6,2,8,1,4,5,3,7]
[(1,6),(2,4),(3,7),(5,8)]=>[6,4,7,2,8,1,3,5]=>[1,6,2,5,8,3,4,7]=>[1,6,8,3,2,5,4,7]
[(1,8),(2,3),(4,6),(5,7)]=>[8,3,2,6,7,4,5,1]=>[8,4,2,1,7,3,6,5]=>[2,4,8,7,6,1,3,5]
[(1,7),(2,3),(4,6),(5,8)]=>[7,3,2,6,8,4,1,5]=>[7,4,2,1,8,5,3,6]=>[2,4,7,5,8,1,3,6]
[(1,6),(2,3),(4,7),(5,8)]=>[6,3,2,7,8,1,4,5]=>[1,5,4,2,8,3,6,7]=>[1,4,5,8,3,2,6,7]
[(1,5),(2,3),(4,7),(6,8)]=>[5,3,2,7,1,8,4,6]=>[6,4,3,1,5,8,2,7]=>[3,4,6,8,1,5,2,7]
[(1,4),(2,3),(5,7),(6,8)]=>[4,3,2,1,7,8,5,6]=>[5,4,3,2,1,8,6,7]=>[2,3,4,5,1,8,6,7]
[(1,3),(2,4),(5,7),(6,8)]=>[3,4,1,2,7,8,5,6]=>[2,5,3,4,1,8,6,7]=>[4,2,5,3,1,8,6,7]
[(1,2),(3,4),(5,7),(6,8)]=>[2,1,4,3,7,8,5,6]=>[4,3,5,2,1,8,6,7]=>[2,5,4,3,1,8,6,7]
[(1,2),(3,4),(5,8),(6,7)]=>[2,1,4,3,8,7,6,5]=>[6,5,7,4,8,3,2,1]=>[2,3,8,7,6,5,4,1]
[(1,3),(2,4),(5,8),(6,7)]=>[3,4,1,2,8,7,6,5]=>[4,7,5,6,8,3,2,1]=>[2,3,8,4,7,5,6,1]
[(1,4),(2,3),(5,8),(6,7)]=>[4,3,2,1,8,7,6,5]=>[7,6,5,4,8,3,2,1]=>[2,3,8,5,6,7,4,1]
[(1,5),(2,3),(4,8),(6,7)]=>[5,3,2,8,1,7,6,4]=>[3,7,6,8,4,5,2,1]=>[2,5,3,8,4,7,6,1]
[(1,7),(2,3),(4,8),(5,6)]=>[7,3,2,8,6,5,1,4]=>[1,7,6,8,5,4,2,3]=>[1,4,2,5,8,7,6,3]
[(1,8),(2,3),(4,7),(5,6)]=>[8,3,2,7,6,5,4,1]=>[8,6,4,7,5,3,2,1]=>[2,3,5,7,6,8,1,4]
[(1,6),(2,4),(3,8),(5,7)]=>[6,4,8,2,7,1,5,3]=>[2,4,8,6,7,3,5,1]=>[5,2,7,4,3,8,6,1]
[(1,5),(2,4),(3,8),(6,7)]=>[5,4,8,2,1,7,6,3]=>[7,3,8,5,4,6,2,1]=>[2,6,7,5,8,4,3,1]
[(1,4),(2,5),(3,8),(6,7)]=>[4,5,8,1,2,7,6,3]=>[3,4,8,5,6,7,2,1]=>[2,7,3,4,8,5,6,1]
[(1,3),(2,5),(4,8),(6,7)]=>[3,5,1,8,2,7,6,4]=>[6,3,5,8,4,7,2,1]=>[2,7,6,8,3,5,4,1]
[(1,2),(3,5),(4,8),(6,7)]=>[2,1,5,8,3,7,6,4]=>[5,4,3,8,6,7,2,1]=>[2,7,4,5,3,8,6,1]
[(1,4),(2,6),(3,8),(5,7)]=>[4,6,8,1,7,2,5,3]=>[3,2,8,5,7,4,6,1]=>[6,3,2,7,8,4,5,1]
[(1,5),(2,6),(3,8),(4,7)]=>[5,6,8,7,1,2,4,3]=>[2,3,8,7,4,5,6,1]=>[6,2,3,7,4,5,8,1]
[(1,6),(2,5),(3,8),(4,7)]=>[6,5,8,7,2,1,4,3]=>[7,2,8,6,4,3,5,1]=>[5,7,4,6,3,8,2,1]
[(1,8),(2,6),(3,7),(4,5)]=>[8,6,7,5,4,2,3,1]=>[8,1,7,6,5,2,4,3]=>[8,5,4,2,6,7,1,3]
[(1,7),(2,6),(3,8),(4,5)]=>[7,6,8,5,4,2,1,3]=>[7,1,8,6,5,3,2,4]=>[7,3,5,2,6,8,1,4]
[(1,6),(2,7),(3,8),(4,5)]=>[6,7,8,5,4,1,2,3]=>[1,2,8,7,6,3,4,5]=>[1,2,6,3,4,7,8,5]
[(1,5),(2,7),(3,8),(4,6)]=>[5,7,8,6,1,4,2,3]=>[2,1,8,7,4,6,3,5]=>[2,1,6,7,3,4,8,5]
[(1,4),(2,7),(3,8),(5,6)]=>[4,7,8,1,6,5,2,3]=>[3,1,8,5,7,6,2,4]=>[3,6,1,2,7,8,4,5]
[(1,3),(2,7),(4,8),(5,6)]=>[3,7,1,8,6,5,2,4]=>[6,1,5,8,7,4,2,3]=>[6,4,2,7,1,5,8,3]
[(1,2),(3,7),(4,8),(5,6)]=>[2,1,7,8,6,5,3,4]=>[5,4,1,8,7,6,2,3]=>[4,5,1,6,2,7,8,3]
[(1,2),(3,8),(4,7),(5,6)]=>[2,1,8,7,6,5,4,3]=>[7,6,8,5,4,3,2,1]=>[2,3,4,5,8,7,6,1]
[(1,3),(2,8),(4,7),(5,6)]=>[3,8,1,7,6,5,4,2]=>[5,8,6,7,4,3,2,1]=>[2,3,4,7,5,8,6,1]
[(1,4),(2,8),(3,7),(5,6)]=>[4,8,7,1,6,5,3,2]=>[4,8,7,5,6,3,2,1]=>[2,3,6,4,7,5,8,1]
[(1,5),(2,8),(3,7),(4,6)]=>[5,8,7,6,1,4,3,2]=>[3,8,7,6,4,5,2,1]=>[2,5,3,6,4,7,8,1]
[(1,6),(2,8),(3,7),(4,5)]=>[6,8,7,5,4,1,3,2]=>[2,8,7,6,5,3,4,1]=>[4,2,5,3,6,7,8,1]
[(1,8),(2,7),(3,6),(4,5)]=>[8,7,6,5,4,3,2,1]=>[8,7,6,5,4,3,2,1]=>[2,3,4,5,6,7,8,1]
[(1,2),(3,4),(5,6),(7,8),(9,10)]=>[2,1,4,3,6,5,8,7,10,9]=>[6,5,7,4,8,3,9,2,10,1]=>[10,9,8,7,6,5,4,3,2,1]
[(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]=>[2,1,4,3,6,5,8,7,10,9,12,11]=>[7,6,8,5,9,4,10,3,11,2,12,1]=>[12,11,10,9,8,7,6,5,4,3,2,1]
Map
to permutation
Description
Returns the fixed point free involution whose transpositions are the pairs in the perfect matching.
Map
major-index to inversion-number bijection
Description
Return the permutation whose Lehmer code equals the major code of the preimage.
This map sends the major index to the number of inversions.
This map sends the major index to the number of inversions.
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 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 $\chi(\pi)$.
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