Identifier
Mp00058:
Perfect matchings
—to permutation⟶
Permutations
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
Mp00237: Permutations —descent views to invisible inversion bottoms⟶ Permutations
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
Mp00237: 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]=>[2,1,4,3]=>[2,1,4,3]
[(1,3),(2,4)]=>[3,4,1,2]=>[2,4,1,3]=>[4,2,1,3]
[(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]=>[2,1,4,3,6,5]=>[2,1,4,3,6,5]
[(1,3),(2,4),(5,6)]=>[3,4,1,2,6,5]=>[2,4,1,3,6,5]=>[4,2,1,3,6,5]
[(1,4),(2,3),(5,6)]=>[4,3,2,1,6,5]=>[4,3,2,1,6,5]=>[2,3,4,1,6,5]
[(1,5),(2,3),(4,6)]=>[5,3,2,6,1,4]=>[4,6,1,5,3,2]=>[6,3,5,4,1,2]
[(1,6),(2,3),(4,5)]=>[6,3,2,5,4,1]=>[5,4,1,6,3,2]=>[4,3,5,6,2,1]
[(1,6),(2,4),(3,5)]=>[6,4,5,2,3,1]=>[3,1,5,2,6,4]=>[3,5,1,6,2,4]
[(1,5),(2,4),(3,6)]=>[5,4,6,2,1,3]=>[3,6,2,1,5,4]=>[2,6,3,5,1,4]
[(1,4),(2,5),(3,6)]=>[4,5,6,1,2,3]=>[2,3,6,1,4,5]=>[6,2,3,1,4,5]
[(1,3),(2,5),(4,6)]=>[3,5,1,6,2,4]=>[4,6,2,5,1,3]=>[5,6,1,4,2,3]
[(1,2),(3,5),(4,6)]=>[2,1,5,6,3,4]=>[2,1,4,6,3,5]=>[2,1,6,4,3,5]
[(1,2),(3,6),(4,5)]=>[2,1,6,5,4,3]=>[2,1,6,5,4,3]=>[2,1,4,5,6,3]
[(1,3),(2,6),(4,5)]=>[3,6,1,5,4,2]=>[5,4,2,6,1,3]=>[6,4,1,5,2,3]
[(1,4),(2,6),(3,5)]=>[4,6,5,1,3,2]=>[3,2,6,5,1,4]=>[5,3,2,1,6,4]
[(1,5),(2,6),(3,4)]=>[5,6,4,3,1,2]=>[2,6,4,3,1,5]=>[3,2,4,6,1,5]
[(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]=>[2,1,4,3,6,5,8,7]=>[2,1,4,3,6,5,8,7]
[(1,3),(2,4),(5,6),(7,8)]=>[3,4,1,2,6,5,8,7]=>[2,4,1,3,6,5,8,7]=>[4,2,1,3,6,5,8,7]
[(1,4),(2,3),(5,6),(7,8)]=>[4,3,2,1,6,5,8,7]=>[4,3,2,1,6,5,8,7]=>[2,3,4,1,6,5,8,7]
[(1,6),(2,3),(4,5),(7,8)]=>[6,3,2,5,4,1,8,7]=>[5,4,1,6,3,2,8,7]=>[4,3,5,6,2,1,8,7]
[(1,7),(2,3),(4,5),(6,8)]=>[7,3,2,5,4,8,1,6]=>[6,8,1,5,4,7,3,2]=>[8,3,7,5,1,6,4,2]
[(1,6),(2,4),(3,5),(7,8)]=>[6,4,5,2,3,1,8,7]=>[3,1,5,2,6,4,8,7]=>[3,5,1,6,2,4,8,7]
[(1,4),(2,5),(3,6),(7,8)]=>[4,5,6,1,2,3,8,7]=>[2,3,6,1,4,5,8,7]=>[6,2,3,1,4,5,8,7]
[(1,3),(2,5),(4,6),(7,8)]=>[3,5,1,6,2,4,8,7]=>[4,6,2,5,1,3,8,7]=>[5,6,1,4,2,3,8,7]
[(1,2),(3,5),(4,6),(7,8)]=>[2,1,5,6,3,4,8,7]=>[2,1,4,6,3,5,8,7]=>[2,1,6,4,3,5,8,7]
[(1,2),(3,6),(4,5),(7,8)]=>[2,1,6,5,4,3,8,7]=>[2,1,6,5,4,3,8,7]=>[2,1,4,5,6,3,8,7]
[(1,5),(2,6),(3,4),(7,8)]=>[5,6,4,3,1,2,8,7]=>[2,6,4,3,1,5,8,7]=>[3,2,4,6,1,5,8,7]
[(1,6),(2,5),(3,4),(7,8)]=>[6,5,4,3,2,1,8,7]=>[6,5,4,3,2,1,8,7]=>[2,3,4,5,6,1,8,7]
[(1,7),(2,5),(3,4),(6,8)]=>[7,5,4,3,2,8,1,6]=>[6,8,1,7,5,4,3,2]=>[8,3,4,5,7,6,1,2]
[(1,8),(2,5),(3,4),(6,7)]=>[8,5,4,3,2,7,6,1]=>[7,6,1,8,5,4,3,2]=>[6,3,4,5,7,8,2,1]
[(1,8),(2,6),(3,4),(5,7)]=>[8,6,4,3,7,2,5,1]=>[5,1,7,2,8,6,4,3]=>[5,7,4,6,1,8,2,3]
[(1,7),(2,6),(3,4),(5,8)]=>[7,6,4,3,8,2,1,5]=>[5,8,2,1,7,6,4,3]=>[2,8,4,6,5,7,1,3]
[(1,5),(2,7),(3,4),(6,8)]=>[5,7,4,3,1,8,2,6]=>[6,8,2,7,4,3,1,5]=>[3,4,7,8,1,6,2,5]
[(1,3),(2,7),(4,5),(6,8)]=>[3,7,1,5,4,8,2,6]=>[6,8,2,5,4,7,1,3]=>[7,8,1,5,3,6,2,4]
[(1,2),(3,8),(4,5),(6,7)]=>[2,1,8,5,4,7,6,3]=>[2,1,7,6,3,8,5,4]=>[2,1,6,5,7,8,4,3]
[(1,4),(2,8),(3,5),(6,7)]=>[4,8,5,1,3,7,6,2]=>[7,6,2,3,8,5,1,4]=>[5,8,2,1,6,7,4,3]
[(1,5),(2,8),(3,4),(6,7)]=>[5,8,4,3,1,7,6,2]=>[7,6,2,8,4,3,1,5]=>[3,4,8,6,1,7,2,5]
[(1,6),(2,8),(3,4),(5,7)]=>[6,8,4,3,7,1,5,2]=>[5,2,7,1,8,4,3,6]=>[7,5,4,8,2,3,1,6]
[(1,7),(2,8),(3,4),(5,6)]=>[7,8,4,3,6,5,1,2]=>[2,6,5,1,8,4,3,7]=>[5,2,4,6,8,3,1,7]
[(1,8),(2,7),(3,4),(5,6)]=>[8,7,4,3,6,5,2,1]=>[6,5,2,1,8,7,4,3]=>[2,5,4,6,7,3,8,1]
[(1,8),(2,7),(3,5),(4,6)]=>[8,7,5,6,3,4,2,1]=>[4,2,1,6,3,8,7,5]=>[2,4,6,1,7,3,8,5]
[(1,7),(2,8),(3,5),(4,6)]=>[7,8,5,6,3,4,1,2]=>[2,4,1,6,3,8,5,7]=>[4,2,6,1,8,3,5,7]
[(1,4),(2,8),(3,6),(5,7)]=>[4,8,6,1,7,3,5,2]=>[5,2,7,3,8,6,1,4]=>[6,5,8,1,2,7,4,3]
[(1,2),(3,8),(4,6),(5,7)]=>[2,1,8,6,7,4,5,3]=>[2,1,5,3,7,4,8,6]=>[2,1,5,7,3,8,4,6]
[(1,4),(2,7),(3,6),(5,8)]=>[4,7,6,1,8,3,2,5]=>[5,8,3,2,7,6,1,4]=>[6,3,7,1,5,8,4,2]
[(1,6),(2,7),(3,5),(4,8)]=>[6,7,5,8,3,1,2,4]=>[2,4,8,3,1,7,5,6]=>[3,2,8,4,7,5,1,6]
[(1,7),(2,6),(3,5),(4,8)]=>[7,6,5,8,3,2,1,4]=>[4,8,3,2,1,7,6,5]=>[2,3,8,4,6,7,1,5]
[(1,7),(2,5),(3,6),(4,8)]=>[7,5,6,8,2,3,1,4]=>[3,1,4,8,2,6,7,5]=>[3,8,1,4,7,2,6,5]
[(1,5),(2,6),(3,7),(4,8)]=>[5,6,7,8,1,2,3,4]=>[2,3,4,8,1,5,6,7]=>[8,2,3,4,1,5,6,7]
[(1,4),(2,6),(3,7),(5,8)]=>[4,6,7,1,8,2,3,5]=>[3,5,8,2,7,1,4,6]=>[7,8,3,1,5,2,6,4]
[(1,3),(2,6),(4,7),(5,8)]=>[3,6,1,7,8,2,4,5]=>[4,5,8,2,6,1,3,7]=>[6,8,1,4,5,2,3,7]
[(1,2),(3,6),(4,7),(5,8)]=>[2,1,6,7,8,3,4,5]=>[2,1,4,5,8,3,6,7]=>[2,1,8,4,5,3,6,7]
[(1,2),(3,5),(4,7),(6,8)]=>[2,1,5,7,3,8,4,6]=>[2,1,6,8,4,7,3,5]=>[2,1,7,8,3,6,4,5]
[(1,3),(2,5),(4,7),(6,8)]=>[3,5,1,7,2,8,4,6]=>[6,8,4,7,2,5,1,3]=>[5,7,1,8,2,6,4,3]
[(1,4),(2,5),(3,7),(6,8)]=>[4,5,7,1,2,8,3,6]=>[2,6,8,3,7,1,4,5]=>[7,2,8,1,3,6,5,4]
[(1,5),(2,4),(3,7),(6,8)]=>[5,4,7,2,1,8,3,6]=>[6,8,3,7,2,1,5,4]=>[2,7,8,5,1,6,3,4]
[(1,7),(2,4),(3,6),(5,8)]=>[7,4,6,2,8,3,1,5]=>[5,8,3,1,6,2,7,4]=>[3,8,6,7,5,2,1,4]
[(1,8),(2,4),(3,6),(5,7)]=>[8,4,6,2,7,3,5,1]=>[5,1,7,3,6,2,8,4]=>[5,6,7,8,1,3,2,4]
[(1,7),(2,3),(4,6),(5,8)]=>[7,3,2,6,8,4,1,5]=>[5,8,4,1,6,7,3,2]=>[4,3,8,7,5,1,2,6]
[(1,6),(2,3),(4,7),(5,8)]=>[6,3,2,7,8,1,4,5]=>[4,5,8,1,6,3,2,7]=>[8,3,6,4,5,1,2,7]
[(1,5),(2,3),(4,7),(6,8)]=>[5,3,2,7,1,8,4,6]=>[6,8,4,7,1,5,3,2]=>[7,3,5,8,1,6,4,2]
[(1,4),(2,3),(5,7),(6,8)]=>[4,3,2,1,7,8,5,6]=>[4,3,2,1,6,8,5,7]=>[2,3,4,1,8,6,5,7]
[(1,3),(2,4),(5,7),(6,8)]=>[3,4,1,2,7,8,5,6]=>[2,4,1,3,6,8,5,7]=>[4,2,1,3,8,6,5,7]
[(1,2),(3,4),(5,7),(6,8)]=>[2,1,4,3,7,8,5,6]=>[2,1,4,3,6,8,5,7]=>[2,1,4,3,8,6,5,7]
[(1,2),(3,4),(5,8),(6,7)]=>[2,1,4,3,8,7,6,5]=>[2,1,4,3,8,7,6,5]=>[2,1,4,3,6,7,8,5]
[(1,3),(2,4),(5,8),(6,7)]=>[3,4,1,2,8,7,6,5]=>[2,4,1,3,8,7,6,5]=>[4,2,1,3,6,7,8,5]
[(1,4),(2,3),(5,8),(6,7)]=>[4,3,2,1,8,7,6,5]=>[4,3,2,1,8,7,6,5]=>[2,3,4,1,6,7,8,5]
[(1,6),(2,3),(4,8),(5,7)]=>[6,3,2,8,7,1,5,4]=>[5,4,8,7,1,6,3,2]=>[7,3,6,5,4,1,8,2]
[(1,7),(2,3),(4,8),(5,6)]=>[7,3,2,8,6,5,1,4]=>[4,8,6,5,1,7,3,2]=>[5,3,6,4,8,7,1,2]
[(1,8),(2,4),(3,7),(5,6)]=>[8,4,7,2,6,5,3,1]=>[6,5,3,1,7,2,8,4]=>[3,5,6,7,8,1,4,2]
[(1,7),(2,4),(3,8),(5,6)]=>[7,4,8,2,6,5,1,3]=>[3,6,5,1,8,2,7,4]=>[5,6,3,8,7,2,4,1]
[(1,5),(2,4),(3,8),(6,7)]=>[5,4,8,2,1,7,6,3]=>[7,6,3,8,2,1,5,4]=>[2,8,6,5,1,7,3,4]
[(1,3),(2,5),(4,8),(6,7)]=>[3,5,1,8,2,7,6,4]=>[7,6,4,8,2,5,1,3]=>[5,8,1,6,2,7,4,3]
[(1,2),(3,6),(4,8),(5,7)]=>[2,1,6,8,7,3,5,4]=>[2,1,5,4,8,7,3,6]=>[2,1,7,5,4,3,8,6]
[(1,4),(2,6),(3,8),(5,7)]=>[4,6,8,1,7,2,5,3]=>[5,3,7,2,8,1,4,6]=>[8,7,5,1,3,2,6,4]
[(1,8),(2,5),(3,7),(4,6)]=>[8,5,7,6,2,4,3,1]=>[4,3,1,7,6,2,8,5]=>[3,4,6,2,7,8,5,1]
[(1,8),(2,6),(3,7),(4,5)]=>[8,6,7,5,4,2,3,1]=>[3,1,7,5,4,2,8,6]=>[3,4,1,5,7,8,2,6]
[(1,7),(2,6),(3,8),(4,5)]=>[7,6,8,5,4,2,1,3]=>[3,8,5,4,2,1,7,6]=>[2,4,3,5,8,7,1,6]
[(1,6),(2,7),(3,8),(4,5)]=>[6,7,8,5,4,1,2,3]=>[2,3,8,5,4,1,6,7]=>[4,2,3,5,8,1,6,7]
[(1,5),(2,7),(3,8),(4,6)]=>[5,7,8,6,1,4,2,3]=>[3,4,2,8,6,1,5,7]=>[6,4,3,2,1,8,5,7]
[(1,4),(2,7),(3,8),(5,6)]=>[4,7,8,1,6,5,2,3]=>[3,6,5,2,8,1,4,7]=>[8,5,3,1,6,2,4,7]
[(1,2),(3,7),(4,8),(5,6)]=>[2,1,7,8,6,5,3,4]=>[2,1,4,8,6,5,3,7]=>[2,1,5,4,6,8,3,7]
[(1,2),(3,8),(4,7),(5,6)]=>[2,1,8,7,6,5,4,3]=>[2,1,8,7,6,5,4,3]=>[2,1,4,5,6,7,8,3]
[(1,4),(2,8),(3,7),(5,6)]=>[4,8,7,1,6,5,3,2]=>[6,5,3,2,8,7,1,4]=>[7,3,5,1,6,2,8,4]
[(1,5),(2,8),(3,7),(4,6)]=>[5,8,7,6,1,4,3,2]=>[4,3,2,8,7,6,1,5]=>[6,3,4,2,1,7,8,5]
[(1,6),(2,8),(3,7),(4,5)]=>[6,8,7,5,4,1,3,2]=>[3,2,8,7,5,4,1,6]=>[4,3,2,5,7,1,8,6]
[(1,7),(2,8),(3,6),(4,5)]=>[7,8,6,5,4,3,1,2]=>[2,8,6,5,4,3,1,7]=>[3,2,4,5,6,8,1,7]
[(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]=>[2,1,4,3,6,5,8,7,10,9]=>[2,1,4,3,6,5,8,7,10,9]
[(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]=>[2,1,4,3,6,5,8,7,10,9,12,11]=>[2,1,4,3,6,5,8,7,10,9,12,11]=>[2,1,4,3,6,5,8,7,10,9,12,11]
Map
to permutation
Description
Returns the fixed point free involution whose transpositions are the pairs in the perfect matching.
Map
invert Laguerre heap
Description
The permutation obtained by inverting the corresponding Laguerre heap, according to Viennot.
Let $\pi$ be a permutation. Following Viennot [1], we associate to $\pi$ a heap of pieces, by considering each decreasing run $(\pi_i, \pi_{i+1}, \dots, \pi_j)$ of $\pi$ as one piece, beginning with the left most run. Two pieces commute if and only if the minimal element of one piece is larger than the maximal element of the other piece.
This map yields the permutation corresponding to the heap obtained by reversing the reading direction of the heap.
Equivalently, this is the permutation obtained by flipping the noncrossing arc diagram of Reading [2] vertically.
By definition, this map preserves the set of decreasing runs.
Let $\pi$ be a permutation. Following Viennot [1], we associate to $\pi$ a heap of pieces, by considering each decreasing run $(\pi_i, \pi_{i+1}, \dots, \pi_j)$ of $\pi$ as one piece, beginning with the left most run. Two pieces commute if and only if the minimal element of one piece is larger than the maximal element of the other piece.
This map yields the permutation corresponding to the heap obtained by reversing the reading direction of the heap.
Equivalently, this is the permutation obtained by flipping the noncrossing arc diagram of Reading [2] vertically.
By definition, this map preserves the set of decreasing runs.
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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