Identifier
Mp00058:
Perfect matchings
—to permutation⟶
Permutations
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Images
=>
Cc0012;cc-rep-0Cc0014;cc-rep-3
[(1,2)]=>[2,1]=>[2,1]=>([],2)
[(1,2),(3,4)]=>[2,1,4,3]=>[2,1,4,3]=>([(0,2),(0,3),(1,2),(1,3)],4)
[(1,3),(2,4)]=>[3,4,1,2]=>[4,1,3,2]=>([(1,2),(1,3)],4)
[(1,4),(2,3)]=>[4,3,2,1]=>[2,3,4,1]=>([(1,2),(2,3)],4)
[(1,2),(3,4),(5,6)]=>[2,1,4,3,6,5]=>[2,1,4,3,6,5]=>([(0,4),(0,5),(1,4),(1,5),(4,2),(4,3),(5,2),(5,3)],6)
[(1,3),(2,4),(5,6)]=>[3,4,1,2,6,5]=>[4,1,3,2,6,5]=>([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
[(1,4),(2,3),(5,6)]=>[4,3,2,1,6,5]=>[2,3,4,1,6,5]=>([(0,4),(0,5),(1,2),(2,3),(3,4),(3,5)],6)
[(1,5),(2,3),(4,6)]=>[5,3,2,6,1,4]=>[6,3,5,1,2,4]=>([(1,4),(2,3),(2,5),(4,5)],6)
[(1,6),(2,3),(4,5)]=>[6,3,2,5,4,1]=>[4,3,5,6,1,2]=>([(0,5),(1,5),(2,3),(5,4)],6)
[(1,6),(2,4),(3,5)]=>[6,4,5,2,3,1]=>[3,5,2,6,4,1]=>([(1,4),(1,5),(2,3),(2,4),(3,5)],6)
[(1,5),(2,4),(3,6)]=>[5,4,6,2,1,3]=>[2,6,1,5,4,3]=>([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5)],6)
[(1,4),(2,5),(3,6)]=>[4,5,6,1,2,3]=>[6,1,2,4,5,3]=>([(1,5),(4,3),(5,2),(5,4)],6)
[(1,3),(2,5),(4,6)]=>[3,5,1,6,2,4]=>[5,6,3,2,1,4]=>([(0,5),(1,5),(2,5),(3,4)],6)
[(1,2),(3,5),(4,6)]=>[2,1,5,6,3,4]=>[2,1,6,3,5,4]=>([(0,4),(0,5),(1,4),(1,5),(5,2),(5,3)],6)
[(1,2),(3,6),(4,5)]=>[2,1,6,5,4,3]=>[2,1,4,5,6,3]=>([(0,4),(0,5),(1,4),(1,5),(2,3),(5,2)],6)
[(1,3),(2,6),(4,5)]=>[3,6,1,5,4,2]=>[6,4,3,5,1,2]=>([(1,5),(2,5),(3,4)],6)
[(1,4),(2,6),(3,5)]=>[4,6,5,1,3,2]=>[5,3,1,4,6,2]=>([(0,5),(1,4),(2,3),(2,4),(4,5)],6)
[(1,5),(2,6),(3,4)]=>[5,6,4,3,1,2]=>[3,1,4,6,5,2]=>([(0,5),(1,2),(1,5),(5,3),(5,4)],6)
[(1,6),(2,5),(3,4)]=>[6,5,4,3,2,1]=>[2,3,4,5,6,1]=>([(1,5),(3,4),(4,2),(5,3)],6)
[(1,2),(3,4),(5,6),(7,8)]=>[2,1,4,3,6,5,8,7]=>[2,1,4,3,6,5,8,7]=>([(0,6),(0,7),(1,6),(1,7),(4,2),(4,3),(5,2),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
[(1,4),(2,3),(5,8),(6,7)]=>[4,3,2,1,8,7,6,5]=>[2,3,4,1,6,7,8,5]=>([(0,6),(0,7),(1,4),(3,6),(3,7),(4,3),(5,2),(7,5)],8)
Map
to permutation
Description
Returns the fixed point free involution whose transpositions are the pairs in the perfect matching.
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 $\chi(\pi)$.
Map
permutation poset
Description
Sends a permutation to its permutation poset.
For a permutation $\pi$ of length $n$, this poset has vertices
$$\{ (i,\pi(i))\ :\ 1 \leq i \leq n \}$$
and the cover relation is given by $(w, x) \leq (y, z)$ if $w \leq y$ and $x \leq z$.
For example, the permutation $[3,1,5,4,2]$ is mapped to the poset with cover relations
$$\{ (2, 1) \prec (5, 2),\ (2, 1) \prec (4, 4),\ (2, 1) \prec (3, 5),\ (1, 3) \prec (4, 4),\ (1, 3) \prec (3, 5) \}.$$
For a permutation $\pi$ of length $n$, this poset has vertices
$$\{ (i,\pi(i))\ :\ 1 \leq i \leq n \}$$
and the cover relation is given by $(w, x) \leq (y, z)$ if $w \leq y$ and $x \leq z$.
For example, the permutation $[3,1,5,4,2]$ is mapped to the poset with cover relations
$$\{ (2, 1) \prec (5, 2),\ (2, 1) \prec (4, 4),\ (2, 1) \prec (3, 5),\ (1, 3) \prec (4, 4),\ (1, 3) \prec (3, 5) \}.$$
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