Identifier
Mp00058: to permutationPermutations
Mp00237: Permutations descent views to invisible inversion bottomsPermutations
Mp00204: Permutations LLPS
Images
=>
Cc0012;cc-rep-0Cc0002;cc-rep-3
[(1,2)]=>[2,1]=>[2,1]=>[2] [(1,2),(3,4)]=>[2,1,4,3]=>[2,1,4,3]=>[2,2] [(1,3),(2,4)]=>[3,4,1,2]=>[4,1,3,2]=>[3,1] [(1,4),(2,3)]=>[4,3,2,1]=>[2,3,4,1]=>[2,1,1] [(1,2),(3,4),(5,6)]=>[2,1,4,3,6,5]=>[2,1,4,3,6,5]=>[2,2,2] [(1,3),(2,4),(5,6)]=>[3,4,1,2,6,5]=>[4,1,3,2,6,5]=>[3,2,1] [(1,4),(2,3),(5,6)]=>[4,3,2,1,6,5]=>[2,3,4,1,6,5]=>[2,2,1,1] [(1,5),(2,3),(4,6)]=>[5,3,2,6,1,4]=>[6,3,5,1,2,4]=>[3,1,1,1] [(1,6),(2,3),(4,5)]=>[6,3,2,5,4,1]=>[4,3,5,6,1,2]=>[3,1,1,1] [(1,6),(2,4),(3,5)]=>[6,4,5,2,3,1]=>[3,5,2,6,4,1]=>[3,2,1] [(1,5),(2,4),(3,6)]=>[5,4,6,2,1,3]=>[2,6,1,5,4,3]=>[4,1,1] [(1,4),(2,5),(3,6)]=>[4,5,6,1,2,3]=>[6,1,2,4,5,3]=>[3,1,1,1] [(1,3),(2,5),(4,6)]=>[3,5,1,6,2,4]=>[5,6,3,2,1,4]=>[4,1,1] [(1,2),(3,5),(4,6)]=>[2,1,5,6,3,4]=>[2,1,6,3,5,4]=>[3,2,1] [(1,2),(3,6),(4,5)]=>[2,1,6,5,4,3]=>[2,1,4,5,6,3]=>[2,2,1,1] [(1,3),(2,6),(4,5)]=>[3,6,1,5,4,2]=>[6,4,3,5,1,2]=>[4,1,1] [(1,4),(2,6),(3,5)]=>[4,6,5,1,3,2]=>[5,3,1,4,6,2]=>[3,2,1] [(1,5),(2,6),(3,4)]=>[5,6,4,3,1,2]=>[3,1,4,6,5,2]=>[3,2,1] [(1,6),(2,5),(3,4)]=>[6,5,4,3,2,1]=>[2,3,4,5,6,1]=>[2,1,1,1,1] [(1,2),(3,4),(5,6),(7,8)]=>[2,1,4,3,6,5,8,7]=>[2,1,4,3,6,5,8,7]=>[2,2,2,2] [(1,3),(2,4),(5,6),(7,8)]=>[3,4,1,2,6,5,8,7]=>[4,1,3,2,6,5,8,7]=>[3,2,2,1] [(1,4),(2,3),(5,6),(7,8)]=>[4,3,2,1,6,5,8,7]=>[2,3,4,1,6,5,8,7]=>[2,2,2,1,1] [(1,5),(2,3),(4,6),(7,8)]=>[5,3,2,6,1,4,8,7]=>[6,3,5,1,2,4,8,7]=>[3,2,1,1,1] [(1,6),(2,3),(4,5),(7,8)]=>[6,3,2,5,4,1,8,7]=>[4,3,5,6,1,2,8,7]=>[3,2,1,1,1] [(1,7),(2,3),(4,5),(6,8)]=>[7,3,2,5,4,8,1,6]=>[8,3,7,5,2,1,4,6]=>[5,1,1,1] [(1,8),(2,3),(4,5),(6,7)]=>[8,3,2,5,4,7,6,1]=>[6,3,7,5,2,8,1,4]=>[4,2,1,1] [(1,8),(2,4),(3,5),(6,7)]=>[8,4,5,2,3,7,6,1]=>[6,5,2,7,4,8,1,3]=>[4,2,1,1] [(1,6),(2,4),(3,5),(7,8)]=>[6,4,5,2,3,1,8,7]=>[3,5,2,6,4,1,8,7]=>[3,2,2,1] [(1,4),(2,5),(3,6),(7,8)]=>[4,5,6,1,2,3,8,7]=>[6,1,2,4,5,3,8,7]=>[3,2,1,1,1] [(1,3),(2,5),(4,6),(7,8)]=>[3,5,1,6,2,4,8,7]=>[5,6,3,2,1,4,8,7]=>[4,2,1,1] [(1,2),(3,5),(4,6),(7,8)]=>[2,1,5,6,3,4,8,7]=>[2,1,6,3,5,4,8,7]=>[3,2,2,1] [(1,2),(3,6),(4,5),(7,8)]=>[2,1,6,5,4,3,8,7]=>[2,1,4,5,6,3,8,7]=>[2,2,2,1,1] [(1,3),(2,6),(4,5),(7,8)]=>[3,6,1,5,4,2,8,7]=>[6,4,3,5,1,2,8,7]=>[4,2,1,1] [(1,4),(2,6),(3,5),(7,8)]=>[4,6,5,1,3,2,8,7]=>[5,3,1,4,6,2,8,7]=>[3,2,2,1] [(1,6),(2,5),(3,4),(7,8)]=>[6,5,4,3,2,1,8,7]=>[2,3,4,5,6,1,8,7]=>[2,2,1,1,1,1] [(1,8),(2,5),(3,4),(6,7)]=>[8,5,4,3,2,7,6,1]=>[6,3,4,5,7,8,1,2]=>[3,1,1,1,1,1] [(1,8),(2,6),(3,4),(5,7)]=>[8,6,4,3,7,2,5,1]=>[5,7,4,6,2,8,3,1]=>[4,2,1,1] [(1,7),(2,6),(3,4),(5,8)]=>[7,6,4,3,8,2,1,5]=>[2,8,4,6,1,7,3,5]=>[3,2,1,1,1] [(1,5),(2,7),(3,4),(6,8)]=>[5,7,4,3,1,8,2,6]=>[3,4,7,8,5,2,1,6]=>[4,1,1,1,1] [(1,2),(3,7),(4,5),(6,8)]=>[2,1,7,5,4,8,3,6]=>[2,1,8,5,7,3,4,6]=>[3,2,1,1,1] [(1,2),(3,8),(4,5),(6,7)]=>[2,1,8,5,4,7,6,3]=>[2,1,6,5,7,8,3,4]=>[3,2,1,1,1] [(1,3),(2,8),(4,5),(6,7)]=>[3,8,1,5,4,7,6,2]=>[8,6,3,5,1,7,4,2]=>[5,2,1] [(1,4),(2,8),(3,5),(6,7)]=>[4,8,5,1,3,7,6,2]=>[5,8,1,4,6,7,2,3]=>[3,1,1,1,1,1] [(1,7),(2,8),(3,4),(5,6)]=>[7,8,4,3,6,5,1,2]=>[5,1,4,6,8,2,7,3]=>[3,2,1,1,1] [(1,8),(2,7),(3,4),(5,6)]=>[8,7,4,3,6,5,2,1]=>[2,5,4,6,7,1,8,3]=>[3,2,1,1,1] [(1,8),(2,7),(3,5),(4,6)]=>[8,7,5,6,3,4,2,1]=>[2,4,6,3,7,5,8,1]=>[3,2,1,1,1] [(1,7),(2,8),(3,5),(4,6)]=>[7,8,5,6,3,4,1,2]=>[4,1,6,3,8,5,7,2]=>[3,2,2,1] [(1,6),(2,8),(3,5),(4,7)]=>[6,8,5,7,3,1,4,2]=>[3,7,4,2,8,6,5,1]=>[4,3,1] [(1,3),(2,8),(4,6),(5,7)]=>[3,8,1,6,7,4,5,2]=>[8,5,3,7,4,1,6,2]=>[4,2,2] [(1,2),(3,8),(4,6),(5,7)]=>[2,1,8,6,7,4,5,3]=>[2,1,5,7,4,8,6,3]=>[3,2,2,1] [(1,4),(2,7),(3,6),(5,8)]=>[4,7,6,1,8,3,2,5]=>[6,3,7,4,2,8,5,1]=>[4,2,2] [(1,7),(2,6),(3,5),(4,8)]=>[7,6,5,8,3,2,1,4]=>[2,3,8,1,6,7,5,4]=>[4,1,1,1,1] [(1,8),(2,6),(3,5),(4,7)]=>[8,6,5,7,3,2,4,1]=>[4,3,7,2,6,8,5,1]=>[4,2,1,1] [(1,6),(2,5),(3,7),(4,8)]=>[6,5,7,8,2,1,3,4]=>[2,8,1,3,6,5,7,4]=>[4,1,1,1,1] [(1,5),(2,6),(3,7),(4,8)]=>[5,6,7,8,1,2,3,4]=>[8,1,2,3,5,6,7,4]=>[3,1,1,1,1,1] [(1,4),(2,6),(3,7),(5,8)]=>[4,6,7,1,8,2,3,5]=>[7,8,2,4,3,6,1,5]=>[4,1,1,1,1] [(1,3),(2,6),(4,7),(5,8)]=>[3,6,1,7,8,2,4,5]=>[6,8,3,2,4,1,7,5]=>[4,2,1,1] [(1,2),(3,6),(4,7),(5,8)]=>[2,1,6,7,8,3,4,5]=>[2,1,8,3,4,6,7,5]=>[3,2,1,1,1] [(1,2),(3,5),(4,7),(6,8)]=>[2,1,5,7,3,8,4,6]=>[2,1,7,8,5,4,3,6]=>[4,2,1,1] [(1,3),(2,5),(4,7),(6,8)]=>[3,5,1,7,2,8,4,6]=>[5,7,3,8,1,4,2,6]=>[3,2,1,1,1] [(1,4),(2,5),(3,7),(6,8)]=>[4,5,7,1,2,8,3,6]=>[7,1,8,4,5,3,2,6]=>[4,2,1,1] [(1,6),(2,4),(3,7),(5,8)]=>[6,4,7,2,8,1,3,5]=>[8,7,1,6,2,4,5,3]=>[5,1,1,1] [(1,8),(2,4),(3,6),(5,7)]=>[8,4,6,2,7,3,5,1]=>[5,6,7,8,3,4,2,1]=>[4,1,1,1,1] [(1,5),(2,3),(4,7),(6,8)]=>[5,3,2,7,1,8,4,6]=>[7,3,5,8,2,4,1,6]=>[4,1,1,1,1] [(1,4),(2,3),(5,7),(6,8)]=>[4,3,2,1,7,8,5,6]=>[2,3,4,1,8,5,7,6]=>[3,2,1,1,1] [(1,3),(2,4),(5,7),(6,8)]=>[3,4,1,2,7,8,5,6]=>[4,1,3,2,8,5,7,6]=>[3,3,1,1] [(1,2),(3,4),(5,7),(6,8)]=>[2,1,4,3,7,8,5,6]=>[2,1,4,3,8,5,7,6]=>[3,2,2,1] [(1,2),(3,4),(5,8),(6,7)]=>[2,1,4,3,8,7,6,5]=>[2,1,4,3,6,7,8,5]=>[2,2,2,1,1] [(1,3),(2,4),(5,8),(6,7)]=>[3,4,1,2,8,7,6,5]=>[4,1,3,2,6,7,8,5]=>[3,2,1,1,1] [(1,4),(2,3),(5,8),(6,7)]=>[4,3,2,1,8,7,6,5]=>[2,3,4,1,6,7,8,5]=>[2,2,1,1,1,1] [(1,5),(2,3),(4,8),(6,7)]=>[5,3,2,8,1,7,6,4]=>[8,3,5,6,2,7,1,4]=>[4,1,1,1,1] [(1,7),(2,3),(4,8),(5,6)]=>[7,3,2,8,6,5,1,4]=>[5,3,6,1,8,7,2,4]=>[3,3,1,1] [(1,8),(2,3),(4,7),(5,6)]=>[8,3,2,7,6,5,4,1]=>[4,3,5,6,7,8,1,2]=>[3,1,1,1,1,1] [(1,8),(2,4),(3,7),(5,6)]=>[8,4,7,2,6,5,3,1]=>[3,5,6,7,8,2,1,4]=>[3,1,1,1,1,1] [(1,7),(2,4),(3,8),(5,6)]=>[7,4,8,2,6,5,1,3]=>[5,6,1,8,7,3,2,4]=>[4,2,1,1] [(1,6),(2,4),(3,8),(5,7)]=>[6,4,8,2,7,1,5,3]=>[7,8,5,6,1,4,2,3]=>[4,1,1,1,1] [(1,4),(2,5),(3,8),(6,7)]=>[4,5,8,1,2,7,6,3]=>[8,1,6,4,5,7,2,3]=>[4,1,1,1,1] [(1,3),(2,5),(4,8),(6,7)]=>[3,5,1,8,2,7,6,4]=>[5,8,3,6,1,7,2,4]=>[3,2,1,1,1] [(1,2),(3,5),(4,8),(6,7)]=>[2,1,5,8,3,7,6,4]=>[2,1,8,6,5,7,3,4]=>[4,2,1,1] [(1,2),(3,6),(4,8),(5,7)]=>[2,1,6,8,7,3,5,4]=>[2,1,7,5,3,6,8,4]=>[3,2,2,1] [(1,4),(2,6),(3,8),(5,7)]=>[4,6,8,1,7,2,5,3]=>[8,7,5,4,2,6,1,3]=>[6,1,1] [(1,5),(2,6),(3,8),(4,7)]=>[5,6,8,7,1,2,4,3]=>[7,1,4,2,5,6,8,3]=>[3,2,1,1,1] [(1,6),(2,5),(3,8),(4,7)]=>[6,5,8,7,2,1,4,3]=>[2,7,4,1,6,5,8,3]=>[4,2,1,1] [(1,8),(2,5),(3,7),(4,6)]=>[8,5,7,6,2,4,3,1]=>[3,4,6,1,7,8,2,5]=>[2,2,1,1,1,1] [(1,2),(3,8),(4,7),(5,6)]=>[2,1,8,7,6,5,4,3]=>[2,1,4,5,6,7,8,3]=>[2,2,1,1,1,1] [(1,3),(2,8),(4,7),(5,6)]=>[3,8,1,7,6,5,4,2]=>[8,4,3,5,6,7,1,2]=>[4,1,1,1,1] [(1,4),(2,8),(3,7),(5,6)]=>[4,8,7,1,6,5,3,2]=>[7,3,5,4,6,1,8,2]=>[4,2,1,1] [(1,5),(2,8),(3,7),(4,6)]=>[5,8,7,6,1,4,3,2]=>[6,3,4,1,5,7,8,2]=>[3,2,1,1,1] [(1,7),(2,8),(3,6),(4,5)]=>[7,8,6,5,4,3,1,2]=>[3,1,4,5,6,8,7,2]=>[3,2,1,1,1] [(1,8),(2,7),(3,6),(4,5)]=>[8,7,6,5,4,3,2,1]=>[2,3,4,5,6,7,8,1]=>[2,1,1,1,1,1,1] [(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]=>[2,2,2,2,2] [(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]=>[2,2,2,2,2,2]
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.
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 $\chi(\pi)$.
Map
LLPS
Description
The Lewis-Lyu-Pylyavskyy-Sen shape of a permutation.
An ascent in a sequence $u = (u_1, u_2, \ldots)$ is an index $i$ such that $u_i < u_{i+1}$. Let $\mathrm{asc}(u)$ denote the number of ascents of $u$, and let
$$\mathrm{asc}^{*}(u) := \begin{cases} 0 &\textrm{if u is empty}, \\ 1 + \mathrm{asc}(u) &\textrm{otherwise}.\end{cases}$$
Given a permutation $w$ in the symmetric group $\mathfrak{S}_n$, define
$A'_k := \max_{u_1, \ldots, u_k} (\mathrm{asc}^{*}(u_1) + \cdots + \mathrm{asc}^{*}(u_k))$
where the maximum is taken over disjoint subsequences ${u_i}$ of $w$.
Then $A'_1, A'_2-A'_1, A'_3-A'_2,\dots$ is a partition of $n$. Its conjugate is the Lewis-Lyu-Pylyavskyy-Sen shape of a permutation.