Identifier
Mp00058: to permutationPermutations
Mp00237: Permutations descent views to invisible inversion bottomsPermutations
Mp00238: Permutations Clarke-Steingrimsson-ZengPermutations
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]=>[4,1,3,2]=>[3,4,1,2] [(1,4),(2,3)]=>[4,3,2,1]=>[2,3,4,1]=>[4,2,3,1] [(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]=>[4,1,3,2,6,5]=>[3,4,1,2,6,5] [(1,4),(2,3),(5,6)]=>[4,3,2,1,6,5]=>[2,3,4,1,6,5]=>[4,2,3,1,6,5] [(1,5),(2,3),(4,6)]=>[5,3,2,6,1,4]=>[6,3,5,1,2,4]=>[5,1,6,3,2,4] [(1,6),(2,3),(4,5)]=>[6,3,2,5,4,1]=>[4,3,5,6,1,2]=>[6,1,4,3,5,2] [(1,6),(2,4),(3,5)]=>[6,4,5,2,3,1]=>[3,5,2,6,4,1]=>[4,6,3,5,2,1] [(1,5),(2,4),(3,6)]=>[5,4,6,2,1,3]=>[2,6,1,5,4,3]=>[4,2,5,6,1,3] [(1,4),(2,5),(3,6)]=>[4,5,6,1,2,3]=>[6,1,2,4,5,3]=>[5,1,6,2,4,3] [(1,3),(2,5),(4,6)]=>[3,5,1,6,2,4]=>[5,6,3,2,1,4]=>[2,3,6,1,5,4] [(1,2),(3,5),(4,6)]=>[2,1,5,6,3,4]=>[2,1,6,3,5,4]=>[2,1,5,6,3,4] [(1,2),(3,6),(4,5)]=>[2,1,6,5,4,3]=>[2,1,4,5,6,3]=>[2,1,6,4,5,3] [(1,3),(2,6),(4,5)]=>[3,6,1,5,4,2]=>[6,4,3,5,1,2]=>[5,1,4,6,3,2] [(1,4),(2,6),(3,5)]=>[4,6,5,1,3,2]=>[5,3,1,4,6,2]=>[6,3,5,1,4,2] [(1,5),(2,6),(3,4)]=>[5,6,4,3,1,2]=>[3,1,4,6,5,2]=>[5,3,1,4,6,2] [(1,6),(2,5),(3,4)]=>[6,5,4,3,2,1]=>[2,3,4,5,6,1]=>[6,2,3,4,5,1] [(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]=>[4,1,3,2,6,5,8,7]=>[3,4,1,2,6,5,8,7] [(1,4),(2,3),(5,6),(7,8)]=>[4,3,2,1,6,5,8,7]=>[2,3,4,1,6,5,8,7]=>[4,2,3,1,6,5,8,7] [(1,5),(2,3),(4,6),(7,8)]=>[5,3,2,6,1,4,8,7]=>[6,3,5,1,2,4,8,7]=>[5,1,6,3,2,4,8,7] [(1,6),(2,3),(4,5),(7,8)]=>[6,3,2,5,4,1,8,7]=>[4,3,5,6,1,2,8,7]=>[6,1,4,3,5,2,8,7] [(1,7),(2,3),(4,5),(6,8)]=>[7,3,2,5,4,8,1,6]=>[8,3,7,5,2,1,4,6]=>[2,5,7,3,8,1,4,6] [(1,8),(2,3),(4,5),(6,7)]=>[8,3,2,5,4,7,6,1]=>[6,3,7,5,2,8,1,4]=>[8,5,7,3,6,2,1,4] [(1,8),(2,4),(3,5),(6,7)]=>[8,4,5,2,3,7,6,1]=>[6,5,2,7,4,8,1,3]=>[8,7,2,5,6,1,4,3] [(1,6),(2,4),(3,5),(7,8)]=>[6,4,5,2,3,1,8,7]=>[3,5,2,6,4,1,8,7]=>[4,6,3,5,2,1,8,7] [(1,4),(2,5),(3,6),(7,8)]=>[4,5,6,1,2,3,8,7]=>[6,1,2,4,5,3,8,7]=>[5,1,6,2,4,3,8,7] [(1,3),(2,5),(4,6),(7,8)]=>[3,5,1,6,2,4,8,7]=>[5,6,3,2,1,4,8,7]=>[2,3,6,1,5,4,8,7] [(1,2),(3,5),(4,6),(7,8)]=>[2,1,5,6,3,4,8,7]=>[2,1,6,3,5,4,8,7]=>[2,1,5,6,3,4,8,7] [(1,2),(3,6),(4,5),(7,8)]=>[2,1,6,5,4,3,8,7]=>[2,1,4,5,6,3,8,7]=>[2,1,6,4,5,3,8,7] [(1,3),(2,6),(4,5),(7,8)]=>[3,6,1,5,4,2,8,7]=>[6,4,3,5,1,2,8,7]=>[5,1,4,6,3,2,8,7] [(1,4),(2,6),(3,5),(7,8)]=>[4,6,5,1,3,2,8,7]=>[5,3,1,4,6,2,8,7]=>[6,3,5,1,4,2,8,7] [(1,6),(2,5),(3,4),(7,8)]=>[6,5,4,3,2,1,8,7]=>[2,3,4,5,6,1,8,7]=>[6,2,3,4,5,1,8,7] [(1,8),(2,5),(3,4),(6,7)]=>[8,5,4,3,2,7,6,1]=>[6,3,4,5,7,8,1,2]=>[8,1,6,3,4,5,7,2] [(1,8),(2,6),(3,4),(5,7)]=>[8,6,4,3,7,2,5,1]=>[5,7,4,6,2,8,3,1]=>[3,8,6,7,5,4,2,1] [(1,7),(2,6),(3,4),(5,8)]=>[7,6,4,3,8,2,1,5]=>[2,8,4,6,1,7,3,5]=>[7,2,6,8,4,1,3,5] [(1,5),(2,7),(3,4),(6,8)]=>[5,7,4,3,1,8,2,6]=>[3,4,7,8,5,2,1,6]=>[2,5,3,4,8,1,7,6] [(1,2),(3,7),(4,5),(6,8)]=>[2,1,7,5,4,8,3,6]=>[2,1,8,5,7,3,4,6]=>[2,1,7,3,8,5,4,6] [(1,2),(3,8),(4,5),(6,7)]=>[2,1,8,5,4,7,6,3]=>[2,1,6,5,7,8,3,4]=>[2,1,8,3,6,5,7,4] [(1,3),(2,8),(4,5),(6,7)]=>[3,8,1,5,4,7,6,2]=>[8,6,3,5,1,7,4,2]=>[4,7,5,6,3,8,1,2] [(1,4),(2,8),(3,5),(6,7)]=>[4,8,5,1,3,7,6,2]=>[5,8,1,4,6,7,2,3]=>[7,8,1,2,5,4,6,3] [(1,5),(2,8),(3,4),(6,7)]=>[5,8,4,3,1,7,6,2]=>[3,4,8,6,5,7,1,2]=>[7,1,3,4,6,8,5,2] [(1,7),(2,8),(3,4),(5,6)]=>[7,8,4,3,6,5,1,2]=>[5,1,4,6,8,2,7,3]=>[7,8,5,1,4,6,2,3] [(1,8),(2,7),(3,4),(5,6)]=>[8,7,4,3,6,5,2,1]=>[2,5,4,6,7,1,8,3]=>[8,2,7,5,4,6,1,3] [(1,8),(2,7),(3,5),(4,6)]=>[8,7,5,6,3,4,2,1]=>[2,4,6,3,7,5,8,1]=>[8,2,7,4,6,3,5,1] [(1,7),(2,8),(3,5),(4,6)]=>[7,8,5,6,3,4,1,2]=>[4,1,6,3,8,5,7,2]=>[7,8,4,1,6,3,5,2] [(1,6),(2,8),(3,5),(4,7)]=>[6,8,5,7,3,1,4,2]=>[3,7,4,2,8,6,5,1]=>[5,4,3,6,8,7,2,1] [(1,3),(2,8),(4,6),(5,7)]=>[3,8,1,6,7,4,5,2]=>[8,5,3,7,4,1,6,2]=>[6,4,7,5,8,3,1,2] [(1,2),(3,8),(4,6),(5,7)]=>[2,1,8,6,7,4,5,3]=>[2,1,5,7,4,8,6,3]=>[2,1,6,8,5,7,4,3] [(1,4),(2,7),(3,6),(5,8)]=>[4,7,6,1,8,3,2,5]=>[6,3,7,4,2,8,5,1]=>[5,4,8,7,6,3,2,1] [(1,6),(2,7),(3,5),(4,8)]=>[6,7,5,8,3,1,2,4]=>[3,1,8,2,7,6,5,4]=>[5,3,1,6,7,8,2,4] [(1,7),(2,6),(3,5),(4,8)]=>[7,6,5,8,3,2,1,4]=>[2,3,8,1,6,7,5,4]=>[5,2,3,7,8,1,6,4] [(1,8),(2,6),(3,5),(4,7)]=>[8,6,5,7,3,2,4,1]=>[4,3,7,2,6,8,5,1]=>[5,8,4,3,7,2,6,1] [(1,6),(2,5),(3,7),(4,8)]=>[6,5,7,8,2,1,3,4]=>[2,8,1,3,6,5,7,4]=>[7,2,1,6,8,3,5,4] [(1,5),(2,6),(3,7),(4,8)]=>[5,6,7,8,1,2,3,4]=>[8,1,2,3,5,6,7,4]=>[7,1,2,8,3,5,6,4] [(1,4),(2,6),(3,7),(5,8)]=>[4,6,7,1,8,2,3,5]=>[7,8,2,4,3,6,1,5]=>[6,4,8,2,3,1,7,5] [(1,3),(2,6),(4,7),(5,8)]=>[3,6,1,7,8,2,4,5]=>[6,8,3,2,4,1,7,5]=>[4,3,7,2,8,6,1,5] [(1,2),(3,6),(4,7),(5,8)]=>[2,1,6,7,8,3,4,5]=>[2,1,8,3,4,6,7,5]=>[2,1,7,3,8,4,6,5] [(1,2),(3,5),(4,7),(6,8)]=>[2,1,5,7,3,8,4,6]=>[2,1,7,8,5,4,3,6]=>[2,1,4,5,8,3,7,6] [(1,3),(2,5),(4,7),(6,8)]=>[3,5,1,7,2,8,4,6]=>[5,7,3,8,1,4,2,6]=>[4,8,7,3,5,1,2,6] [(1,4),(2,5),(3,7),(6,8)]=>[4,5,7,1,2,8,3,6]=>[7,1,8,4,5,3,2,6]=>[3,5,8,7,1,4,2,6] [(1,6),(2,4),(3,7),(5,8)]=>[6,4,7,2,8,1,3,5]=>[8,7,1,6,2,4,5,3]=>[5,6,7,1,2,4,8,3] [(1,8),(2,4),(3,6),(5,7)]=>[8,4,6,2,7,3,5,1]=>[5,6,7,8,3,4,2,1]=>[2,4,8,3,5,6,7,1] [(1,5),(2,3),(4,7),(6,8)]=>[5,3,2,7,1,8,4,6]=>[7,3,5,8,2,4,1,6]=>[4,8,7,3,2,5,1,6] [(1,4),(2,3),(5,7),(6,8)]=>[4,3,2,1,7,8,5,6]=>[2,3,4,1,8,5,7,6]=>[4,2,3,1,7,8,5,6] [(1,3),(2,4),(5,7),(6,8)]=>[3,4,1,2,7,8,5,6]=>[4,1,3,2,8,5,7,6]=>[3,4,1,2,7,8,5,6] [(1,2),(3,4),(5,7),(6,8)]=>[2,1,4,3,7,8,5,6]=>[2,1,4,3,8,5,7,6]=>[2,1,4,3,7,8,5,6] [(1,2),(3,4),(5,8),(6,7)]=>[2,1,4,3,8,7,6,5]=>[2,1,4,3,6,7,8,5]=>[2,1,4,3,8,6,7,5] [(1,3),(2,4),(5,8),(6,7)]=>[3,4,1,2,8,7,6,5]=>[4,1,3,2,6,7,8,5]=>[3,4,1,2,8,6,7,5] [(1,4),(2,3),(5,8),(6,7)]=>[4,3,2,1,8,7,6,5]=>[2,3,4,1,6,7,8,5]=>[4,2,3,1,8,6,7,5] [(1,5),(2,3),(4,8),(6,7)]=>[5,3,2,8,1,7,6,4]=>[8,3,5,6,2,7,1,4]=>[7,6,8,3,2,5,1,4] [(1,7),(2,3),(4,8),(5,6)]=>[7,3,2,8,6,5,1,4]=>[5,3,6,1,8,7,2,4]=>[7,6,5,3,1,2,8,4] [(1,8),(2,3),(4,7),(5,6)]=>[8,3,2,7,6,5,4,1]=>[4,3,5,6,7,8,1,2]=>[8,1,4,3,5,6,7,2] [(1,8),(2,4),(3,7),(5,6)]=>[8,4,7,2,6,5,3,1]=>[3,5,6,7,8,2,1,4]=>[2,8,3,1,5,6,7,4] [(1,7),(2,4),(3,8),(5,6)]=>[7,4,8,2,6,5,1,3]=>[5,6,1,8,7,3,2,4]=>[3,7,6,1,5,2,8,4] [(1,6),(2,4),(3,8),(5,7)]=>[6,4,8,2,7,1,5,3]=>[7,8,5,6,1,4,2,3]=>[4,6,1,2,8,5,7,3] [(1,4),(2,5),(3,8),(6,7)]=>[4,5,8,1,2,7,6,3]=>[8,1,6,4,5,7,2,3]=>[7,6,1,8,2,4,5,3] [(1,3),(2,5),(4,8),(6,7)]=>[3,5,1,8,2,7,6,4]=>[5,8,3,6,1,7,2,4]=>[7,6,8,3,5,1,2,4] [(1,2),(3,5),(4,8),(6,7)]=>[2,1,5,8,3,7,6,4]=>[2,1,8,6,5,7,3,4]=>[2,1,7,3,6,8,5,4] [(1,2),(3,6),(4,8),(5,7)]=>[2,1,6,8,7,3,5,4]=>[2,1,7,5,3,6,8,4]=>[2,1,8,5,7,3,6,4] [(1,4),(2,6),(3,8),(5,7)]=>[4,6,8,1,7,2,5,3]=>[8,7,5,4,2,6,1,3]=>[6,4,2,5,7,1,8,3] [(1,5),(2,6),(3,8),(4,7)]=>[5,6,8,7,1,2,4,3]=>[7,1,4,2,5,6,8,3]=>[8,4,7,1,2,5,6,3] [(1,6),(2,5),(3,8),(4,7)]=>[6,5,8,7,2,1,4,3]=>[2,7,4,1,6,5,8,3]=>[8,2,4,6,7,1,5,3] [(1,8),(2,5),(3,7),(4,6)]=>[8,5,7,6,2,4,3,1]=>[3,4,6,1,7,8,2,5]=>[8,6,3,4,1,2,7,5] [(1,6),(2,7),(3,8),(4,5)]=>[6,7,8,5,4,1,2,3]=>[4,1,2,5,8,6,7,3]=>[7,1,4,2,5,8,6,3] [(1,2),(3,8),(4,7),(5,6)]=>[2,1,8,7,6,5,4,3]=>[2,1,4,5,6,7,8,3]=>[2,1,8,4,5,6,7,3] [(1,3),(2,8),(4,7),(5,6)]=>[3,8,1,7,6,5,4,2]=>[8,4,3,5,6,7,1,2]=>[7,1,4,8,3,5,6,2] [(1,4),(2,8),(3,7),(5,6)]=>[4,8,7,1,6,5,3,2]=>[7,3,5,4,6,1,8,2]=>[8,6,5,7,3,4,1,2] [(1,5),(2,8),(3,7),(4,6)]=>[5,8,7,6,1,4,3,2]=>[6,3,4,1,5,7,8,2]=>[8,4,6,3,1,5,7,2] [(1,7),(2,8),(3,6),(4,5)]=>[7,8,6,5,4,3,1,2]=>[3,1,4,5,6,8,7,2]=>[7,3,1,4,5,6,8,2] [(1,8),(2,7),(3,6),(4,5)]=>[8,7,6,5,4,3,2,1]=>[2,3,4,5,6,7,8,1]=>[8,2,3,4,5,6,7,1] [(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
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
Clarke-Steingrimsson-Zeng
Description
The Clarke-Steingrimsson-Zeng map sending descents to excedances.
This is the map $\Phi$ in [1, sec.3]. In particular, it satisfies
$$(des, Dbot, Ddif, Res)\pi = (exc, Ebot, Edif, Ine)\Phi(\pi),$$
where
• $des$ is the number of descents, St000021The number of descents of a permutation.,
• $exc$ is the number of (strict) excedances, St000155The number of exceedances (also excedences) of a permutation.,
• $Dbot$ is the sum of the descent bottoms, St000154The sum of the descent bottoms of a permutation.,
• $Ebot$ is the sum of the excedance bottoms,
• $Ddif$ is the sum of the descent differences, St000030The sum of the descent differences of a permutations.,
• $Edif$ is the sum of the excedance differences (or depth), St000029The depth of a permutation.,
• $Res$ is the sum of the (right) embracing numbers,
• $Ine$ is the sum of the side numbers.