Identifier
Mp00058: to permutationPermutations
Mp00235: Permutations descent views to invisible inversion bottomsPermutations
Mp00248: Permutations DEX compositionInteger compositions
Images
=>
Cc0012;cc-rep-0
[(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]=>[4] [(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]=>[3,1,2] [(1,4),(2,3),(5,6)]=>[4,3,2,1,6,5]=>[2,3,4,1,6,5]=>[4,2] [(1,5),(2,3),(4,6)]=>[5,3,2,6,1,4]=>[6,3,5,1,2,4]=>[1,5] [(1,6),(2,3),(4,5)]=>[6,3,2,5,4,1]=>[4,3,5,6,1,2]=>[1,5] [(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]=>[3,2,1] [(1,4),(2,5),(3,6)]=>[4,5,6,1,2,3]=>[6,1,2,4,5,3]=>[5,1] [(1,3),(2,5),(4,6)]=>[3,5,1,6,2,4]=>[5,6,3,2,1,4]=>[3,1,2] [(1,2),(3,5),(4,6)]=>[2,1,5,6,3,4]=>[2,1,6,3,5,4]=>[2,3,1] [(1,2),(3,6),(4,5)]=>[2,1,6,5,4,3]=>[2,1,4,5,6,3]=>[2,4] [(1,3),(2,6),(4,5)]=>[3,6,1,5,4,2]=>[6,4,3,5,1,2]=>[1,2,3] [(1,4),(2,6),(3,5)]=>[4,6,5,1,3,2]=>[5,3,1,4,6,2]=>[1,3,2] [(1,5),(2,6),(3,4)]=>[5,6,4,3,1,2]=>[3,1,4,6,5,2]=>[2,3,1] [(1,6),(2,5),(3,4)]=>[6,5,4,3,2,1]=>[2,3,4,5,6,1]=>[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]=>[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]=>[3,1,2,2] [(1,4),(2,3),(5,6),(7,8)]=>[4,3,2,1,6,5,8,7]=>[2,3,4,1,6,5,8,7]=>[4,2,2] [(1,5),(2,3),(4,6),(7,8)]=>[5,3,2,6,1,4,8,7]=>[6,3,5,1,2,4,8,7]=>[1,5,2] [(1,6),(2,4),(3,5),(7,8)]=>[6,4,5,2,3,1,8,7]=>[3,5,2,6,4,1,8,7]=>[3,2,1,2] [(1,4),(2,5),(3,6),(7,8)]=>[4,5,6,1,2,3,8,7]=>[6,1,2,4,5,3,8,7]=>[5,1,2] [(1,3),(2,5),(4,6),(7,8)]=>[3,5,1,6,2,4,8,7]=>[5,6,3,2,1,4,8,7]=>[3,1,2,2] [(1,2),(3,5),(4,6),(7,8)]=>[2,1,5,6,3,4,8,7]=>[2,1,6,3,5,4,8,7]=>[2,3,1,2] [(1,7),(2,6),(3,4),(5,8)]=>[7,6,4,3,8,2,1,5]=>[2,8,4,6,1,7,3,5]=>[2,3,3] [(1,2),(3,8),(4,5),(6,7)]=>[2,1,8,5,4,7,6,3]=>[2,1,6,5,7,8,3,4]=>[2,1,5] [(1,7),(2,8),(3,4),(5,6)]=>[7,8,4,3,6,5,1,2]=>[5,1,4,6,8,2,7,3]=>[2,5,1] [(1,7),(2,8),(3,5),(4,6)]=>[7,8,5,6,3,4,1,2]=>[4,1,6,3,8,5,7,2]=>[2,2,3,1] [(1,2),(3,8),(4,6),(5,7)]=>[2,1,8,6,7,4,5,3]=>[2,1,5,7,4,8,6,3]=>[2,3,2,1] [(1,6),(2,7),(3,5),(4,8)]=>[6,7,5,8,3,1,2,4]=>[3,1,8,2,7,6,5,4]=>[2,2,2,1,1] [(1,7),(2,6),(3,5),(4,8)]=>[7,6,5,8,3,2,1,4]=>[2,3,8,1,6,7,5,4]=>[4,3,1] [(1,6),(2,5),(3,7),(4,8)]=>[6,5,7,8,2,1,3,4]=>[2,8,1,3,6,5,7,4]=>[4,3,1] [(1,5),(2,6),(3,7),(4,8)]=>[5,6,7,8,1,2,3,4]=>[8,1,2,3,5,6,7,4]=>[7,1] [(1,4),(2,6),(3,7),(5,8)]=>[4,6,7,1,8,2,3,5]=>[7,8,2,4,3,6,1,5]=>[4,2,2] [(1,3),(2,6),(4,7),(5,8)]=>[3,6,1,7,8,2,4,5]=>[6,8,3,2,4,1,7,5]=>[3,2,2,1] [(1,2),(3,6),(4,7),(5,8)]=>[2,1,6,7,8,3,4,5]=>[2,1,8,3,4,6,7,5]=>[2,5,1] [(1,2),(3,5),(4,7),(6,8)]=>[2,1,5,7,3,8,4,6]=>[2,1,7,8,5,4,3,6]=>[2,3,1,2] [(1,3),(2,5),(4,7),(6,8)]=>[3,5,1,7,2,8,4,6]=>[5,7,3,8,1,4,2,6]=>[3,3,2] [(1,4),(2,5),(3,7),(6,8)]=>[4,5,7,1,2,8,3,6]=>[7,1,8,4,5,3,2,6]=>[2,3,1,2] [(1,6),(2,4),(3,7),(5,8)]=>[6,4,7,2,8,1,3,5]=>[8,7,1,6,3,4,2,5]=>[1,2,3,2] [(1,8),(2,4),(3,6),(5,7)]=>[8,4,6,2,7,3,5,1]=>[5,6,7,8,3,4,2,1]=>[6,1,1] [(1,5),(2,3),(4,7),(6,8)]=>[5,3,2,7,1,8,4,6]=>[7,3,5,8,2,4,1,6]=>[1,5,2] [(1,4),(2,3),(5,7),(6,8)]=>[4,3,2,1,7,8,5,6]=>[2,3,4,1,8,5,7,6]=>[4,3,1] [(1,3),(2,4),(5,7),(6,8)]=>[3,4,1,2,7,8,5,6]=>[4,1,3,2,8,5,7,6]=>[3,1,3,1] [(1,2),(3,4),(5,7),(6,8)]=>[2,1,4,3,7,8,5,6]=>[2,1,4,3,8,5,7,6]=>[2,2,3,1] [(1,2),(3,4),(5,8),(6,7)]=>[2,1,4,3,8,7,6,5]=>[2,1,4,3,6,7,8,5]=>[2,2,4] [(1,3),(2,4),(5,8),(6,7)]=>[3,4,1,2,8,7,6,5]=>[4,1,3,2,6,7,8,5]=>[3,1,4] [(1,4),(2,3),(5,8),(6,7)]=>[4,3,2,1,8,7,6,5]=>[2,3,4,1,6,7,8,5]=>[4,4] [(1,5),(2,3),(4,8),(6,7)]=>[5,3,2,8,1,7,6,4]=>[8,3,5,6,2,7,1,4]=>[1,4,3] [(1,7),(2,3),(4,8),(5,6)]=>[7,3,2,8,6,5,1,4]=>[5,3,6,1,8,7,2,4]=>[1,3,1,3] [(1,8),(2,4),(3,7),(5,6)]=>[8,4,7,2,6,5,3,1]=>[3,5,6,7,8,4,2,1]=>[6,1,1] [(1,6),(2,4),(3,8),(5,7)]=>[6,4,8,2,7,1,5,3]=>[7,8,5,6,1,4,2,3]=>[2,4,2] [(1,3),(2,5),(4,8),(6,7)]=>[3,5,1,8,2,7,6,4]=>[5,8,3,6,1,7,2,4]=>[3,2,3] [(1,2),(3,5),(4,8),(6,7)]=>[2,1,5,8,3,7,6,4]=>[2,1,8,6,5,7,3,4]=>[2,1,2,3] [(1,4),(2,6),(3,8),(5,7)]=>[4,6,8,1,7,2,5,3]=>[8,7,5,4,2,6,1,3]=>[1,1,2,2,2] [(1,5),(2,6),(3,8),(4,7)]=>[5,6,8,7,1,2,4,3]=>[7,1,4,2,5,6,8,3]=>[2,4,2] [(1,6),(2,5),(3,8),(4,7)]=>[6,5,8,7,2,1,4,3]=>[2,7,4,1,6,5,8,3]=>[2,2,2,2] [(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] [(1,6),(2,4),(3,5),(7,8),(9,10)]=>[6,4,5,2,3,1,8,7,10,9]=>[3,5,2,6,4,1,8,7,10,9]=>[3,2,1,2,2] [(1,2),(3,8),(4,6),(5,7),(9,10)]=>[2,1,8,6,7,4,5,3,10,9]=>[2,1,5,7,4,8,6,3,10,9]=>[2,3,2,1,2] [(1,10),(2,8),(3,9),(4,6),(5,7)]=>[10,8,9,6,7,4,5,2,3,1]=>[3,5,2,7,4,9,6,10,8,1]=>[3,2,2,2,1] [(1,2),(3,4),(5,10),(6,8),(7,9)]=>[2,1,4,3,10,8,9,6,7,5]=>[2,1,4,3,7,9,6,10,8,5]=>[2,2,3,2,1] [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]=>[2,1,4,3,6,5,8,7,10,9,12,11]=>[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.
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
DEX composition
Description
The DEX composition of a permutation.
Let $\pi$ be a permutation in $\mathfrak S_n$. Let $\bar\pi$ be the word in the ordered set $\bar 1 < \dots < \bar n < 1 \dots < n$ obtained from $\pi$ by replacing every excedance $\pi(i) > i$ by $\overline{\pi(i)}$. Then the DEX set of $\pi$ is the set of indices $1 \leq i < n$ such that $\bar\pi(i) > \bar\pi(i+1)$. Finally, the DEX composition $c_1, \dots, c_k$ of $n$ corresponds to the DEX subset $\{c_1, c_1 + c_2, \dots, c_1 + \dots + c_{k-1}\}$.
The (quasi)symmetric function
$$\sum_{\pi\in\mathfrak S_{\lambda, j}} F_{DEX(\pi)},$$
where the sum is over the set of permutations of cycle type $\lambda$ with $j$ excedances, is the Eulerian quasisymmetric function.