Identifier
Mp00058: Perfect matchings —to permutation⟶ Permutations
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
Images
=>
Cc0012;cc-rep-0Cc0007;cc-rep-3
[(1,2)]=>[2,1]=>[2,1]=>[[1],[2]] [(1,2),(3,4)]=>[2,1,4,3]=>[2,1,4,3]=>[[1,3],[2,4]] [(1,3),(2,4)]=>[3,4,1,2]=>[4,1,3,2]=>[[1,2],[3],[4]] [(1,4),(2,3)]=>[4,3,2,1]=>[2,3,4,1]=>[[1,3,4],[2]] [(1,2),(3,4),(5,6)]=>[2,1,4,3,6,5]=>[2,1,4,3,6,5]=>[[1,3,5],[2,4,6]] [(1,3),(2,4),(5,6)]=>[3,4,1,2,6,5]=>[4,1,3,2,6,5]=>[[1,2,5],[3,6],[4]] [(1,4),(2,3),(5,6)]=>[4,3,2,1,6,5]=>[2,3,4,1,6,5]=>[[1,3,4,5],[2,6]] [(1,5),(2,3),(4,6)]=>[5,3,2,6,1,4]=>[6,3,5,1,2,4]=>[[1,2,4],[3,5],[6]] [(1,6),(2,3),(4,5)]=>[6,3,2,5,4,1]=>[4,3,5,6,1,2]=>[[1,2,6],[3,5],[4]] [(1,6),(2,4),(3,5)]=>[6,4,5,2,3,1]=>[3,5,2,6,4,1]=>[[1,4,6],[2,5],[3]] [(1,5),(2,4),(3,6)]=>[5,4,6,2,1,3]=>[2,6,1,5,4,3]=>[[1,3],[2,4],[5],[6]] [(1,4),(2,5),(3,6)]=>[4,5,6,1,2,3]=>[6,1,2,4,5,3]=>[[1,2,3,5],[4],[6]] [(1,3),(2,5),(4,6)]=>[3,5,1,6,2,4]=>[5,6,3,2,1,4]=>[[1,4],[2,6],[3],[5]] [(1,2),(3,5),(4,6)]=>[2,1,5,6,3,4]=>[2,1,6,3,5,4]=>[[1,3,4],[2,5],[6]] [(1,2),(3,6),(4,5)]=>[2,1,6,5,4,3]=>[2,1,4,5,6,3]=>[[1,3,5,6],[2,4]] [(1,3),(2,6),(4,5)]=>[3,6,1,5,4,2]=>[6,4,3,5,1,2]=>[[1,2],[3,5],[4],[6]] [(1,4),(2,6),(3,5)]=>[4,6,5,1,3,2]=>[5,3,1,4,6,2]=>[[1,2,6],[3,4],[5]] [(1,5),(2,6),(3,4)]=>[5,6,4,3,1,2]=>[3,1,4,6,5,2]=>[[1,2,5],[3,4],[6]] [(1,6),(2,5),(3,4)]=>[6,5,4,3,2,1]=>[2,3,4,5,6,1]=>[[1,3,4,5,6],[2]] [(1,2),(3,4),(5,6),(7,8)]=>[2,1,4,3,6,5,8,7]=>[2,1,4,3,6,5,8,7]=>[[1,3,5,7],[2,4,6,8]] [(1,3),(2,4),(5,6),(7,8)]=>[3,4,1,2,6,5,8,7]=>[4,1,3,2,6,5,8,7]=>[[1,2,5,7],[3,6,8],[4]] [(1,4),(2,3),(5,6),(7,8)]=>[4,3,2,1,6,5,8,7]=>[2,3,4,1,6,5,8,7]=>[[1,3,4,5,7],[2,6,8]] [(1,5),(2,3),(4,6),(7,8)]=>[5,3,2,6,1,4,8,7]=>[6,3,5,1,2,4,8,7]=>[[1,2,4,7],[3,5,8],[6]] [(1,6),(2,3),(4,5),(7,8)]=>[6,3,2,5,4,1,8,7]=>[4,3,5,6,1,2,8,7]=>[[1,2,6,7],[3,5,8],[4]] [(1,7),(2,3),(4,5),(6,8)]=>[7,3,2,5,4,8,1,6]=>[8,3,7,5,2,1,4,6]=>[[1,4,6],[2,5],[3],[7],[8]] [(1,8),(2,3),(4,5),(6,7)]=>[8,3,2,5,4,7,6,1]=>[6,3,7,5,1,8,4,2]=>[[1,2,8],[3,4],[5,7],[6]] [(1,8),(2,4),(3,5),(6,7)]=>[8,4,5,2,3,7,6,1]=>[6,5,2,7,1,8,3,4]=>[[1,3,4],[2,7,8],[5],[6]] [(1,6),(2,4),(3,5),(7,8)]=>[6,4,5,2,3,1,8,7]=>[3,5,2,6,4,1,8,7]=>[[1,4,6,7],[2,5,8],[3]] [(1,4),(2,5),(3,6),(7,8)]=>[4,5,6,1,2,3,8,7]=>[6,1,2,4,5,3,8,7]=>[[1,2,3,5,7],[4,8],[6]] [(1,3),(2,5),(4,6),(7,8)]=>[3,5,1,6,2,4,8,7]=>[5,6,3,2,1,4,8,7]=>[[1,4,7],[2,6,8],[3],[5]] [(1,2),(3,5),(4,6),(7,8)]=>[2,1,5,6,3,4,8,7]=>[2,1,6,3,5,4,8,7]=>[[1,3,4,7],[2,5,8],[6]] [(1,2),(3,6),(4,5),(7,8)]=>[2,1,6,5,4,3,8,7]=>[2,1,4,5,6,3,8,7]=>[[1,3,5,6,7],[2,4,8]] [(1,3),(2,6),(4,5),(7,8)]=>[3,6,1,5,4,2,8,7]=>[6,4,3,5,1,2,8,7]=>[[1,2,7],[3,5,8],[4],[6]] [(1,6),(2,5),(3,4),(7,8)]=>[6,5,4,3,2,1,8,7]=>[2,3,4,5,6,1,8,7]=>[[1,3,4,5,6,7],[2,8]] [(1,8),(2,5),(3,4),(6,7)]=>[8,5,4,3,2,7,6,1]=>[6,3,4,5,7,8,1,2]=>[[1,2,5,7,8],[3,4],[6]] [(1,8),(2,6),(3,4),(5,7)]=>[8,6,4,3,7,2,5,1]=>[5,7,4,6,2,8,3,1]=>[[1,3,8],[2,6],[4,7],[5]] [(1,7),(2,6),(3,4),(5,8)]=>[7,6,4,3,8,2,1,5]=>[2,8,4,6,1,7,3,5]=>[[1,3,5,7],[2,4,6],[8]] [(1,3),(2,7),(4,5),(6,8)]=>[3,7,1,5,4,8,2,6]=>[7,8,3,5,1,2,4,6]=>[[1,2,4,6],[3,5],[7,8]] [(1,2),(3,7),(4,5),(6,8)]=>[2,1,7,5,4,8,3,6]=>[2,1,8,5,7,3,4,6]=>[[1,3,4,6],[2,5,7],[8]] [(1,2),(3,8),(4,5),(6,7)]=>[2,1,8,5,4,7,6,3]=>[2,1,6,5,7,8,3,4]=>[[1,3,4,8],[2,5,7],[6]] [(1,3),(2,8),(4,5),(6,7)]=>[3,8,1,5,4,7,6,2]=>[8,6,3,5,1,7,4,2]=>[[1,2,7],[3,4],[5],[6],[8]] [(1,5),(2,8),(3,4),(6,7)]=>[5,8,4,3,1,7,6,2]=>[3,4,8,6,5,7,1,2]=>[[1,2,5,7],[3,4],[6],[8]] [(1,7),(2,8),(3,4),(5,6)]=>[7,8,4,3,6,5,1,2]=>[5,1,4,6,8,2,7,3]=>[[1,2,3,7],[4,6],[5,8]] [(1,8),(2,7),(3,4),(5,6)]=>[8,7,4,3,6,5,2,1]=>[2,5,4,6,7,1,8,3]=>[[1,3,6,7,8],[2,4],[5]] [(1,7),(2,8),(3,5),(4,6)]=>[7,8,5,6,3,4,1,2]=>[4,1,6,3,8,5,7,2]=>[[1,2,5,7],[3,6,8],[4]] [(1,6),(2,8),(3,5),(4,7)]=>[6,8,5,7,3,1,4,2]=>[3,7,4,2,8,6,5,1]=>[[1,4,5],[2,6],[3,8],[7]] [(1,2),(3,8),(4,6),(5,7)]=>[2,1,8,6,7,4,5,3]=>[2,1,5,7,4,8,6,3]=>[[1,3,6,8],[2,4,7],[5]] [(1,6),(2,7),(3,5),(4,8)]=>[6,7,5,8,3,1,2,4]=>[3,1,8,2,7,6,5,4]=>[[1,2,4],[3,5],[6],[7],[8]] [(1,8),(2,6),(3,5),(4,7)]=>[8,6,5,7,3,2,4,1]=>[4,3,7,2,6,8,5,1]=>[[1,5,8],[2,6],[3,7],[4]] [(1,6),(2,5),(3,7),(4,8)]=>[6,5,7,8,2,1,3,4]=>[2,8,1,3,6,5,7,4]=>[[1,3,4,7],[2,5],[6],[8]] [(1,5),(2,6),(3,7),(4,8)]=>[5,6,7,8,1,2,3,4]=>[8,1,2,3,5,6,7,4]=>[[1,2,3,4,6,7],[5],[8]] [(1,4),(2,6),(3,7),(5,8)]=>[4,6,7,1,8,2,3,5]=>[7,8,2,4,3,6,1,5]=>[[1,3,5],[2,6],[4,8],[7]] [(1,3),(2,6),(4,7),(5,8)]=>[3,6,1,7,8,2,4,5]=>[6,8,3,2,4,1,7,5]=>[[1,4,5],[2,7],[3,8],[6]] [(1,2),(3,6),(4,7),(5,8)]=>[2,1,6,7,8,3,4,5]=>[2,1,8,3,4,6,7,5]=>[[1,3,4,5,7],[2,6],[8]] [(1,2),(3,5),(4,7),(6,8)]=>[2,1,5,7,3,8,4,6]=>[2,1,7,8,5,4,3,6]=>[[1,3,6],[2,4,8],[5],[7]] [(1,3),(2,5),(4,7),(6,8)]=>[3,5,1,7,2,8,4,6]=>[5,7,3,8,1,4,2,6]=>[[1,2,6],[3,4,8],[5,7]] [(1,4),(2,5),(3,7),(6,8)]=>[4,5,7,1,2,8,3,6]=>[7,1,8,4,5,3,2,6]=>[[1,2,5,6],[3,8],[4],[7]] [(1,6),(2,4),(3,7),(5,8)]=>[6,4,7,2,8,1,3,5]=>[8,7,1,6,3,4,2,5]=>[[1,2,4,5],[3],[6],[7],[8]] [(1,8),(2,4),(3,6),(5,7)]=>[8,4,6,2,7,3,5,1]=>[5,6,7,8,3,4,2,1]=>[[1,4,7,8],[2,6],[3],[5]] [(1,8),(2,3),(4,6),(5,7)]=>[8,3,2,6,7,4,5,1]=>[5,3,8,7,4,2,6,1]=>[[1,4,6],[2,7],[3],[5],[8]] [(1,5),(2,3),(4,7),(6,8)]=>[5,3,2,7,1,8,4,6]=>[7,3,5,8,2,4,1,6]=>[[1,4,6],[2,5,8],[3],[7]] [(1,4),(2,3),(5,7),(6,8)]=>[4,3,2,1,7,8,5,6]=>[2,3,4,1,8,5,7,6]=>[[1,3,4,5,6],[2,7],[8]] [(1,3),(2,4),(5,7),(6,8)]=>[3,4,1,2,7,8,5,6]=>[4,1,3,2,8,5,7,6]=>[[1,2,5,6],[3,7],[4,8]] [(1,2),(3,4),(5,7),(6,8)]=>[2,1,4,3,7,8,5,6]=>[2,1,4,3,8,5,7,6]=>[[1,3,5,6],[2,4,7],[8]] [(1,2),(3,4),(5,8),(6,7)]=>[2,1,4,3,8,7,6,5]=>[2,1,4,3,6,7,8,5]=>[[1,3,5,7,8],[2,4,6]] [(1,3),(2,4),(5,8),(6,7)]=>[3,4,1,2,8,7,6,5]=>[4,1,3,2,6,7,8,5]=>[[1,2,5,7,8],[3,6],[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]=>[[1,3,4,5,7,8],[2,6]] [(1,5),(2,3),(4,8),(6,7)]=>[5,3,2,8,1,7,6,4]=>[8,3,5,6,2,7,1,4]=>[[1,4,6,7],[2,5],[3],[8]] [(1,7),(2,3),(4,8),(5,6)]=>[7,3,2,8,6,5,1,4]=>[5,3,6,1,8,7,2,4]=>[[1,2,4],[3,6,7],[5,8]] [(1,8),(2,3),(4,7),(5,6)]=>[8,3,2,7,6,5,4,1]=>[4,3,5,6,7,8,1,2]=>[[1,2,6,7,8],[3,5],[4]] [(1,8),(2,4),(3,7),(5,6)]=>[8,4,7,2,6,5,3,1]=>[3,5,6,7,8,4,2,1]=>[[1,4,6,7,8],[2],[3],[5]] [(1,7),(2,4),(3,8),(5,6)]=>[7,4,8,2,6,5,1,3]=>[5,6,2,8,7,4,1,3]=>[[1,3,7],[2,4],[5,6],[8]] [(1,6),(2,4),(3,8),(5,7)]=>[6,4,8,2,7,1,5,3]=>[7,8,5,6,1,4,2,3]=>[[1,2,3],[4,6],[5,8],[7]] [(1,4),(2,5),(3,8),(6,7)]=>[4,5,8,1,2,7,6,3]=>[8,1,6,4,5,7,2,3]=>[[1,2,3,7],[4,5],[6],[8]] [(1,3),(2,5),(4,8),(6,7)]=>[3,5,1,8,2,7,6,4]=>[5,8,3,6,1,7,2,4]=>[[1,2,4],[3,6,7],[5,8]] [(1,2),(3,5),(4,8),(6,7)]=>[2,1,5,8,3,7,6,4]=>[2,1,8,6,5,7,3,4]=>[[1,3,4],[2,5,7],[6],[8]] [(1,2),(3,6),(4,8),(5,7)]=>[2,1,6,8,7,3,5,4]=>[2,1,7,5,3,6,8,4]=>[[1,3,4,8],[2,5,6],[7]] [(1,4),(2,6),(3,8),(5,7)]=>[4,6,8,1,7,2,5,3]=>[8,7,5,4,2,6,1,3]=>[[1,3],[2,6],[4],[5],[7],[8]] [(1,5),(2,6),(3,8),(4,7)]=>[5,6,8,7,1,2,4,3]=>[7,1,4,2,5,6,8,3]=>[[1,2,3,6,8],[4,5],[7]] [(1,6),(2,5),(3,8),(4,7)]=>[6,5,8,7,2,1,4,3]=>[2,7,4,1,6,5,8,3]=>[[1,3,5,8],[2,4],[6],[7]] [(1,2),(3,8),(4,7),(5,6)]=>[2,1,8,7,6,5,4,3]=>[2,1,4,5,6,7,8,3]=>[[1,3,5,6,7,8],[2,4]] [(1,3),(2,8),(4,7),(5,6)]=>[3,8,1,7,6,5,4,2]=>[8,4,3,5,6,7,1,2]=>[[1,2,6,7],[3,5],[4],[8]] [(1,4),(2,8),(3,7),(5,6)]=>[4,8,7,1,6,5,3,2]=>[7,3,5,4,6,1,8,2]=>[[1,2,6,8],[3,4],[5],[7]] [(1,5),(2,8),(3,7),(4,6)]=>[5,8,7,6,1,4,3,2]=>[6,3,4,1,5,7,8,2]=>[[1,2,5,7,8],[3,4],[6]] [(1,7),(2,8),(3,6),(4,5)]=>[7,8,6,5,4,3,1,2]=>[3,1,4,5,6,8,7,2]=>[[1,2,5,6,7],[3,4],[8]] [(1,8),(2,7),(3,6),(4,5)]=>[8,7,6,5,4,3,2,1]=>[2,3,4,5,6,7,8,1]=>[[1,3,4,5,6,7,8],[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]=>[[1,3,5,7,9],[2,4,6,8,10]] [(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]=>[[1,4,6,7,9],[2,5,8,10],[3]] [(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]=>[[1,3,6,8,9],[2,4,7,10],[5]] [(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]=>[[1,4,6,8,10],[2,5,7,9],[3]] [(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]=>[[1,3,5,8,10],[2,4,6,9],[7]] [(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]=>[[1,3,5,7,9,11],[2,4,6,8,10,12]]
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
Robinson-Schensted insertion tableau
Description
Sends a permutation to its Robinson-Schensted insertion tableau.
The Robinson-Schensted corrspondence is a bijection between permutations of length $n$ and pairs of standard Young tableaux of the same shape and of size $n$, see [1]. These two tableaux are the insertion tableau and the recording tableau.
This map sends a permutation to its corresponding insertion tableau.