Identifier
Mp00058: Perfect matchings —to permutation⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Images
=>
Cc0012;cc-rep-0Cc0020;cc-rep-3Cc0002;cc-rep-4
[(1,2)]=>[2,1]=>[2,1]=>([(0,1)],2)=>[2] [(1,2),(3,4)]=>[2,1,4,3]=>[2,4,1,3]=>([(0,3),(1,2),(2,3)],4)=>[4] [(1,3),(2,4)]=>[3,4,1,2]=>[3,1,4,2]=>([(0,3),(1,2),(2,3)],4)=>[4] [(1,4),(2,3)]=>[4,3,2,1]=>[4,3,2,1]=>([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)=>[4] [(1,2),(3,4),(5,6)]=>[2,1,4,3,6,5]=>[2,4,6,1,3,5]=>([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)=>[6] [(1,3),(2,4),(5,6)]=>[3,4,1,2,6,5]=>[3,1,4,6,2,5]=>([(0,5),(1,4),(2,3),(3,5),(4,5)],6)=>[6] [(1,4),(2,3),(5,6)]=>[4,3,2,1,6,5]=>[4,3,2,6,1,5]=>([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>[6] [(1,5),(2,3),(4,6)]=>[5,3,2,6,1,4]=>[3,2,5,1,6,4]=>([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)=>[6] [(1,6),(2,3),(4,5)]=>[6,3,2,5,4,1]=>[3,6,2,5,4,1]=>([(0,3),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>[6] [(1,6),(2,4),(3,5)]=>[6,4,5,2,3,1]=>[4,2,6,5,3,1]=>([(0,3),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>[6] [(1,5),(2,4),(3,6)]=>[5,4,6,2,1,3]=>[5,4,2,1,6,3]=>([(0,3),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>[6] [(1,4),(2,5),(3,6)]=>[4,5,6,1,2,3]=>[4,1,5,2,6,3]=>([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)=>[6] [(1,3),(2,5),(4,6)]=>[3,5,1,6,2,4]=>[5,3,1,6,2,4]=>([(0,1),(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>[6] [(1,2),(3,5),(4,6)]=>[2,1,5,6,3,4]=>[5,6,2,1,3,4]=>([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)=>[6] [(1,2),(3,6),(4,5)]=>[2,1,6,5,4,3]=>[6,5,2,4,1,3]=>([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>[6] [(1,3),(2,6),(4,5)]=>[3,6,1,5,4,2]=>[6,3,5,1,4,2]=>([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>[6] [(1,4),(2,6),(3,5)]=>[4,6,5,1,3,2]=>[4,6,1,5,3,2]=>([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)=>[6] [(1,5),(2,6),(3,4)]=>[5,6,4,3,1,2]=>[5,1,6,4,3,2]=>([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>[6] [(1,6),(2,5),(3,4)]=>[6,5,4,3,2,1]=>[6,5,4,3,2,1]=>([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>[6] [(1,2),(3,4),(5,6),(7,8)]=>[2,1,4,3,6,5,8,7]=>[2,4,6,8,1,3,5,7]=>([(0,7),(1,6),(2,5),(2,6),(3,4),(3,7),(4,5),(4,6),(5,7),(6,7)],8)=>[8] [(1,3),(2,4),(5,6),(7,8)]=>[3,4,1,2,6,5,8,7]=>[3,1,4,6,8,2,5,7]=>([(0,7),(1,6),(2,3),(3,7),(4,5),(4,7),(5,6),(6,7)],8)=>[8] [(1,4),(2,5),(3,6),(7,8)]=>[4,5,6,1,2,3,8,7]=>[4,1,5,2,6,8,3,7]=>([(0,7),(1,6),(2,3),(3,7),(4,5),(4,7),(5,6),(6,7)],8)=>[8] [(1,3),(2,5),(4,6),(7,8)]=>[3,5,1,6,2,4,8,7]=>[5,3,1,6,8,2,4,7]=>([(0,5),(1,3),(1,7),(2,4),(2,6),(3,5),(3,6),(4,6),(4,7),(5,7),(6,7)],8)=>[8] [(1,2),(3,5),(4,6),(7,8)]=>[2,1,5,6,3,4,8,7]=>[5,6,2,8,1,3,4,7]=>([(0,7),(1,4),(1,5),(1,6),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)=>[8] [(1,5),(2,6),(3,7),(4,8)]=>[5,6,7,8,1,2,3,4]=>[5,1,6,2,7,3,8,4]=>([(0,7),(1,6),(2,5),(2,6),(3,4),(3,7),(4,5),(4,6),(5,7),(6,7)],8)=>[8] [(1,4),(2,6),(3,7),(5,8)]=>[4,6,7,1,8,2,3,5]=>[6,1,7,4,2,8,3,5]=>([(0,7),(1,3),(1,6),(2,4),(2,5),(2,7),(3,4),(3,7),(4,5),(4,6),(5,6),(5,7),(6,7)],8)=>[8] [(1,3),(2,6),(4,7),(5,8)]=>[3,6,1,7,8,2,4,5]=>[3,6,1,7,2,8,4,5]=>([(0,4),(0,5),(1,2),(1,6),(2,7),(3,4),(3,5),(3,6),(4,7),(5,7),(6,7)],8)=>[8] [(1,2),(3,6),(4,7),(5,8)]=>[2,1,6,7,8,3,4,5]=>[6,7,2,1,8,3,4,5]=>([(0,3),(0,4),(0,5),(1,2),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)=>[8] [(1,2),(3,5),(4,7),(6,8)]=>[2,1,5,7,3,8,4,6]=>[7,5,8,2,1,3,4,6]=>([(0,6),(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7)],8)=>[8] [(1,3),(2,5),(4,7),(6,8)]=>[3,5,1,7,2,8,4,6]=>[5,7,3,1,8,2,4,6]=>([(0,3),(0,7),(1,3),(1,4),(1,7),(2,4),(2,5),(2,7),(3,6),(4,5),(4,6),(5,6),(5,7),(6,7)],8)=>[8] [(1,4),(2,5),(3,7),(6,8)]=>[4,5,7,1,2,8,3,6]=>[4,7,1,5,2,8,3,6]=>([(0,1),(0,7),(1,6),(2,3),(2,7),(3,5),(3,6),(4,5),(4,6),(4,7),(5,7),(6,7)],8)=>[8] [(1,8),(2,4),(3,6),(5,7)]=>[8,4,6,2,7,3,5,1]=>[2,4,8,6,3,7,5,1]=>([(0,7),(1,5),(1,7),(2,4),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)=>[8] [(1,5),(2,3),(4,7),(6,8)]=>[5,3,2,7,1,8,4,6]=>[3,5,2,7,1,8,4,6]=>([(0,4),(0,7),(1,2),(1,6),(2,5),(3,4),(3,5),(3,7),(4,7),(5,6),(6,7)],8)=>[8] [(1,3),(2,4),(5,7),(6,8)]=>[3,4,1,2,7,8,5,6]=>[3,4,7,1,8,2,5,6]=>([(0,5),(0,7),(1,5),(1,7),(2,4),(2,6),(3,4),(3,6),(4,7),(5,6),(6,7)],8)=>[8] [(1,2),(3,4),(5,7),(6,8)]=>[2,1,4,3,7,8,5,6]=>[7,8,2,4,1,3,5,6]=>([(0,6),(0,7),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7)],8)=>[8] [(1,3),(2,4),(5,8),(6,7)]=>[3,4,1,2,8,7,6,5]=>[8,3,7,1,4,6,2,5]=>([(0,1),(0,6),(0,7),(1,5),(1,7),(2,5),(2,6),(2,7),(3,4),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)=>[8] [(1,3),(2,5),(4,8),(6,7)]=>[3,5,1,8,2,7,6,4]=>[5,8,3,7,1,6,2,4]=>([(0,1),(0,2),(0,3),(0,6),(1,2),(1,5),(1,7),(2,6),(2,7),(3,4),(3,5),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)=>[8] [(1,4),(2,6),(3,8),(5,7)]=>[4,6,8,1,7,2,5,3]=>[6,4,8,1,7,2,5,3]=>([(0,4),(0,6),(0,7),(1,2),(1,3),(1,5),(1,6),(2,5),(2,6),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7)],8)=>[8]
Map
to permutation
Description
Returns the fixed point free involution whose transpositions are the pairs in the perfect matching.
Map
inverse Foata bijection
Description
The inverse of Foata's bijection.
See Mp00067Foata bijection.
Map
graph of inversions
Description
The graph of inversions of a permutation.
For a permutation of $\{1,\dots,n\}$, this is the graph with vertices $\{1,\dots,n\}$, where $(i,j)$ is an edge if and only if it is an inversion of the permutation.
Map
to partition of connected components
Description
Return the partition of the sizes of the connected components of the graph.