Identifier
Mp00058: Perfect matchings —to permutation⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Images
=>
Cc0012;cc-rep-0Cc0020;cc-rep-3
[(1,2)]=>[2,1]=>[2,1]=>([(0,1)],2) [(1,2),(3,4)]=>[2,1,4,3]=>[2,1,4,3]=>([(0,3),(1,2)],4) [(1,3),(2,4)]=>[3,4,1,2]=>[4,3,2,1]=>([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) [(1,4),(2,3)]=>[4,3,2,1]=>[3,4,1,2]=>([(0,2),(0,3),(1,2),(1,3)],4) [(1,2),(3,4),(5,6)]=>[2,1,4,3,6,5]=>[2,1,4,3,6,5]=>([(0,5),(1,4),(2,3)],6) [(1,3),(2,4),(5,6)]=>[3,4,1,2,6,5]=>[4,3,2,1,6,5]=>([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) [(1,4),(2,3),(5,6)]=>[4,3,2,1,6,5]=>[3,4,1,2,6,5]=>([(0,1),(2,4),(2,5),(3,4),(3,5)],6) [(1,5),(2,3),(4,6)]=>[5,3,2,6,1,4]=>[3,6,1,5,4,2]=>([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) [(1,6),(2,3),(4,5)]=>[6,3,2,5,4,1]=>[3,5,1,6,2,4]=>([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6) [(1,6),(2,4),(3,5)]=>[6,4,5,2,3,1]=>[4,6,5,2,1,3]=>([(0,1),(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) [(1,5),(2,4),(3,6)]=>[5,4,6,2,1,3]=>[5,6,4,2,3,1]=>([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) [(1,4),(2,5),(3,6)]=>[4,5,6,1,2,3]=>[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) [(1,3),(2,5),(4,6)]=>[3,5,1,6,2,4]=>[6,3,2,5,4,1]=>([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) [(1,2),(3,5),(4,6)]=>[2,1,5,6,3,4]=>[2,1,6,5,4,3]=>([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) [(1,2),(3,6),(4,5)]=>[2,1,6,5,4,3]=>[2,1,5,6,3,4]=>([(0,1),(2,4),(2,5),(3,4),(3,5)],6) [(1,3),(2,6),(4,5)]=>[3,6,1,5,4,2]=>[5,3,2,6,1,4]=>([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) [(1,4),(2,6),(3,5)]=>[4,6,5,1,3,2]=>[6,4,5,3,1,2]=>([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) [(1,5),(2,6),(3,4)]=>[5,6,4,3,1,2]=>[5,4,6,1,3,2]=>([(0,1),(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) [(1,6),(2,5),(3,4)]=>[6,5,4,3,2,1]=>[4,5,6,1,2,3]=>([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],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]=>([(0,7),(1,6),(2,5),(3,4)],8) [(1,3),(2,4),(5,6),(7,8)]=>[3,4,1,2,6,5,8,7]=>[4,3,2,1,6,5,8,7]=>([(0,3),(1,2),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8) [(1,4),(2,3),(5,6),(7,8)]=>[4,3,2,1,6,5,8,7]=>[3,4,1,2,6,5,8,7]=>([(0,3),(1,2),(4,6),(4,7),(5,6),(5,7)],8) [(1,5),(2,3),(4,6),(7,8)]=>[5,3,2,6,1,4,8,7]=>[3,6,1,5,4,2,8,7]=>([(0,1),(2,3),(2,7),(3,6),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8) [(1,6),(2,3),(4,5),(7,8)]=>[6,3,2,5,4,1,8,7]=>[3,5,1,6,2,4,8,7]=>([(0,1),(2,5),(2,7),(3,4),(3,7),(4,6),(5,6),(6,7)],8) [(1,7),(2,3),(4,5),(6,8)]=>[7,3,2,5,4,8,1,6]=>[3,5,1,8,2,7,6,4]=>([(0,1),(0,5),(1,4),(2,3),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(5,7),(6,7)],8) [(1,8),(2,3),(4,5),(6,7)]=>[8,3,2,5,4,7,6,1]=>[3,5,1,7,2,8,4,6]=>([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8) [(1,4),(2,5),(3,6),(7,8)]=>[4,5,6,1,2,3,8,7]=>[6,5,4,3,2,1,8,7]=>([(0,1),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8) [(1,3),(2,5),(4,6),(7,8)]=>[3,5,1,6,2,4,8,7]=>[6,3,2,5,4,1,8,7]=>([(0,1),(2,5),(2,6),(2,7),(3,4),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8) [(1,2),(3,5),(4,6),(7,8)]=>[2,1,5,6,3,4,8,7]=>[2,1,6,5,4,3,8,7]=>([(0,3),(1,2),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8) [(1,2),(3,6),(4,5),(7,8)]=>[2,1,6,5,4,3,8,7]=>[2,1,5,6,3,4,8,7]=>([(0,3),(1,2),(4,6),(4,7),(5,6),(5,7)],8) [(1,3),(2,6),(4,5),(7,8)]=>[3,6,1,5,4,2,8,7]=>[5,3,2,6,1,4,8,7]=>([(0,1),(2,3),(2,7),(3,6),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8) [(1,6),(2,5),(3,4),(7,8)]=>[6,5,4,3,2,1,8,7]=>[4,5,6,1,2,3,8,7]=>([(0,1),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8) [(1,7),(2,5),(3,4),(6,8)]=>[7,5,4,3,2,8,1,6]=>[4,5,8,1,2,7,6,3]=>([(0,2),(0,3),(0,7),(1,2),(1,3),(1,7),(2,6),(3,6),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8) [(1,8),(2,5),(3,4),(6,7)]=>[8,5,4,3,2,7,6,1]=>[4,5,7,1,2,8,3,6]=>([(0,1),(0,7),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,7),(5,7),(6,7)],8) [(1,3),(2,7),(4,5),(6,8)]=>[3,7,1,5,4,8,2,6]=>[5,3,2,8,1,7,6,4]=>([(0,3),(0,6),(0,7),(1,2),(1,4),(1,5),(2,4),(2,5),(3,6),(3,7),(4,5),(4,7),(5,6),(6,7)],8) [(1,2),(3,7),(4,5),(6,8)]=>[2,1,7,5,4,8,3,6]=>[2,1,5,8,3,7,6,4]=>([(0,1),(2,3),(2,7),(3,6),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8) [(1,2),(3,8),(4,5),(6,7)]=>[2,1,8,5,4,7,6,3]=>[2,1,5,7,3,8,4,6]=>([(0,1),(2,5),(2,7),(3,4),(3,7),(4,6),(5,6),(6,7)],8) [(1,3),(2,8),(4,5),(6,7)]=>[3,8,1,5,4,7,6,2]=>[5,3,2,7,1,8,4,6]=>([(0,1),(0,5),(1,4),(2,3),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(5,7),(6,7)],8) [(1,8),(2,7),(3,4),(5,6)]=>[8,7,4,3,6,5,2,1]=>[4,6,7,1,8,2,3,5]=>([(0,3),(0,6),(0,7),(1,2),(1,4),(1,5),(2,6),(2,7),(3,4),(3,5),(4,6),(4,7),(5,6),(5,7)],8) [(1,7),(2,8),(3,5),(4,6)]=>[7,8,5,6,3,4,1,2]=>[6,5,8,7,2,1,4,3]=>([(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,7),(5,6)],8) [(1,6),(2,8),(3,5),(4,7)]=>[6,8,5,7,3,1,4,2]=>[7,5,8,6,2,4,1,3]=>([(0,2),(0,3),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,6),(1,7),(2,4),(2,5),(2,7),(3,4),(3,5),(3,6),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8) [(1,5),(2,7),(3,6),(4,8)]=>[5,7,6,8,1,3,2,4]=>[8,6,7,5,4,2,3,1]=>([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8) [(1,5),(2,6),(3,7),(4,8)]=>[5,6,7,8,1,2,3,4]=>[8,7,6,5,4,3,2,1]=>([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8) [(1,4),(2,6),(3,7),(5,8)]=>[4,6,7,1,8,2,3,5]=>[8,7,4,3,6,5,2,1]=>([(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8) [(1,3),(2,6),(4,7),(5,8)]=>[3,6,1,7,8,2,4,5]=>[8,3,2,7,6,5,4,1]=>([(0,1),(0,6),(0,7),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8) [(1,2),(3,6),(4,7),(5,8)]=>[2,1,6,7,8,3,4,5]=>[2,1,8,7,6,5,4,3]=>([(0,1),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8) [(1,2),(3,5),(4,7),(6,8)]=>[2,1,5,7,3,8,4,6]=>[2,1,8,5,4,7,6,3]=>([(0,1),(2,5),(2,6),(2,7),(3,4),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8) [(1,3),(2,5),(4,7),(6,8)]=>[3,5,1,7,2,8,4,6]=>[8,3,2,5,4,7,6,1]=>([(0,5),(0,6),(0,7),(1,4),(1,6),(1,7),(2,3),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8) [(1,4),(2,5),(3,7),(6,8)]=>[4,5,7,1,2,8,3,6]=>[8,5,4,3,2,7,6,1]=>([(0,1),(0,6),(0,7),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8) [(1,6),(2,3),(4,7),(5,8)]=>[6,3,2,7,8,1,4,5]=>[3,8,1,7,6,5,4,2]=>([(0,1),(0,7),(1,6),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8) [(1,5),(2,3),(4,7),(6,8)]=>[5,3,2,7,1,8,4,6]=>[3,8,1,5,4,7,6,2]=>([(0,1),(0,7),(1,6),(2,5),(2,6),(2,7),(3,4),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8) [(1,4),(2,3),(5,7),(6,8)]=>[4,3,2,1,7,8,5,6]=>[3,4,1,2,8,7,6,5]=>([(0,2),(0,3),(1,2),(1,3),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8) [(1,3),(2,4),(5,7),(6,8)]=>[3,4,1,2,7,8,5,6]=>[4,3,2,1,8,7,6,5]=>([(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4),(5,6),(5,7),(6,7)],8) [(1,2),(3,4),(5,7),(6,8)]=>[2,1,4,3,7,8,5,6]=>[2,1,4,3,8,7,6,5]=>([(0,3),(1,2),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8) [(1,2),(3,4),(5,8),(6,7)]=>[2,1,4,3,8,7,6,5]=>[2,1,4,3,7,8,5,6]=>([(0,3),(1,2),(4,6),(4,7),(5,6),(5,7)],8) [(1,3),(2,4),(5,8),(6,7)]=>[3,4,1,2,8,7,6,5]=>[4,3,2,1,7,8,5,6]=>([(0,2),(0,3),(1,2),(1,3),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8) [(1,4),(2,3),(5,8),(6,7)]=>[4,3,2,1,8,7,6,5]=>[3,4,1,2,7,8,5,6]=>([(0,6),(0,7),(1,3),(1,4),(2,3),(2,4),(5,6),(5,7)],8) [(1,5),(2,3),(4,8),(6,7)]=>[5,3,2,8,1,7,6,4]=>[3,7,1,5,4,8,2,6]=>([(0,3),(0,7),(1,2),(1,7),(2,6),(3,6),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8) [(1,8),(2,3),(4,7),(5,6)]=>[8,3,2,7,6,5,4,1]=>[3,6,1,7,8,2,4,5]=>([(0,1),(0,7),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,7),(5,7),(6,7)],8) [(1,4),(2,5),(3,8),(6,7)]=>[4,5,8,1,2,7,6,3]=>[7,5,4,3,2,8,1,6]=>([(0,1),(0,7),(1,6),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8) [(1,3),(2,5),(4,8),(6,7)]=>[3,5,1,8,2,7,6,4]=>[7,3,2,5,4,8,1,6]=>([(0,1),(0,7),(1,6),(2,5),(2,6),(2,7),(3,4),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8) [(1,2),(3,5),(4,8),(6,7)]=>[2,1,5,8,3,7,6,4]=>[2,1,7,5,4,8,3,6]=>([(0,1),(2,3),(2,7),(3,6),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8) [(1,6),(2,5),(3,8),(4,7)]=>[6,5,8,7,2,1,4,3]=>[7,8,5,6,3,4,1,2]=>([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8) [(1,7),(2,5),(3,8),(4,6)]=>[7,5,8,6,2,4,1,3]=>[6,8,5,7,3,1,4,2]=>([(0,2),(0,3),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,6),(1,7),(2,4),(2,5),(2,7),(3,4),(3,5),(3,6),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8) [(1,8),(2,6),(3,7),(4,5)]=>[8,6,7,5,4,2,3,1]=>[5,7,6,8,1,3,2,4]=>([(0,2),(0,3),(0,6),(0,7),(1,2),(1,3),(1,6),(1,7),(2,4),(2,5),(3,4),(3,5),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8) [(1,2),(3,8),(4,7),(5,6)]=>[2,1,8,7,6,5,4,3]=>[2,1,6,7,8,3,4,5]=>([(0,1),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8) [(1,3),(2,8),(4,7),(5,6)]=>[3,8,1,7,6,5,4,2]=>[6,3,2,7,8,1,4,5]=>([(0,2),(0,3),(0,7),(1,2),(1,3),(1,7),(2,6),(3,6),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8) [(1,8),(2,7),(3,6),(4,5)]=>[8,7,6,5,4,3,2,1]=>[5,6,7,8,1,2,3,4]=>([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7)],8)
Map
to permutation
Description
Returns the fixed point free involution whose transpositions are the pairs in the perfect matching.
Map
Corteel
Description
Corteel's map interchanging the number of crossings and the number of nestings of a permutation.
This involution creates a labelled bicoloured Motzkin path, using the Foata-Zeilberger map. In the corresponding bump diagram, each label records the number of arcs nesting the given arc. Then each label is replaced by its complement, and the inverse of the Foata-Zeilberger map is applied.
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.