Identifier
Mp00058:
Perfect matchings
—to permutation⟶
Permutations
Mp00239: Permutations —Corteel⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00239: Permutations —Corteel⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Images
=>
Cc0012;cc-rep-0
[(1,2)]=>[2,1]=>[2,1]=>[2,1]=>[1,1]
[(1,2),(3,4)]=>[2,1,4,3]=>[2,1,4,3]=>[2,1,4,3]=>[1,2,1]
[(1,3),(2,4)]=>[3,4,1,2]=>[4,3,2,1]=>[3,2,4,1]=>[1,2,1]
[(1,4),(2,3)]=>[4,3,2,1]=>[3,4,1,2]=>[4,1,3,2]=>[1,2,1]
[(1,2),(3,4),(5,6)]=>[2,1,4,3,6,5]=>[2,1,4,3,6,5]=>[2,1,4,3,6,5]=>[1,2,2,1]
[(1,3),(2,4),(5,6)]=>[3,4,1,2,6,5]=>[4,3,2,1,6,5]=>[3,2,4,1,6,5]=>[1,2,2,1]
[(1,4),(2,3),(5,6)]=>[4,3,2,1,6,5]=>[3,4,1,2,6,5]=>[4,1,3,2,6,5]=>[1,2,2,1]
[(1,5),(2,3),(4,6)]=>[5,3,2,6,1,4]=>[3,6,1,5,4,2]=>[5,4,6,1,3,2]=>[1,2,2,1]
[(1,6),(2,3),(4,5)]=>[6,3,2,5,4,1]=>[3,5,1,6,2,4]=>[6,1,3,2,5,4]=>[1,2,2,1]
[(1,6),(2,4),(3,5)]=>[6,4,5,2,3,1]=>[4,6,5,2,1,3]=>[5,2,6,1,4,3]=>[1,2,2,1]
[(1,5),(2,4),(3,6)]=>[5,4,6,2,1,3]=>[5,6,4,2,3,1]=>[4,2,6,3,5,1]=>[1,2,2,1]
[(1,4),(2,5),(3,6)]=>[4,5,6,1,2,3]=>[6,5,4,3,2,1]=>[4,3,5,2,6,1]=>[1,2,2,1]
[(1,3),(2,5),(4,6)]=>[3,5,1,6,2,4]=>[6,3,2,5,4,1]=>[3,2,5,4,6,1]=>[1,2,2,1]
[(1,2),(3,5),(4,6)]=>[2,1,5,6,3,4]=>[2,1,6,5,4,3]=>[2,1,5,4,6,3]=>[1,2,2,1]
[(1,2),(3,6),(4,5)]=>[2,1,6,5,4,3]=>[2,1,5,6,3,4]=>[2,1,6,3,5,4]=>[1,2,2,1]
[(1,3),(2,6),(4,5)]=>[3,6,1,5,4,2]=>[5,3,2,6,1,4]=>[3,2,6,1,5,4]=>[1,2,2,1]
[(1,4),(2,6),(3,5)]=>[4,6,5,1,3,2]=>[6,4,5,3,1,2]=>[5,3,4,1,6,2]=>[1,2,2,1]
[(1,5),(2,6),(3,4)]=>[5,6,4,3,1,2]=>[5,4,6,1,3,2]=>[6,1,4,3,5,2]=>[1,2,2,1]
[(1,6),(2,5),(3,4)]=>[6,5,4,3,2,1]=>[4,5,6,1,2,3]=>[6,1,5,2,4,3]=>[1,2,2,1]
[(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]=>[1,2,2,2,1]
[(1,3),(2,4),(5,6),(7,8)]=>[3,4,1,2,6,5,8,7]=>[4,3,2,1,6,5,8,7]=>[3,2,4,1,6,5,8,7]=>[1,2,2,2,1]
[(1,4),(2,3),(5,6),(7,8)]=>[4,3,2,1,6,5,8,7]=>[3,4,1,2,6,5,8,7]=>[4,1,3,2,6,5,8,7]=>[1,2,2,2,1]
[(1,5),(2,3),(4,6),(7,8)]=>[5,3,2,6,1,4,8,7]=>[3,6,1,5,4,2,8,7]=>[5,4,6,1,3,2,8,7]=>[1,2,2,2,1]
[(1,6),(2,3),(4,5),(7,8)]=>[6,3,2,5,4,1,8,7]=>[3,5,1,6,2,4,8,7]=>[6,1,3,2,5,4,8,7]=>[1,2,2,2,1]
[(1,7),(2,3),(4,5),(6,8)]=>[7,3,2,5,4,8,1,6]=>[3,5,1,8,2,7,6,4]=>[7,6,8,1,3,2,5,4]=>[1,2,2,2,1]
[(1,8),(2,3),(4,5),(6,7)]=>[8,3,2,5,4,7,6,1]=>[3,5,1,7,2,8,4,6]=>[8,1,3,2,5,4,7,6]=>[1,2,2,2,1]
[(1,6),(2,4),(3,5),(7,8)]=>[6,4,5,2,3,1,8,7]=>[4,6,5,2,1,3,8,7]=>[5,2,6,1,4,3,8,7]=>[1,2,2,2,1]
[(1,5),(2,4),(3,6),(7,8)]=>[5,4,6,2,1,3,8,7]=>[5,6,4,2,3,1,8,7]=>[4,2,6,3,5,1,8,7]=>[1,2,2,2,1]
[(1,4),(2,5),(3,6),(7,8)]=>[4,5,6,1,2,3,8,7]=>[6,5,4,3,2,1,8,7]=>[4,3,5,2,6,1,8,7]=>[1,2,2,2,1]
[(1,3),(2,5),(4,6),(7,8)]=>[3,5,1,6,2,4,8,7]=>[6,3,2,5,4,1,8,7]=>[3,2,5,4,6,1,8,7]=>[1,2,2,2,1]
[(1,2),(3,5),(4,6),(7,8)]=>[2,1,5,6,3,4,8,7]=>[2,1,6,5,4,3,8,7]=>[2,1,5,4,6,3,8,7]=>[1,2,2,2,1]
[(1,2),(3,6),(4,5),(7,8)]=>[2,1,6,5,4,3,8,7]=>[2,1,5,6,3,4,8,7]=>[2,1,6,3,5,4,8,7]=>[1,2,2,2,1]
[(1,3),(2,6),(4,5),(7,8)]=>[3,6,1,5,4,2,8,7]=>[5,3,2,6,1,4,8,7]=>[3,2,6,1,5,4,8,7]=>[1,2,2,2,1]
[(1,5),(2,6),(3,4),(7,8)]=>[5,6,4,3,1,2,8,7]=>[5,4,6,1,3,2,8,7]=>[6,1,4,3,5,2,8,7]=>[1,2,2,2,1]
[(1,6),(2,5),(3,4),(7,8)]=>[6,5,4,3,2,1,8,7]=>[4,5,6,1,2,3,8,7]=>[6,1,5,2,4,3,8,7]=>[1,2,2,2,1]
[(1,7),(2,5),(3,4),(6,8)]=>[7,5,4,3,2,8,1,6]=>[4,5,8,1,2,7,6,3]=>[7,6,8,1,5,2,4,3]=>[1,2,2,2,1]
[(1,8),(2,5),(3,4),(6,7)]=>[8,5,4,3,2,7,6,1]=>[4,5,7,1,2,8,3,6]=>[8,1,5,2,4,3,7,6]=>[1,2,2,2,1]
[(1,7),(2,6),(3,4),(5,8)]=>[7,6,4,3,8,2,1,5]=>[4,7,8,1,6,3,5,2]=>[6,1,8,5,7,3,4,2]=>[1,2,2,2,1]
[(1,6),(2,7),(3,4),(5,8)]=>[6,7,4,3,8,1,2,5]=>[7,4,8,1,6,5,3,2]=>[6,5,8,1,4,3,7,2]=>[1,2,2,2,1]
[(1,3),(2,7),(4,5),(6,8)]=>[3,7,1,5,4,8,2,6]=>[5,3,2,8,1,7,6,4]=>[3,2,7,6,8,1,5,4]=>[1,2,2,2,1]
[(1,2),(3,7),(4,5),(6,8)]=>[2,1,7,5,4,8,3,6]=>[2,1,5,8,3,7,6,4]=>[2,1,7,6,8,3,5,4]=>[1,2,2,2,1]
[(1,2),(3,8),(4,5),(6,7)]=>[2,1,8,5,4,7,6,3]=>[2,1,5,7,3,8,4,6]=>[2,1,8,3,5,4,7,6]=>[1,2,2,2,1]
[(1,3),(2,8),(4,5),(6,7)]=>[3,8,1,5,4,7,6,2]=>[5,3,2,7,1,8,4,6]=>[3,2,8,1,5,4,7,6]=>[1,2,2,2,1]
[(1,4),(2,8),(3,5),(6,7)]=>[4,8,5,1,3,7,6,2]=>[7,4,5,3,1,8,2,6]=>[5,3,4,1,8,2,7,6]=>[1,2,2,2,1]
[(1,6),(2,8),(3,4),(5,7)]=>[6,8,4,3,7,1,5,2]=>[6,4,8,1,7,5,2,3]=>[7,5,8,1,4,2,6,3]=>[1,2,2,2,1]
[(1,7),(2,8),(3,4),(5,6)]=>[7,8,4,3,6,5,1,2]=>[6,4,7,1,8,2,5,3]=>[8,1,4,2,7,5,6,3]=>[1,2,2,2,1]
[(1,8),(2,7),(3,4),(5,6)]=>[8,7,4,3,6,5,2,1]=>[4,6,7,1,8,2,3,5]=>[8,1,7,2,4,3,6,5]=>[1,2,2,2,1]
[(1,7),(2,8),(3,5),(4,6)]=>[7,8,5,6,3,4,1,2]=>[6,5,8,7,2,1,4,3]=>[7,2,8,1,5,4,6,3]=>[1,2,2,2,1]
[(1,6),(2,8),(3,5),(4,7)]=>[6,8,5,7,3,1,4,2]=>[7,5,8,6,2,4,1,3]=>[6,2,8,4,5,1,7,3]=>[1,2,2,2,1]
[(1,5),(2,8),(3,6),(4,7)]=>[5,8,6,7,1,3,4,2]=>[8,5,7,6,4,2,1,3]=>[6,4,7,2,5,1,8,3]=>[1,2,2,2,1]
[(1,2),(3,8),(4,6),(5,7)]=>[2,1,8,6,7,4,5,3]=>[2,1,6,8,7,4,3,5]=>[2,1,7,4,8,3,6,5]=>[1,2,2,2,1]
[(1,2),(3,7),(4,6),(5,8)]=>[2,1,7,6,8,4,3,5]=>[2,1,7,8,6,4,5,3]=>[2,1,6,4,8,5,7,3]=>[1,2,2,2,1]
[(1,4),(2,7),(3,6),(5,8)]=>[4,7,6,1,8,3,2,5]=>[8,4,7,3,6,2,5,1]=>[6,3,4,2,7,5,8,1]=>[1,2,2,2,1]
[(1,5),(2,7),(3,6),(4,8)]=>[5,7,6,8,1,3,2,4]=>[8,6,7,5,4,2,3,1]=>[5,4,7,2,6,3,8,1]=>[1,2,2,2,1]
[(1,6),(2,7),(3,5),(4,8)]=>[6,7,5,8,3,1,2,4]=>[7,6,8,5,2,4,3,1]=>[5,2,8,4,6,3,7,1]=>[1,2,2,2,1]
[(1,7),(2,6),(3,5),(4,8)]=>[7,6,5,8,3,2,1,4]=>[6,7,8,5,2,3,4,1]=>[5,2,8,3,7,4,6,1]=>[1,2,2,2,1]
[(1,8),(2,6),(3,5),(4,7)]=>[8,6,5,7,3,2,4,1]=>[5,7,8,6,2,3,1,4]=>[6,2,8,3,7,1,5,4]=>[1,2,2,2,1]
[(1,8),(2,5),(3,6),(4,7)]=>[8,5,6,7,2,3,4,1]=>[5,8,7,6,3,2,1,4]=>[6,3,7,2,8,1,5,4]=>[1,2,2,2,1]
[(1,7),(2,5),(3,6),(4,8)]=>[7,5,6,8,2,3,1,4]=>[6,8,7,5,3,2,4,1]=>[5,3,7,2,8,4,6,1]=>[1,2,2,2,1]
[(1,6),(2,5),(3,7),(4,8)]=>[6,5,7,8,2,1,3,4]=>[7,8,6,5,3,4,2,1]=>[5,3,6,4,8,2,7,1]=>[1,2,2,2,1]
[(1,5),(2,6),(3,7),(4,8)]=>[5,6,7,8,1,2,3,4]=>[8,7,6,5,4,3,2,1]=>[5,4,6,3,7,2,8,1]=>[1,2,2,2,1]
[(1,4),(2,6),(3,7),(5,8)]=>[4,6,7,1,8,2,3,5]=>[8,7,4,3,6,5,2,1]=>[4,3,6,5,7,2,8,1]=>[1,2,2,2,1]
[(1,3),(2,6),(4,7),(5,8)]=>[3,6,1,7,8,2,4,5]=>[8,3,2,7,6,5,4,1]=>[3,2,6,5,7,4,8,1]=>[1,2,2,2,1]
[(1,2),(3,6),(4,7),(5,8)]=>[2,1,6,7,8,3,4,5]=>[2,1,8,7,6,5,4,3]=>[2,1,6,5,7,4,8,3]=>[1,2,2,2,1]
[(1,2),(3,5),(4,7),(6,8)]=>[2,1,5,7,3,8,4,6]=>[2,1,8,5,4,7,6,3]=>[2,1,5,4,7,6,8,3]=>[1,2,2,2,1]
[(1,3),(2,5),(4,7),(6,8)]=>[3,5,1,7,2,8,4,6]=>[8,3,2,5,4,7,6,1]=>[3,2,5,4,7,6,8,1]=>[1,2,2,2,1]
[(1,4),(2,5),(3,7),(6,8)]=>[4,5,7,1,2,8,3,6]=>[8,5,4,3,2,7,6,1]=>[4,3,5,2,7,6,8,1]=>[1,2,2,2,1]
[(1,6),(2,4),(3,7),(5,8)]=>[6,4,7,2,8,1,3,5]=>[7,8,4,2,6,5,3,1]=>[4,2,6,5,8,3,7,1]=>[1,2,2,2,1]
[(1,8),(2,4),(3,6),(5,7)]=>[8,4,6,2,7,3,5,1]=>[4,8,6,2,7,3,1,5]=>[7,2,6,3,4,1,8,5]=>[1,2,2,2,1]
[(1,6),(2,3),(4,7),(5,8)]=>[6,3,2,7,8,1,4,5]=>[3,8,1,7,6,5,4,2]=>[6,5,7,4,8,1,3,2]=>[1,2,2,2,1]
[(1,5),(2,3),(4,7),(6,8)]=>[5,3,2,7,1,8,4,6]=>[3,8,1,5,4,7,6,2]=>[5,4,7,6,8,1,3,2]=>[1,2,2,2,1]
[(1,4),(2,3),(5,7),(6,8)]=>[4,3,2,1,7,8,5,6]=>[3,4,1,2,8,7,6,5]=>[4,1,3,2,7,6,8,5]=>[1,2,2,2,1]
[(1,3),(2,4),(5,7),(6,8)]=>[3,4,1,2,7,8,5,6]=>[4,3,2,1,8,7,6,5]=>[3,2,4,1,7,6,8,5]=>[1,2,2,2,1]
[(1,2),(3,4),(5,7),(6,8)]=>[2,1,4,3,7,8,5,6]=>[2,1,4,3,8,7,6,5]=>[2,1,4,3,7,6,8,5]=>[1,2,2,2,1]
[(1,2),(3,4),(5,8),(6,7)]=>[2,1,4,3,8,7,6,5]=>[2,1,4,3,7,8,5,6]=>[2,1,4,3,8,5,7,6]=>[1,2,2,2,1]
[(1,3),(2,4),(5,8),(6,7)]=>[3,4,1,2,8,7,6,5]=>[4,3,2,1,7,8,5,6]=>[3,2,4,1,8,5,7,6]=>[1,2,2,2,1]
[(1,4),(2,3),(5,8),(6,7)]=>[4,3,2,1,8,7,6,5]=>[3,4,1,2,7,8,5,6]=>[4,1,3,2,8,5,7,6]=>[1,2,2,2,1]
[(1,8),(2,3),(4,7),(5,6)]=>[8,3,2,7,6,5,4,1]=>[3,6,1,7,8,2,4,5]=>[8,1,3,2,7,4,6,5]=>[1,2,2,2,1]
[(1,6),(2,4),(3,8),(5,7)]=>[6,4,8,2,7,1,5,3]=>[6,8,4,2,7,5,1,3]=>[4,2,7,5,8,1,6,3]=>[1,2,2,2,1]
[(1,5),(2,4),(3,8),(6,7)]=>[5,4,8,2,1,7,6,3]=>[5,7,4,2,3,8,1,6]=>[4,2,8,3,5,1,7,6]=>[1,2,2,2,1]
[(1,4),(2,5),(3,8),(6,7)]=>[4,5,8,1,2,7,6,3]=>[7,5,4,3,2,8,1,6]=>[4,3,5,2,8,1,7,6]=>[1,2,2,2,1]
[(1,3),(2,5),(4,8),(6,7)]=>[3,5,1,8,2,7,6,4]=>[7,3,2,5,4,8,1,6]=>[3,2,5,4,8,1,7,6]=>[1,2,2,2,1]
[(1,2),(3,6),(4,8),(5,7)]=>[2,1,6,8,7,3,5,4]=>[2,1,8,6,7,5,3,4]=>[2,1,7,5,6,3,8,4]=>[1,2,2,2,1]
[(1,5),(2,6),(3,8),(4,7)]=>[5,6,8,7,1,2,4,3]=>[8,7,5,6,4,3,1,2]=>[6,4,5,3,7,1,8,2]=>[1,2,2,2,1]
[(1,6),(2,5),(3,8),(4,7)]=>[6,5,8,7,2,1,4,3]=>[7,8,5,6,3,4,1,2]=>[6,3,5,4,8,1,7,2]=>[1,2,2,2,1]
[(1,7),(2,5),(3,8),(4,6)]=>[7,5,8,6,2,4,1,3]=>[6,8,5,7,3,1,4,2]=>[7,3,5,1,8,4,6,2]=>[1,2,2,2,1]
[(1,8),(2,6),(3,7),(4,5)]=>[8,6,7,5,4,2,3,1]=>[5,7,6,8,1,3,2,4]=>[8,1,6,3,7,2,5,4]=>[1,2,2,2,1]
[(1,7),(2,6),(3,8),(4,5)]=>[7,6,8,5,4,2,1,3]=>[6,7,5,8,1,3,4,2]=>[8,1,5,3,7,4,6,2]=>[1,2,2,2,1]
[(1,6),(2,7),(3,8),(4,5)]=>[6,7,8,5,4,1,2,3]=>[7,6,5,8,1,4,3,2]=>[8,1,5,4,6,3,7,2]=>[1,2,2,2,1]
[(1,5),(2,7),(3,8),(4,6)]=>[5,7,8,6,1,4,2,3]=>[8,6,5,7,4,1,3,2]=>[7,4,5,1,6,3,8,2]=>[1,2,2,2,1]
[(1,4),(2,7),(3,8),(5,6)]=>[4,7,8,1,6,5,2,3]=>[7,6,4,3,8,1,5,2]=>[4,3,8,1,6,5,7,2]=>[1,2,2,2,1]
[(1,2),(3,7),(4,8),(5,6)]=>[2,1,7,8,6,5,3,4]=>[2,1,7,6,8,3,5,4]=>[2,1,8,3,6,5,7,4]=>[1,2,2,2,1]
[(1,2),(3,8),(4,7),(5,6)]=>[2,1,8,7,6,5,4,3]=>[2,1,6,7,8,3,4,5]=>[2,1,8,3,7,4,6,5]=>[1,2,2,2,1]
[(1,3),(2,8),(4,7),(5,6)]=>[3,8,1,7,6,5,4,2]=>[6,3,2,7,8,1,4,5]=>[3,2,8,1,7,4,6,5]=>[1,2,2,2,1]
[(1,5),(2,8),(3,7),(4,6)]=>[5,8,7,6,1,4,3,2]=>[8,5,6,7,4,1,2,3]=>[7,4,6,1,5,2,8,3]=>[1,2,2,2,1]
[(1,6),(2,8),(3,7),(4,5)]=>[6,8,7,5,4,1,3,2]=>[7,5,6,8,1,4,2,3]=>[8,1,6,4,5,2,7,3]=>[1,2,2,2,1]
[(1,8),(2,7),(3,6),(4,5)]=>[8,7,6,5,4,3,2,1]=>[5,6,7,8,1,2,3,4]=>[8,1,7,2,6,3,5,4]=>[1,2,2,2,1]
[(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]=>[1,2,2,2,2,1]
[(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]=>[2,1,4,3,6,5,8,7,10,9,12,11]=>[1,2,2,2,2,2,1]
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.
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
Clarke-Steingrimsson-Zeng inverse
Description
The inverse of the Clarke-Steingrimsson-Zeng map, sending excedances to descents.
This is the inverse of the map $\Phi$ in [1, sec.3].
This is the inverse of the map $\Phi$ in [1, sec.3].
Map
descent composition
Description
The descent composition of a permutation.
The descent composition of a permutation $\pi$ of length $n$ is the integer composition of $n$ whose descent set equals the descent set of $\pi$. The descent set of a permutation $\pi$ is $\{i \mid 1 \leq i < n, \pi(i) > \pi(i+1)\}$. The descent set of a composition $c = (i_1, i_2, \ldots, i_k)$ is the set $\{ i_1, i_1 + i_2, i_1 + i_2 + i_3, \ldots, i_1 + i_2 + \cdots + i_{k-1} \}$.
The descent composition of a permutation $\pi$ of length $n$ is the integer composition of $n$ whose descent set equals the descent set of $\pi$. The descent set of a permutation $\pi$ is $\{i \mid 1 \leq i < n, \pi(i) > \pi(i+1)\}$. The descent set of a composition $c = (i_1, i_2, \ldots, i_k)$ is the set $\{ i_1, i_1 + i_2, i_1 + i_2 + i_3, \ldots, i_1 + i_2 + \cdots + i_{k-1} \}$.
searching the database
Sorry, this map was not found in the database.