Identifier
Mp00058: to permutationPermutations
Mp00239: Permutations CorteelPermutations
Mp00236: Permutations Clarke-Steingrimsson-Zeng inversePermutations
Mp00071: Permutations descent compositionInteger 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.
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].
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 $\{ i_1, i_1 + i_2, i_1 + i_2 + i_3, \ldots, i_1 + i_2 + \cdots + i_{k-1} \}$.