Identifier
Mp00058: to permutationPermutations
Mp00239: Permutations CorteelPermutations
Mp00087: Permutations inverse first fundamental transformationPermutations
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]=>[3,1,4,2]=>[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]=>[3,1,4,2,6,5]=>[1,2,2,1] [(1,5),(2,3),(4,6)]=>[5,3,2,6,1,4]=>[3,6,1,5,4,2]=>[3,1,5,4,6,2]=>[1,2,2,1] [(1,6),(2,3),(4,5)]=>[6,3,2,5,4,1]=>[3,5,1,6,2,4]=>[3,1,5,2,6,4]=>[1,2,2,1] [(1,6),(2,4),(3,5)]=>[6,4,5,2,3,1]=>[4,6,5,2,1,3]=>[6,3,5,1,4,2]=>[1,2,2,1] [(1,5),(2,4),(3,6)]=>[5,4,6,2,1,3]=>[5,6,4,2,3,1]=>[6,1,5,3,4,2]=>[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]=>[2,1,5,3,6,4]=>[1,2,2,1] [(1,3),(2,6),(4,5)]=>[3,6,1,5,4,2]=>[5,3,2,6,1,4]=>[3,2,5,1,6,4]=>[1,2,2,1] [(1,4),(2,6),(3,5)]=>[4,6,5,1,3,2]=>[6,4,5,3,1,2]=>[6,2,4,3,5,1]=>[1,2,2,1] [(1,5),(2,6),(3,4)]=>[5,6,4,3,1,2]=>[5,4,6,1,3,2]=>[6,2,4,1,5,3]=>[1,2,2,1] [(1,6),(2,5),(3,4)]=>[6,5,4,3,2,1]=>[4,5,6,1,2,3]=>[4,1,5,2,6,3]=>[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]=>[3,1,4,2,6,5,8,7]=>[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]=>[3,1,5,4,6,2,8,7]=>[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]=>[3,1,5,2,6,4,8,7]=>[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]=>[3,1,5,2,7,6,8,4]=>[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]=>[3,1,5,2,7,4,8,6]=>[1,2,2,2,1] [(1,8),(2,4),(3,5),(6,7)]=>[8,4,5,2,3,7,6,1]=>[4,7,5,2,1,8,3,6]=>[7,3,5,1,4,2,8,6]=>[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]=>[6,3,5,1,4,2,8,7]=>[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]=>[2,1,5,3,6,4,8,7]=>[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]=>[3,2,5,1,6,4,8,7]=>[1,2,2,2,1] [(1,4),(2,6),(3,5),(7,8)]=>[4,6,5,1,3,2,8,7]=>[6,4,5,3,1,2,8,7]=>[6,2,4,3,5,1,8,7]=>[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]=>[4,1,5,2,6,3,8,7]=>[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]=>[4,1,5,2,7,6,8,3]=>[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]=>[4,1,5,2,7,3,8,6]=>[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]=>[3,2,5,1,7,6,8,4]=>[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]=>[2,1,5,3,7,6,8,4]=>[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]=>[2,1,5,3,7,4,8,6]=>[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]=>[3,2,5,1,7,4,8,6]=>[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]=>[7,2,4,3,5,1,8,6]=>[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]=>[7,2,4,1,6,5,8,3]=>[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]=>[6,2,4,1,8,3,7,5]=>[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]=>[4,1,6,2,7,3,8,5]=>[1,2,2,2,1] [(1,8),(2,7),(3,5),(4,6)]=>[8,7,5,6,3,4,2,1]=>[5,6,8,7,2,1,3,4]=>[6,1,5,2,8,4,7,3]=>[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]=>[5,2,6,1,7,4,8,3]=>[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]=>[5,2,6,4,7,1,8,3]=>[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]=>[6,2,5,4,8,3,7,1]=>[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]=>[2,1,8,5,7,3,6,4]=>[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]=>[2,1,8,3,7,5,6,4]=>[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]=>[7,5,6,2,4,3,8,1]=>[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]=>[5,4,6,2,7,3,8,1]=>[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]=>[6,4,5,2,8,1,7,3]=>[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]=>[7,4,5,2,8,1,6,3]=>[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]=>[7,1,5,2,8,4,6,3]=>[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]=>[7,1,5,3,8,4,6,2]=>[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]=>[6,5,8,1,7,3,4,2]=>[1,2,2,2,1] [(1,7),(2,4),(3,6),(5,8)]=>[7,4,6,2,8,3,1,5]=>[4,8,7,2,6,3,5,1]=>[7,5,6,3,8,1,4,2]=>[1,2,2,2,1] [(1,7),(2,3),(4,6),(5,8)]=>[7,3,2,6,8,4,1,5]=>[3,7,1,8,6,4,5,2]=>[3,1,8,2,7,5,6,4]=>[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]=>[3,1,6,5,7,4,8,2]=>[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]=>[3,1,5,4,7,6,8,2]=>[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]=>[3,1,4,2,7,6,8,5]=>[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]=>[2,1,4,3,7,5,8,6]=>[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]=>[3,2,4,1,7,5,8,6]=>[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]=>[3,1,4,2,7,5,8,6]=>[1,2,2,2,1] [(1,5),(2,3),(4,8),(6,7)]=>[5,3,2,8,1,7,6,4]=>[3,7,1,5,4,8,2,6]=>[3,1,5,4,7,2,8,6]=>[1,2,2,2,1] [(1,6),(2,3),(4,8),(5,7)]=>[6,3,2,8,7,1,5,4]=>[3,8,1,6,7,5,2,4]=>[3,1,8,4,6,5,7,2]=>[1,2,2,2,1] [(1,7),(2,3),(4,8),(5,6)]=>[7,3,2,8,6,5,1,4]=>[3,7,1,6,8,2,5,4]=>[3,1,8,4,6,2,7,5]=>[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]=>[3,1,6,2,7,4,8,5]=>[1,2,2,2,1] [(1,8),(2,4),(3,7),(5,6)]=>[8,4,7,2,6,5,3,1]=>[4,7,6,2,8,1,3,5]=>[7,3,6,1,4,2,8,5]=>[1,2,2,2,1] [(1,7),(2,4),(3,8),(5,6)]=>[7,4,8,2,6,5,1,3]=>[6,7,4,2,8,1,5,3]=>[6,1,8,3,4,2,7,5]=>[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]=>[7,1,6,5,8,3,4,2]=>[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]=>[7,1,5,3,4,2,8,6]=>[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]=>[4,3,5,2,7,1,8,6]=>[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]=>[3,2,5,4,7,1,8,6]=>[1,2,2,2,1] [(1,2),(3,5),(4,8),(6,7)]=>[2,1,5,8,3,7,6,4]=>[2,1,7,5,4,8,3,6]=>[2,1,5,4,7,3,8,6]=>[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]=>[2,1,8,4,6,5,7,3]=>[1,2,2,2,1] [(1,4),(2,6),(3,8),(5,7)]=>[4,6,8,1,7,2,5,3]=>[8,6,4,3,7,5,1,2]=>[4,3,8,2,6,5,7,1]=>[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]=>[6,3,5,4,8,2,7,1]=>[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]=>[5,3,6,4,7,1,8,2]=>[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]=>[5,3,6,1,7,4,8,2]=>[1,2,2,2,1] [(1,8),(2,5),(3,7),(4,6)]=>[8,5,7,6,2,4,3,1]=>[5,8,6,7,3,1,2,4]=>[6,1,5,3,8,4,7,2]=>[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]=>[5,1,6,3,7,2,8,4]=>[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]=>[6,3,5,1,8,2,7,4]=>[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]=>[7,3,5,1,8,2,6,4]=>[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]=>[7,3,5,4,8,2,6,1]=>[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]=>[4,3,8,2,6,1,7,5]=>[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]=>[2,1,8,4,6,3,7,5]=>[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]=>[2,1,6,3,7,4,8,5]=>[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]=>[3,2,6,1,7,4,8,5]=>[1,2,2,2,1] [(1,4),(2,8),(3,7),(5,6)]=>[4,8,7,1,6,5,3,2]=>[7,4,6,3,8,1,2,5]=>[7,2,4,3,6,1,8,5]=>[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]=>[7,2,5,1,8,3,6,4]=>[1,2,2,2,1] [(1,7),(2,8),(3,6),(4,5)]=>[7,8,6,5,4,3,1,2]=>[6,5,7,8,1,2,4,3]=>[6,2,5,1,8,3,7,4]=>[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]=>[5,1,6,2,7,3,8,4]=>[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
inverse first fundamental transformation
Description
Let $\sigma = (i_{11}\cdots i_{1k_1})\cdots(i_{\ell 1}\cdots i_{\ell k_\ell})$ be a permutation given by cycle notation such that every cycle starts with its maximal entry, and all cycles are ordered increasingly by these maximal entries.
Maps $\sigma$ to the permutation $[i_{11},\ldots,i_{1k_1},\ldots,i_{\ell 1},\ldots,i_{\ell k_\ell}]$ in one-line notation.
In other words, this map sends the maximal entries of the cycles to the left-to-right maxima, and the sequences between two left-to-right maxima are given by the cycles.
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 $\{ i_1, i_1 + i_2, i_1 + i_2 + i_3, \ldots, i_1 + i_2 + \cdots + i_{k-1} \}$.