Identifier
Mp00058: to permutationPermutations
Mp00087: Permutations inverse first fundamental transformationPermutations
Mp00238: Permutations Clarke-Steingrimsson-ZengPermutations
Mp00239: Permutations CorteelPermutations
Images
=>
Cc0012;cc-rep-0
[(1,2)]=>[2,1]=>[2,1]=>[2,1]=>[2,1] [(1,2),(3,4)]=>[2,1,4,3]=>[2,1,4,3]=>[2,1,4,3]=>[2,1,4,3] [(1,3),(2,4)]=>[3,4,1,2]=>[3,1,4,2]=>[4,3,1,2]=>[3,4,2,1] [(1,4),(2,3)]=>[4,3,2,1]=>[3,2,4,1]=>[4,3,2,1]=>[3,4,1,2] [(1,2),(3,4),(5,6)]=>[2,1,4,3,6,5]=>[2,1,4,3,6,5]=>[2,1,4,3,6,5]=>[2,1,4,3,6,5] [(1,3),(2,4),(5,6)]=>[3,4,1,2,6,5]=>[3,1,4,2,6,5]=>[4,3,1,2,6,5]=>[3,4,2,1,6,5] [(1,4),(2,3),(5,6)]=>[4,3,2,1,6,5]=>[3,2,4,1,6,5]=>[4,3,2,1,6,5]=>[3,4,1,2,6,5] [(1,5),(2,3),(4,6)]=>[5,3,2,6,1,4]=>[3,2,5,1,6,4]=>[6,3,2,5,1,4]=>[3,5,1,6,4,2] [(1,6),(2,3),(4,5)]=>[6,3,2,5,4,1]=>[3,2,5,4,6,1]=>[6,3,2,5,4,1]=>[3,5,1,6,2,4] [(1,6),(2,4),(3,5)]=>[6,4,5,2,3,1]=>[4,2,5,3,6,1]=>[6,5,4,2,3,1]=>[4,5,6,2,1,3] [(1,5),(2,4),(3,6)]=>[5,4,6,2,1,3]=>[4,2,5,1,6,3]=>[6,5,4,2,1,3]=>[4,5,6,2,3,1] [(1,4),(2,5),(3,6)]=>[4,5,6,1,2,3]=>[4,1,5,2,6,3]=>[6,5,4,1,2,3]=>[4,5,6,3,2,1] [(1,3),(2,5),(4,6)]=>[3,5,1,6,2,4]=>[3,1,5,2,6,4]=>[6,3,1,5,2,4]=>[3,5,2,6,4,1] [(1,2),(3,5),(4,6)]=>[2,1,5,6,3,4]=>[2,1,5,3,6,4]=>[2,1,6,5,3,4]=>[2,1,5,6,4,3] [(1,2),(3,6),(4,5)]=>[2,1,6,5,4,3]=>[2,1,5,4,6,3]=>[2,1,6,5,4,3]=>[2,1,5,6,3,4] [(1,3),(2,6),(4,5)]=>[3,6,1,5,4,2]=>[3,1,5,4,6,2]=>[6,3,1,5,4,2]=>[3,5,2,6,1,4] [(1,4),(2,6),(3,5)]=>[4,6,5,1,3,2]=>[4,1,5,3,6,2]=>[6,5,4,1,3,2]=>[4,5,6,3,1,2] [(1,5),(2,6),(3,4)]=>[5,6,4,3,1,2]=>[4,3,5,1,6,2]=>[6,5,4,3,1,2]=>[4,5,6,1,3,2] [(1,6),(2,5),(3,4)]=>[6,5,4,3,2,1]=>[4,3,5,2,6,1]=>[6,5,4,3,2,1]=>[4,5,6,1,2,3] [(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]=>[2,1,4,3,6,5,8,7] [(1,3),(2,4),(5,6),(7,8)]=>[3,4,1,2,6,5,8,7]=>[3,1,4,2,6,5,8,7]=>[4,3,1,2,6,5,8,7]=>[3,4,2,1,6,5,8,7] [(1,4),(2,3),(5,6),(7,8)]=>[4,3,2,1,6,5,8,7]=>[3,2,4,1,6,5,8,7]=>[4,3,2,1,6,5,8,7]=>[3,4,1,2,6,5,8,7] [(1,5),(2,3),(4,6),(7,8)]=>[5,3,2,6,1,4,8,7]=>[3,2,5,1,6,4,8,7]=>[6,3,2,5,1,4,8,7]=>[3,5,1,6,4,2,8,7] [(1,6),(2,3),(4,5),(7,8)]=>[6,3,2,5,4,1,8,7]=>[3,2,5,4,6,1,8,7]=>[6,3,2,5,4,1,8,7]=>[3,5,1,6,2,4,8,7] [(1,7),(2,3),(4,5),(6,8)]=>[7,3,2,5,4,8,1,6]=>[3,2,5,4,7,1,8,6]=>[8,3,2,5,4,7,1,6]=>[3,5,1,7,2,8,6,4] [(1,8),(2,3),(4,5),(6,7)]=>[8,3,2,5,4,7,6,1]=>[3,2,5,4,7,6,8,1]=>[8,3,2,5,4,7,6,1]=>[3,5,1,7,2,8,4,6] [(1,8),(2,4),(3,5),(6,7)]=>[8,4,5,2,3,7,6,1]=>[4,2,5,3,7,6,8,1]=>[8,5,4,2,3,7,6,1]=>[4,5,7,2,1,8,3,6] [(1,7),(2,4),(3,5),(6,8)]=>[7,4,5,2,3,8,1,6]=>[4,2,5,3,7,1,8,6]=>[8,5,4,2,3,7,1,6]=>[4,5,7,2,1,8,6,3] [(1,6),(2,4),(3,5),(7,8)]=>[6,4,5,2,3,1,8,7]=>[4,2,5,3,6,1,8,7]=>[6,5,4,2,3,1,8,7]=>[4,5,6,2,1,3,8,7] [(1,5),(2,4),(3,6),(7,8)]=>[5,4,6,2,1,3,8,7]=>[4,2,5,1,6,3,8,7]=>[6,5,4,2,1,3,8,7]=>[4,5,6,2,3,1,8,7] [(1,4),(2,5),(3,6),(7,8)]=>[4,5,6,1,2,3,8,7]=>[4,1,5,2,6,3,8,7]=>[6,5,4,1,2,3,8,7]=>[4,5,6,3,2,1,8,7] [(1,3),(2,5),(4,6),(7,8)]=>[3,5,1,6,2,4,8,7]=>[3,1,5,2,6,4,8,7]=>[6,3,1,5,2,4,8,7]=>[3,5,2,6,4,1,8,7] [(1,2),(3,5),(4,6),(7,8)]=>[2,1,5,6,3,4,8,7]=>[2,1,5,3,6,4,8,7]=>[2,1,6,5,3,4,8,7]=>[2,1,5,6,4,3,8,7] [(1,2),(3,6),(4,5),(7,8)]=>[2,1,6,5,4,3,8,7]=>[2,1,5,4,6,3,8,7]=>[2,1,6,5,4,3,8,7]=>[2,1,5,6,3,4,8,7] [(1,3),(2,6),(4,5),(7,8)]=>[3,6,1,5,4,2,8,7]=>[3,1,5,4,6,2,8,7]=>[6,3,1,5,4,2,8,7]=>[3,5,2,6,1,4,8,7] [(1,4),(2,6),(3,5),(7,8)]=>[4,6,5,1,3,2,8,7]=>[4,1,5,3,6,2,8,7]=>[6,5,4,1,3,2,8,7]=>[4,5,6,3,1,2,8,7] [(1,6),(2,5),(3,4),(7,8)]=>[6,5,4,3,2,1,8,7]=>[4,3,5,2,6,1,8,7]=>[6,5,4,3,2,1,8,7]=>[4,5,6,1,2,3,8,7] [(1,7),(2,5),(3,4),(6,8)]=>[7,5,4,3,2,8,1,6]=>[4,3,5,2,7,1,8,6]=>[8,5,4,3,2,7,1,6]=>[4,5,7,1,2,8,6,3] [(1,8),(2,5),(3,4),(6,7)]=>[8,5,4,3,2,7,6,1]=>[4,3,5,2,7,6,8,1]=>[8,5,4,3,2,7,6,1]=>[4,5,7,1,2,8,3,6] [(1,5),(2,7),(3,4),(6,8)]=>[5,7,4,3,1,8,2,6]=>[4,3,5,1,7,2,8,6]=>[8,5,4,3,1,7,2,6]=>[4,5,7,1,3,8,6,2] [(1,4),(2,7),(3,5),(6,8)]=>[4,7,5,1,3,8,2,6]=>[4,1,5,3,7,2,8,6]=>[8,5,4,1,3,7,2,6]=>[4,5,7,3,1,8,6,2] [(1,3),(2,7),(4,5),(6,8)]=>[3,7,1,5,4,8,2,6]=>[3,1,5,4,7,2,8,6]=>[8,3,1,5,4,7,2,6]=>[3,5,2,7,1,8,6,4] [(1,2),(3,7),(4,5),(6,8)]=>[2,1,7,5,4,8,3,6]=>[2,1,5,4,7,3,8,6]=>[2,1,8,5,4,7,3,6]=>[2,1,5,7,3,8,6,4] [(1,2),(3,8),(4,5),(6,7)]=>[2,1,8,5,4,7,6,3]=>[2,1,5,4,7,6,8,3]=>[2,1,8,5,4,7,6,3]=>[2,1,5,7,3,8,4,6] [(1,3),(2,8),(4,5),(6,7)]=>[3,8,1,5,4,7,6,2]=>[3,1,5,4,7,6,8,2]=>[8,3,1,5,4,7,6,2]=>[3,5,2,7,1,8,4,6] [(1,4),(2,8),(3,5),(6,7)]=>[4,8,5,1,3,7,6,2]=>[4,1,5,3,7,6,8,2]=>[8,5,4,1,3,7,6,2]=>[4,5,7,3,1,8,2,6] [(1,7),(2,8),(3,4),(5,6)]=>[7,8,4,3,6,5,1,2]=>[4,3,6,5,7,1,8,2]=>[8,7,4,3,6,5,1,2]=>[4,6,7,1,8,2,5,3] [(1,8),(2,7),(3,4),(5,6)]=>[8,7,4,3,6,5,2,1]=>[4,3,6,5,7,2,8,1]=>[8,7,4,3,6,5,2,1]=>[4,6,7,1,8,2,3,5] [(1,7),(2,8),(3,5),(4,6)]=>[7,8,5,6,3,4,1,2]=>[5,3,6,4,7,1,8,2]=>[8,7,6,5,3,4,1,2]=>[5,6,7,8,2,1,4,3] [(1,5),(2,8),(3,6),(4,7)]=>[5,8,6,7,1,3,4,2]=>[5,1,6,3,7,4,8,2]=>[8,7,6,5,1,3,4,2]=>[5,6,7,8,4,2,1,3] [(1,4),(2,8),(3,6),(5,7)]=>[4,8,6,1,7,3,5,2]=>[4,1,6,3,7,5,8,2]=>[8,7,4,1,6,3,5,2]=>[4,6,7,3,8,2,1,5] [(1,3),(2,8),(4,6),(5,7)]=>[3,8,1,6,7,4,5,2]=>[3,1,6,4,7,5,8,2]=>[8,3,1,7,6,4,5,2]=>[3,6,2,7,8,4,1,5] [(1,2),(3,8),(4,6),(5,7)]=>[2,1,8,6,7,4,5,3]=>[2,1,6,4,7,5,8,3]=>[2,1,8,7,6,4,5,3]=>[2,1,6,7,8,4,3,5] [(1,2),(3,7),(4,6),(5,8)]=>[2,1,7,6,8,4,3,5]=>[2,1,6,4,7,3,8,5]=>[2,1,8,7,6,4,3,5]=>[2,1,6,7,8,4,5,3] [(1,3),(2,7),(4,6),(5,8)]=>[3,7,1,6,8,4,2,5]=>[3,1,6,4,7,2,8,5]=>[8,3,1,7,6,4,2,5]=>[3,6,2,7,8,4,5,1] [(1,4),(2,7),(3,6),(5,8)]=>[4,7,6,1,8,3,2,5]=>[4,1,6,3,7,2,8,5]=>[8,7,4,1,6,3,2,5]=>[4,6,7,3,8,2,5,1] [(1,5),(2,7),(3,6),(4,8)]=>[5,7,6,8,1,3,2,4]=>[5,1,6,3,7,2,8,4]=>[8,7,6,5,1,3,2,4]=>[5,6,7,8,4,2,3,1] [(1,8),(2,6),(3,5),(4,7)]=>[8,6,5,7,3,2,4,1]=>[5,3,6,2,7,4,8,1]=>[8,7,6,5,3,2,4,1]=>[5,6,7,8,2,3,1,4] [(1,8),(2,5),(3,6),(4,7)]=>[8,5,6,7,2,3,4,1]=>[5,2,6,3,7,4,8,1]=>[8,7,6,5,2,3,4,1]=>[5,6,7,8,3,2,1,4] [(1,7),(2,5),(3,6),(4,8)]=>[7,5,6,8,2,3,1,4]=>[5,2,6,3,7,1,8,4]=>[8,7,6,5,2,3,1,4]=>[5,6,7,8,3,2,4,1] [(1,6),(2,5),(3,7),(4,8)]=>[6,5,7,8,2,1,3,4]=>[5,2,6,1,7,3,8,4]=>[8,7,6,5,2,1,3,4]=>[5,6,7,8,3,4,2,1] [(1,5),(2,6),(3,7),(4,8)]=>[5,6,7,8,1,2,3,4]=>[5,1,6,2,7,3,8,4]=>[8,7,6,5,1,2,3,4]=>[5,6,7,8,4,3,2,1] [(1,4),(2,6),(3,7),(5,8)]=>[4,6,7,1,8,2,3,5]=>[4,1,6,2,7,3,8,5]=>[8,7,4,1,6,2,3,5]=>[4,6,7,3,8,5,2,1] [(1,3),(2,6),(4,7),(5,8)]=>[3,6,1,7,8,2,4,5]=>[3,1,6,2,7,4,8,5]=>[8,3,1,7,6,2,4,5]=>[3,6,2,7,8,5,4,1] [(1,2),(3,6),(4,7),(5,8)]=>[2,1,6,7,8,3,4,5]=>[2,1,6,3,7,4,8,5]=>[2,1,8,7,6,3,4,5]=>[2,1,6,7,8,5,4,3] [(1,2),(3,5),(4,7),(6,8)]=>[2,1,5,7,3,8,4,6]=>[2,1,5,3,7,4,8,6]=>[2,1,8,5,3,7,4,6]=>[2,1,5,7,4,8,6,3] [(1,3),(2,5),(4,7),(6,8)]=>[3,5,1,7,2,8,4,6]=>[3,1,5,2,7,4,8,6]=>[8,3,1,5,2,7,4,6]=>[3,5,2,7,4,8,6,1] [(1,4),(2,5),(3,7),(6,8)]=>[4,5,7,1,2,8,3,6]=>[4,1,5,2,7,3,8,6]=>[8,5,4,1,2,7,3,6]=>[4,5,7,3,2,8,6,1] [(1,5),(2,4),(3,7),(6,8)]=>[5,4,7,2,1,8,3,6]=>[4,2,5,1,7,3,8,6]=>[8,5,4,2,1,7,3,6]=>[4,5,7,2,3,8,6,1] [(1,6),(2,4),(3,7),(5,8)]=>[6,4,7,2,8,1,3,5]=>[4,2,6,1,7,3,8,5]=>[8,7,4,2,6,1,3,5]=>[4,6,7,2,8,5,3,1] [(1,7),(2,4),(3,6),(5,8)]=>[7,4,6,2,8,3,1,5]=>[4,2,6,3,7,1,8,5]=>[8,7,4,2,6,3,1,5]=>[4,6,7,2,8,3,5,1] [(1,8),(2,4),(3,6),(5,7)]=>[8,4,6,2,7,3,5,1]=>[4,2,6,3,7,5,8,1]=>[8,7,4,2,6,3,5,1]=>[4,6,7,2,8,3,1,5] [(1,8),(2,3),(4,6),(5,7)]=>[8,3,2,6,7,4,5,1]=>[3,2,6,4,7,5,8,1]=>[8,3,2,7,6,4,5,1]=>[3,6,1,7,8,4,2,5] [(1,7),(2,3),(4,6),(5,8)]=>[7,3,2,6,8,4,1,5]=>[3,2,6,4,7,1,8,5]=>[8,3,2,7,6,4,1,5]=>[3,6,1,7,8,4,5,2] [(1,6),(2,3),(4,7),(5,8)]=>[6,3,2,7,8,1,4,5]=>[3,2,6,1,7,4,8,5]=>[8,3,2,7,6,1,4,5]=>[3,6,1,7,8,5,4,2] [(1,5),(2,3),(4,7),(6,8)]=>[5,3,2,7,1,8,4,6]=>[3,2,5,1,7,4,8,6]=>[8,3,2,5,1,7,4,6]=>[3,5,1,7,4,8,6,2] [(1,4),(2,3),(5,7),(6,8)]=>[4,3,2,1,7,8,5,6]=>[3,2,4,1,7,5,8,6]=>[4,3,2,1,8,7,5,6]=>[3,4,1,2,7,8,6,5] [(1,3),(2,4),(5,7),(6,8)]=>[3,4,1,2,7,8,5,6]=>[3,1,4,2,7,5,8,6]=>[4,3,1,2,8,7,5,6]=>[3,4,2,1,7,8,6,5] [(1,2),(3,4),(5,7),(6,8)]=>[2,1,4,3,7,8,5,6]=>[2,1,4,3,7,5,8,6]=>[2,1,4,3,8,7,5,6]=>[2,1,4,3,7,8,6,5] [(1,2),(3,4),(5,8),(6,7)]=>[2,1,4,3,8,7,6,5]=>[2,1,4,3,7,6,8,5]=>[2,1,4,3,8,7,6,5]=>[2,1,4,3,7,8,5,6] [(1,3),(2,4),(5,8),(6,7)]=>[3,4,1,2,8,7,6,5]=>[3,1,4,2,7,6,8,5]=>[4,3,1,2,8,7,6,5]=>[3,4,2,1,7,8,5,6] [(1,4),(2,3),(5,8),(6,7)]=>[4,3,2,1,8,7,6,5]=>[3,2,4,1,7,6,8,5]=>[4,3,2,1,8,7,6,5]=>[3,4,1,2,7,8,5,6] [(1,5),(2,3),(4,8),(6,7)]=>[5,3,2,8,1,7,6,4]=>[3,2,5,1,7,6,8,4]=>[8,3,2,5,1,7,6,4]=>[3,5,1,7,4,8,2,6] [(1,6),(2,3),(4,8),(5,7)]=>[6,3,2,8,7,1,5,4]=>[3,2,6,1,7,5,8,4]=>[8,3,2,7,6,1,5,4]=>[3,6,1,7,8,5,2,4] [(1,8),(2,3),(4,7),(5,6)]=>[8,3,2,7,6,5,4,1]=>[3,2,6,5,7,4,8,1]=>[8,3,2,7,6,5,4,1]=>[3,6,1,7,8,2,4,5] [(1,6),(2,4),(3,8),(5,7)]=>[6,4,8,2,7,1,5,3]=>[4,2,6,1,7,5,8,3]=>[8,7,4,2,6,1,5,3]=>[4,6,7,2,8,5,1,3] [(1,5),(2,4),(3,8),(6,7)]=>[5,4,8,2,1,7,6,3]=>[4,2,5,1,7,6,8,3]=>[8,5,4,2,1,7,6,3]=>[4,5,7,2,3,8,1,6] [(1,4),(2,5),(3,8),(6,7)]=>[4,5,8,1,2,7,6,3]=>[4,1,5,2,7,6,8,3]=>[8,5,4,1,2,7,6,3]=>[4,5,7,3,2,8,1,6] [(1,3),(2,5),(4,8),(6,7)]=>[3,5,1,8,2,7,6,4]=>[3,1,5,2,7,6,8,4]=>[8,3,1,5,2,7,6,4]=>[3,5,2,7,4,8,1,6] [(1,2),(3,5),(4,8),(6,7)]=>[2,1,5,8,3,7,6,4]=>[2,1,5,3,7,6,8,4]=>[2,1,8,5,3,7,6,4]=>[2,1,5,7,4,8,3,6] [(1,2),(3,6),(4,8),(5,7)]=>[2,1,6,8,7,3,5,4]=>[2,1,6,3,7,5,8,4]=>[2,1,8,7,6,3,5,4]=>[2,1,6,7,8,5,3,4] [(1,3),(2,6),(4,8),(5,7)]=>[3,6,1,8,7,2,5,4]=>[3,1,6,2,7,5,8,4]=>[8,3,1,7,6,2,5,4]=>[3,6,2,7,8,5,1,4] [(1,4),(2,6),(3,8),(5,7)]=>[4,6,8,1,7,2,5,3]=>[4,1,6,2,7,5,8,3]=>[8,7,4,1,6,2,5,3]=>[4,6,7,3,8,5,1,2] [(1,5),(2,6),(3,8),(4,7)]=>[5,6,8,7,1,2,4,3]=>[5,1,6,2,7,4,8,3]=>[8,7,6,5,1,2,4,3]=>[5,6,7,8,4,3,1,2] [(1,6),(2,5),(3,8),(4,7)]=>[6,5,8,7,2,1,4,3]=>[5,2,6,1,7,4,8,3]=>[8,7,6,5,2,1,4,3]=>[5,6,7,8,3,4,1,2] [(1,7),(2,6),(3,8),(4,5)]=>[7,6,8,5,4,2,1,3]=>[5,4,6,2,7,1,8,3]=>[8,7,6,5,4,2,1,3]=>[5,6,7,8,1,3,4,2] [(1,5),(2,7),(3,8),(4,6)]=>[5,7,8,6,1,4,2,3]=>[5,1,6,4,7,2,8,3]=>[8,7,6,5,1,4,2,3]=>[5,6,7,8,4,1,3,2] [(1,3),(2,7),(4,8),(5,6)]=>[3,7,1,8,6,5,2,4]=>[3,1,6,5,7,2,8,4]=>[8,3,1,7,6,5,2,4]=>[3,6,2,7,8,1,5,4] [(1,2),(3,8),(4,7),(5,6)]=>[2,1,8,7,6,5,4,3]=>[2,1,6,5,7,4,8,3]=>[2,1,8,7,6,5,4,3]=>[2,1,6,7,8,3,4,5] [(1,3),(2,8),(4,7),(5,6)]=>[3,8,1,7,6,5,4,2]=>[3,1,6,5,7,4,8,2]=>[8,3,1,7,6,5,4,2]=>[3,6,2,7,8,1,4,5] [(1,4),(2,8),(3,7),(5,6)]=>[4,8,7,1,6,5,3,2]=>[4,1,6,5,7,3,8,2]=>[8,7,4,1,6,5,3,2]=>[4,6,7,3,8,1,2,5] [(1,5),(2,8),(3,7),(4,6)]=>[5,8,7,6,1,4,3,2]=>[5,1,6,4,7,3,8,2]=>[8,7,6,5,1,4,3,2]=>[5,6,7,8,4,1,2,3] [(1,8),(2,7),(3,6),(4,5)]=>[8,7,6,5,4,3,2,1]=>[5,4,6,3,7,2,8,1]=>[8,7,6,5,4,3,2,1]=>[5,6,7,8,1,2,3,4] [(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]=>[2,1,4,3,6,5,8,7,10,9] [(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]=>[2,1,4,3,6,5,8,7,10,9,12,11]
Map
to permutation
Description
Returns the fixed point free involution whose transpositions are the pairs in the perfect matching.
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
Clarke-Steingrimsson-Zeng
Description
The Clarke-Steingrimsson-Zeng map sending descents to excedances.
This is the map $\Phi$ in [1, sec.3]. In particular, it satisfies
$$(des, Dbot, Ddif, Res)\pi = (exc, Ebot, Edif, Ine)\Phi(\pi),$$
where
• $des$ is the number of descents, St000021The number of descents of a permutation.,
• $exc$ is the number of (strict) excedances, St000155The number of exceedances (also excedences) of a permutation.,
• $Dbot$ is the sum of the descent bottoms, St000154The sum of the descent bottoms of a permutation.,
• $Ebot$ is the sum of the excedance bottoms,
• $Ddif$ is the sum of the descent differences, St000030The sum of the descent differences of a permutations.,
• $Edif$ is the sum of the excedance differences (or depth), St000029The depth of a permutation.,
• $Res$ is the sum of the (right) embracing numbers,
• $Ine$ is the sum of the side numbers.
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.