Identifier
Mp00058: to permutationPermutations
Mp00068: Permutations Simion-Schmidt mapPermutations
Mp00151: Permutations to cycle type
Images
=>
Cc0012;cc-rep-0Cc0009;cc-rep-3
[(1,2)]=>[2,1]=>[2,1]=>{{1,2}} [(1,2),(3,4)]=>[2,1,4,3]=>[2,1,4,3]=>{{1,2},{3,4}} [(1,3),(2,4)]=>[3,4,1,2]=>[3,4,1,2]=>{{1,3},{2,4}} [(1,4),(2,3)]=>[4,3,2,1]=>[4,3,2,1]=>{{1,4},{2,3}} [(1,2),(3,4),(5,6)]=>[2,1,4,3,6,5]=>[2,1,6,5,4,3]=>{{1,2},{3,6},{4,5}} [(1,3),(2,4),(5,6)]=>[3,4,1,2,6,5]=>[3,6,1,5,4,2]=>{{1,3},{2,6},{4,5}} [(1,4),(2,3),(5,6)]=>[4,3,2,1,6,5]=>[4,3,2,1,6,5]=>{{1,4},{2,3},{5,6}} [(1,5),(2,3),(4,6)]=>[5,3,2,6,1,4]=>[5,3,2,6,1,4]=>{{1,5},{2,3},{4,6}} [(1,6),(2,3),(4,5)]=>[6,3,2,5,4,1]=>[6,3,2,5,4,1]=>{{1,6},{2,3},{4,5}} [(1,6),(2,4),(3,5)]=>[6,4,5,2,3,1]=>[6,4,5,2,3,1]=>{{1,6},{2,4},{3,5}} [(1,5),(2,4),(3,6)]=>[5,4,6,2,1,3]=>[5,4,6,2,1,3]=>{{1,5},{2,4},{3,6}} [(1,4),(2,5),(3,6)]=>[4,5,6,1,2,3]=>[4,6,5,1,3,2]=>{{1,4},{2,6},{3,5}} [(1,3),(2,5),(4,6)]=>[3,5,1,6,2,4]=>[3,6,1,5,4,2]=>{{1,3},{2,6},{4,5}} [(1,2),(3,5),(4,6)]=>[2,1,5,6,3,4]=>[2,1,6,5,4,3]=>{{1,2},{3,6},{4,5}} [(1,2),(3,6),(4,5)]=>[2,1,6,5,4,3]=>[2,1,6,5,4,3]=>{{1,2},{3,6},{4,5}} [(1,3),(2,6),(4,5)]=>[3,6,1,5,4,2]=>[3,6,1,5,4,2]=>{{1,3},{2,6},{4,5}} [(1,4),(2,6),(3,5)]=>[4,6,5,1,3,2]=>[4,6,5,1,3,2]=>{{1,4},{2,6},{3,5}} [(1,5),(2,6),(3,4)]=>[5,6,4,3,1,2]=>[5,6,4,3,1,2]=>{{1,5},{2,6},{3,4}} [(1,6),(2,5),(3,4)]=>[6,5,4,3,2,1]=>[6,5,4,3,2,1]=>{{1,6},{2,5},{3,4}} [(1,2),(3,4),(5,6),(7,8)]=>[2,1,4,3,6,5,8,7]=>[2,1,8,7,6,5,4,3]=>{{1,2},{3,8},{4,7},{5,6}} [(1,3),(2,4),(5,6),(7,8)]=>[3,4,1,2,6,5,8,7]=>[3,8,1,7,6,5,4,2]=>{{1,3},{2,8},{4,7},{5,6}} [(1,4),(2,3),(5,6),(7,8)]=>[4,3,2,1,6,5,8,7]=>[4,3,2,1,8,7,6,5]=>{{1,4},{2,3},{5,8},{6,7}} [(1,5),(2,3),(4,6),(7,8)]=>[5,3,2,6,1,4,8,7]=>[5,3,2,8,1,7,6,4]=>{{1,5},{2,3},{4,8},{6,7}} [(1,6),(2,3),(4,5),(7,8)]=>[6,3,2,5,4,1,8,7]=>[6,3,2,8,7,1,5,4]=>{{1,6},{2,3},{4,8},{5,7}} [(1,7),(2,3),(4,5),(6,8)]=>[7,3,2,5,4,8,1,6]=>[7,3,2,8,6,5,1,4]=>{{1,7},{2,3},{4,8},{5,6}} [(1,8),(2,3),(4,5),(6,7)]=>[8,3,2,5,4,7,6,1]=>[8,3,2,7,6,5,4,1]=>{{1,8},{2,3},{4,7},{5,6}} [(1,8),(2,4),(3,5),(6,7)]=>[8,4,5,2,3,7,6,1]=>[8,4,7,2,6,5,3,1]=>{{1,8},{2,4},{3,7},{5,6}} [(1,7),(2,4),(3,5),(6,8)]=>[7,4,5,2,3,8,1,6]=>[7,4,8,2,6,5,1,3]=>{{1,7},{2,4},{3,8},{5,6}} [(1,6),(2,4),(3,5),(7,8)]=>[6,4,5,2,3,1,8,7]=>[6,4,8,2,7,1,5,3]=>{{1,6},{2,4},{3,8},{5,7}} [(1,5),(2,4),(3,6),(7,8)]=>[5,4,6,2,1,3,8,7]=>[5,4,8,2,1,7,6,3]=>{{1,5},{2,4},{3,8},{6,7}} [(1,4),(2,5),(3,6),(7,8)]=>[4,5,6,1,2,3,8,7]=>[4,8,7,1,6,5,3,2]=>{{1,4},{2,8},{3,7},{5,6}} [(1,3),(2,5),(4,6),(7,8)]=>[3,5,1,6,2,4,8,7]=>[3,8,1,7,6,5,4,2]=>{{1,3},{2,8},{4,7},{5,6}} [(1,2),(3,5),(4,6),(7,8)]=>[2,1,5,6,3,4,8,7]=>[2,1,8,7,6,5,4,3]=>{{1,2},{3,8},{4,7},{5,6}} [(1,2),(3,6),(4,5),(7,8)]=>[2,1,6,5,4,3,8,7]=>[2,1,8,7,6,5,4,3]=>{{1,2},{3,8},{4,7},{5,6}} [(1,3),(2,6),(4,5),(7,8)]=>[3,6,1,5,4,2,8,7]=>[3,8,1,7,6,5,4,2]=>{{1,3},{2,8},{4,7},{5,6}} [(1,4),(2,6),(3,5),(7,8)]=>[4,6,5,1,3,2,8,7]=>[4,8,7,1,6,5,3,2]=>{{1,4},{2,8},{3,7},{5,6}} [(1,5),(2,6),(3,4),(7,8)]=>[5,6,4,3,1,2,8,7]=>[5,8,4,3,1,7,6,2]=>{{1,5},{2,8},{3,4},{6,7}} [(1,6),(2,5),(3,4),(7,8)]=>[6,5,4,3,2,1,8,7]=>[6,5,4,3,2,1,8,7]=>{{1,6},{2,5},{3,4},{7,8}} [(1,7),(2,5),(3,4),(6,8)]=>[7,5,4,3,2,8,1,6]=>[7,5,4,3,2,8,1,6]=>{{1,7},{2,5},{3,4},{6,8}} [(1,8),(2,5),(3,4),(6,7)]=>[8,5,4,3,2,7,6,1]=>[8,5,4,3,2,7,6,1]=>{{1,8},{2,5},{3,4},{6,7}} [(1,8),(2,6),(3,4),(5,7)]=>[8,6,4,3,7,2,5,1]=>[8,6,4,3,7,2,5,1]=>{{1,8},{2,6},{3,4},{5,7}} [(1,7),(2,6),(3,4),(5,8)]=>[7,6,4,3,8,2,1,5]=>[7,6,4,3,8,2,1,5]=>{{1,7},{2,6},{3,4},{5,8}} [(1,6),(2,7),(3,4),(5,8)]=>[6,7,4,3,8,1,2,5]=>[6,8,4,3,7,1,5,2]=>{{1,6},{2,8},{3,4},{5,7}} [(1,5),(2,7),(3,4),(6,8)]=>[5,7,4,3,1,8,2,6]=>[5,8,4,3,1,7,6,2]=>{{1,5},{2,8},{3,4},{6,7}} [(1,4),(2,7),(3,5),(6,8)]=>[4,7,5,1,3,8,2,6]=>[4,8,7,1,6,5,3,2]=>{{1,4},{2,8},{3,7},{5,6}} [(1,3),(2,7),(4,5),(6,8)]=>[3,7,1,5,4,8,2,6]=>[3,8,1,7,6,5,4,2]=>{{1,3},{2,8},{4,7},{5,6}} [(1,2),(3,7),(4,5),(6,8)]=>[2,1,7,5,4,8,3,6]=>[2,1,8,7,6,5,4,3]=>{{1,2},{3,8},{4,7},{5,6}} [(1,2),(3,8),(4,5),(6,7)]=>[2,1,8,5,4,7,6,3]=>[2,1,8,7,6,5,4,3]=>{{1,2},{3,8},{4,7},{5,6}} [(1,3),(2,8),(4,5),(6,7)]=>[3,8,1,5,4,7,6,2]=>[3,8,1,7,6,5,4,2]=>{{1,3},{2,8},{4,7},{5,6}} [(1,4),(2,8),(3,5),(6,7)]=>[4,8,5,1,3,7,6,2]=>[4,8,7,1,6,5,3,2]=>{{1,4},{2,8},{3,7},{5,6}} [(1,5),(2,8),(3,4),(6,7)]=>[5,8,4,3,1,7,6,2]=>[5,8,4,3,1,7,6,2]=>{{1,5},{2,8},{3,4},{6,7}} [(1,6),(2,8),(3,4),(5,7)]=>[6,8,4,3,7,1,5,2]=>[6,8,4,3,7,1,5,2]=>{{1,6},{2,8},{3,4},{5,7}} [(1,7),(2,8),(3,4),(5,6)]=>[7,8,4,3,6,5,1,2]=>[7,8,4,3,6,5,1,2]=>{{1,7},{2,8},{3,4},{5,6}} [(1,8),(2,7),(3,4),(5,6)]=>[8,7,4,3,6,5,2,1]=>[8,7,4,3,6,5,2,1]=>{{1,8},{2,7},{3,4},{5,6}} [(1,8),(2,7),(3,5),(4,6)]=>[8,7,5,6,3,4,2,1]=>[8,7,5,6,3,4,2,1]=>{{1,8},{2,7},{3,5},{4,6}} [(1,7),(2,8),(3,5),(4,6)]=>[7,8,5,6,3,4,1,2]=>[7,8,5,6,3,4,1,2]=>{{1,7},{2,8},{3,5},{4,6}} [(1,6),(2,8),(3,5),(4,7)]=>[6,8,5,7,3,1,4,2]=>[6,8,5,7,3,1,4,2]=>{{1,6},{2,8},{3,5},{4,7}} [(1,5),(2,8),(3,6),(4,7)]=>[5,8,6,7,1,3,4,2]=>[5,8,7,6,1,4,3,2]=>{{1,5},{2,8},{3,7},{4,6}} [(1,4),(2,8),(3,6),(5,7)]=>[4,8,6,1,7,3,5,2]=>[4,8,7,1,6,5,3,2]=>{{1,4},{2,8},{3,7},{5,6}} [(1,3),(2,8),(4,6),(5,7)]=>[3,8,1,6,7,4,5,2]=>[3,8,1,7,6,5,4,2]=>{{1,3},{2,8},{4,7},{5,6}} [(1,2),(3,8),(4,6),(5,7)]=>[2,1,8,6,7,4,5,3]=>[2,1,8,7,6,5,4,3]=>{{1,2},{3,8},{4,7},{5,6}} [(1,2),(3,7),(4,6),(5,8)]=>[2,1,7,6,8,4,3,5]=>[2,1,8,7,6,5,4,3]=>{{1,2},{3,8},{4,7},{5,6}} [(1,3),(2,7),(4,6),(5,8)]=>[3,7,1,6,8,4,2,5]=>[3,8,1,7,6,5,4,2]=>{{1,3},{2,8},{4,7},{5,6}} [(1,4),(2,7),(3,6),(5,8)]=>[4,7,6,1,8,3,2,5]=>[4,8,7,1,6,5,3,2]=>{{1,4},{2,8},{3,7},{5,6}} [(1,5),(2,7),(3,6),(4,8)]=>[5,7,6,8,1,3,2,4]=>[5,8,7,6,1,4,3,2]=>{{1,5},{2,8},{3,7},{4,6}} [(1,6),(2,7),(3,5),(4,8)]=>[6,7,5,8,3,1,2,4]=>[6,8,5,7,3,1,4,2]=>{{1,6},{2,8},{3,5},{4,7}} [(1,7),(2,6),(3,5),(4,8)]=>[7,6,5,8,3,2,1,4]=>[7,6,5,8,3,2,1,4]=>{{1,7},{2,6},{3,5},{4,8}} [(1,8),(2,6),(3,5),(4,7)]=>[8,6,5,7,3,2,4,1]=>[8,6,5,7,3,2,4,1]=>{{1,8},{2,6},{3,5},{4,7}} [(1,8),(2,5),(3,6),(4,7)]=>[8,5,6,7,2,3,4,1]=>[8,5,7,6,2,4,3,1]=>{{1,8},{2,5},{3,7},{4,6}} [(1,7),(2,5),(3,6),(4,8)]=>[7,5,6,8,2,3,1,4]=>[7,5,8,6,2,4,1,3]=>{{1,7},{2,5},{3,8},{4,6}} [(1,6),(2,5),(3,7),(4,8)]=>[6,5,7,8,2,1,3,4]=>[6,5,8,7,2,1,4,3]=>{{1,6},{2,5},{3,8},{4,7}} [(1,5),(2,6),(3,7),(4,8)]=>[5,6,7,8,1,2,3,4]=>[5,8,7,6,1,4,3,2]=>{{1,5},{2,8},{3,7},{4,6}} [(1,4),(2,6),(3,7),(5,8)]=>[4,6,7,1,8,2,3,5]=>[4,8,7,1,6,5,3,2]=>{{1,4},{2,8},{3,7},{5,6}} [(1,3),(2,6),(4,7),(5,8)]=>[3,6,1,7,8,2,4,5]=>[3,8,1,7,6,5,4,2]=>{{1,3},{2,8},{4,7},{5,6}} [(1,2),(3,6),(4,7),(5,8)]=>[2,1,6,7,8,3,4,5]=>[2,1,8,7,6,5,4,3]=>{{1,2},{3,8},{4,7},{5,6}} [(1,2),(3,5),(4,7),(6,8)]=>[2,1,5,7,3,8,4,6]=>[2,1,8,7,6,5,4,3]=>{{1,2},{3,8},{4,7},{5,6}} [(1,3),(2,5),(4,7),(6,8)]=>[3,5,1,7,2,8,4,6]=>[3,8,1,7,6,5,4,2]=>{{1,3},{2,8},{4,7},{5,6}} [(1,4),(2,5),(3,7),(6,8)]=>[4,5,7,1,2,8,3,6]=>[4,8,7,1,6,5,3,2]=>{{1,4},{2,8},{3,7},{5,6}} [(1,5),(2,4),(3,7),(6,8)]=>[5,4,7,2,1,8,3,6]=>[5,4,8,2,1,7,6,3]=>{{1,5},{2,4},{3,8},{6,7}} [(1,6),(2,4),(3,7),(5,8)]=>[6,4,7,2,8,1,3,5]=>[6,4,8,2,7,1,5,3]=>{{1,6},{2,4},{3,8},{5,7}} [(1,7),(2,4),(3,6),(5,8)]=>[7,4,6,2,8,3,1,5]=>[7,4,8,2,6,5,1,3]=>{{1,7},{2,4},{3,8},{5,6}} [(1,8),(2,4),(3,6),(5,7)]=>[8,4,6,2,7,3,5,1]=>[8,4,7,2,6,5,3,1]=>{{1,8},{2,4},{3,7},{5,6}} [(1,8),(2,3),(4,6),(5,7)]=>[8,3,2,6,7,4,5,1]=>[8,3,2,7,6,5,4,1]=>{{1,8},{2,3},{4,7},{5,6}} [(1,7),(2,3),(4,6),(5,8)]=>[7,3,2,6,8,4,1,5]=>[7,3,2,8,6,5,1,4]=>{{1,7},{2,3},{4,8},{5,6}} [(1,6),(2,3),(4,7),(5,8)]=>[6,3,2,7,8,1,4,5]=>[6,3,2,8,7,1,5,4]=>{{1,6},{2,3},{4,8},{5,7}} [(1,5),(2,3),(4,7),(6,8)]=>[5,3,2,7,1,8,4,6]=>[5,3,2,8,1,7,6,4]=>{{1,5},{2,3},{4,8},{6,7}} [(1,4),(2,3),(5,7),(6,8)]=>[4,3,2,1,7,8,5,6]=>[4,3,2,1,8,7,6,5]=>{{1,4},{2,3},{5,8},{6,7}} [(1,3),(2,4),(5,7),(6,8)]=>[3,4,1,2,7,8,5,6]=>[3,8,1,7,6,5,4,2]=>{{1,3},{2,8},{4,7},{5,6}} [(1,2),(3,4),(5,7),(6,8)]=>[2,1,4,3,7,8,5,6]=>[2,1,8,7,6,5,4,3]=>{{1,2},{3,8},{4,7},{5,6}} [(1,2),(3,4),(5,8),(6,7)]=>[2,1,4,3,8,7,6,5]=>[2,1,8,7,6,5,4,3]=>{{1,2},{3,8},{4,7},{5,6}} [(1,3),(2,4),(5,8),(6,7)]=>[3,4,1,2,8,7,6,5]=>[3,8,1,7,6,5,4,2]=>{{1,3},{2,8},{4,7},{5,6}} [(1,4),(2,3),(5,8),(6,7)]=>[4,3,2,1,8,7,6,5]=>[4,3,2,1,8,7,6,5]=>{{1,4},{2,3},{5,8},{6,7}} [(1,5),(2,3),(4,8),(6,7)]=>[5,3,2,8,1,7,6,4]=>[5,3,2,8,1,7,6,4]=>{{1,5},{2,3},{4,8},{6,7}} [(1,6),(2,3),(4,8),(5,7)]=>[6,3,2,8,7,1,5,4]=>[6,3,2,8,7,1,5,4]=>{{1,6},{2,3},{4,8},{5,7}} [(1,7),(2,3),(4,8),(5,6)]=>[7,3,2,8,6,5,1,4]=>[7,3,2,8,6,5,1,4]=>{{1,7},{2,3},{4,8},{5,6}} [(1,8),(2,3),(4,7),(5,6)]=>[8,3,2,7,6,5,4,1]=>[8,3,2,7,6,5,4,1]=>{{1,8},{2,3},{4,7},{5,6}} [(1,8),(2,4),(3,7),(5,6)]=>[8,4,7,2,6,5,3,1]=>[8,4,7,2,6,5,3,1]=>{{1,8},{2,4},{3,7},{5,6}} [(1,7),(2,4),(3,8),(5,6)]=>[7,4,8,2,6,5,1,3]=>[7,4,8,2,6,5,1,3]=>{{1,7},{2,4},{3,8},{5,6}} [(1,6),(2,4),(3,8),(5,7)]=>[6,4,8,2,7,1,5,3]=>[6,4,8,2,7,1,5,3]=>{{1,6},{2,4},{3,8},{5,7}} [(1,5),(2,4),(3,8),(6,7)]=>[5,4,8,2,1,7,6,3]=>[5,4,8,2,1,7,6,3]=>{{1,5},{2,4},{3,8},{6,7}} [(1,4),(2,5),(3,8),(6,7)]=>[4,5,8,1,2,7,6,3]=>[4,8,7,1,6,5,3,2]=>{{1,4},{2,8},{3,7},{5,6}} [(1,3),(2,5),(4,8),(6,7)]=>[3,5,1,8,2,7,6,4]=>[3,8,1,7,6,5,4,2]=>{{1,3},{2,8},{4,7},{5,6}} [(1,2),(3,5),(4,8),(6,7)]=>[2,1,5,8,3,7,6,4]=>[2,1,8,7,6,5,4,3]=>{{1,2},{3,8},{4,7},{5,6}} [(1,2),(3,6),(4,8),(5,7)]=>[2,1,6,8,7,3,5,4]=>[2,1,8,7,6,5,4,3]=>{{1,2},{3,8},{4,7},{5,6}} [(1,3),(2,6),(4,8),(5,7)]=>[3,6,1,8,7,2,5,4]=>[3,8,1,7,6,5,4,2]=>{{1,3},{2,8},{4,7},{5,6}} [(1,4),(2,6),(3,8),(5,7)]=>[4,6,8,1,7,2,5,3]=>[4,8,7,1,6,5,3,2]=>{{1,4},{2,8},{3,7},{5,6}} [(1,5),(2,6),(3,8),(4,7)]=>[5,6,8,7,1,2,4,3]=>[5,8,7,6,1,4,3,2]=>{{1,5},{2,8},{3,7},{4,6}} [(1,6),(2,5),(3,8),(4,7)]=>[6,5,8,7,2,1,4,3]=>[6,5,8,7,2,1,4,3]=>{{1,6},{2,5},{3,8},{4,7}} [(1,7),(2,5),(3,8),(4,6)]=>[7,5,8,6,2,4,1,3]=>[7,5,8,6,2,4,1,3]=>{{1,7},{2,5},{3,8},{4,6}} [(1,8),(2,5),(3,7),(4,6)]=>[8,5,7,6,2,4,3,1]=>[8,5,7,6,2,4,3,1]=>{{1,8},{2,5},{3,7},{4,6}} [(1,8),(2,6),(3,7),(4,5)]=>[8,6,7,5,4,2,3,1]=>[8,6,7,5,4,2,3,1]=>{{1,8},{2,6},{3,7},{4,5}} [(1,7),(2,6),(3,8),(4,5)]=>[7,6,8,5,4,2,1,3]=>[7,6,8,5,4,2,1,3]=>{{1,7},{2,6},{3,8},{4,5}} [(1,6),(2,7),(3,8),(4,5)]=>[6,7,8,5,4,1,2,3]=>[6,8,7,5,4,1,3,2]=>{{1,6},{2,8},{3,7},{4,5}} [(1,5),(2,7),(3,8),(4,6)]=>[5,7,8,6,1,4,2,3]=>[5,8,7,6,1,4,3,2]=>{{1,5},{2,8},{3,7},{4,6}} [(1,4),(2,7),(3,8),(5,6)]=>[4,7,8,1,6,5,2,3]=>[4,8,7,1,6,5,3,2]=>{{1,4},{2,8},{3,7},{5,6}} [(1,3),(2,7),(4,8),(5,6)]=>[3,7,1,8,6,5,2,4]=>[3,8,1,7,6,5,4,2]=>{{1,3},{2,8},{4,7},{5,6}} [(1,2),(3,7),(4,8),(5,6)]=>[2,1,7,8,6,5,3,4]=>[2,1,8,7,6,5,4,3]=>{{1,2},{3,8},{4,7},{5,6}} [(1,2),(3,8),(4,7),(5,6)]=>[2,1,8,7,6,5,4,3]=>[2,1,8,7,6,5,4,3]=>{{1,2},{3,8},{4,7},{5,6}} [(1,3),(2,8),(4,7),(5,6)]=>[3,8,1,7,6,5,4,2]=>[3,8,1,7,6,5,4,2]=>{{1,3},{2,8},{4,7},{5,6}} [(1,4),(2,8),(3,7),(5,6)]=>[4,8,7,1,6,5,3,2]=>[4,8,7,1,6,5,3,2]=>{{1,4},{2,8},{3,7},{5,6}} [(1,5),(2,8),(3,7),(4,6)]=>[5,8,7,6,1,4,3,2]=>[5,8,7,6,1,4,3,2]=>{{1,5},{2,8},{3,7},{4,6}} [(1,6),(2,8),(3,7),(4,5)]=>[6,8,7,5,4,1,3,2]=>[6,8,7,5,4,1,3,2]=>{{1,6},{2,8},{3,7},{4,5}} [(1,7),(2,8),(3,6),(4,5)]=>[7,8,6,5,4,3,1,2]=>[7,8,6,5,4,3,1,2]=>{{1,7},{2,8},{3,6},{4,5}} [(1,8),(2,7),(3,6),(4,5)]=>[8,7,6,5,4,3,2,1]=>[8,7,6,5,4,3,2,1]=>{{1,8},{2,7},{3,6},{4,5}}
Map
to permutation
Description
Returns the fixed point free involution whose transpositions are the pairs in the perfect matching.
Map
Simion-Schmidt map
Description
The Simion-Schmidt map sends any permutation to a $123$-avoiding permutation.
Details can be found in [1].
In particular, this is a bijection between $132$-avoiding permutations and $123$-avoiding permutations, see [1, Proposition 19].
Map
to cycle type
Description
Let $\pi=c_1\dots c_r$ a permutation of size $n$ decomposed in its cyclic parts. The associated set partition of $[n]$ then is $S=S_1\cup\dots\cup S_r$ such that $S_i$ is the set of integers in the cycle $c_i$.
A permutation is cyclic [1] if and only if its cycle type is a hook partition [2].