Identifier
Mp00058: Perfect matchings to permutationPermutations
Mp00064: Permutations reversePermutations
Mp00151: Permutations to cycle type Set partitions
Images
=>
Cc0012;cc-rep-0Cc0009;cc-rep-3
[(1,2)]=>[2,1]=>[1,2]=>{{1},{2}} [(1,2),(3,4)]=>[2,1,4,3]=>[3,4,1,2]=>{{1,3},{2,4}} [(1,3),(2,4)]=>[3,4,1,2]=>[2,1,4,3]=>{{1,2},{3,4}} [(1,4),(2,3)]=>[4,3,2,1]=>[1,2,3,4]=>{{1},{2},{3},{4}} [(1,2),(3,4),(5,6)]=>[2,1,4,3,6,5]=>[5,6,3,4,1,2]=>{{1,5},{2,6},{3},{4}} [(1,3),(2,4),(5,6)]=>[3,4,1,2,6,5]=>[5,6,2,1,4,3]=>{{1,4,5},{2,3,6}} [(1,4),(2,3),(5,6)]=>[4,3,2,1,6,5]=>[5,6,1,2,3,4]=>{{1,3,5},{2,4,6}} [(1,5),(2,3),(4,6)]=>[5,3,2,6,1,4]=>[4,1,6,2,3,5]=>{{1,2,4},{3,5,6}} [(1,6),(2,3),(4,5)]=>[6,3,2,5,4,1]=>[1,4,5,2,3,6]=>{{1},{2,4},{3,5},{6}} [(1,6),(2,4),(3,5)]=>[6,4,5,2,3,1]=>[1,3,2,5,4,6]=>{{1},{2,3},{4,5},{6}} [(1,5),(2,4),(3,6)]=>[5,4,6,2,1,3]=>[3,1,2,6,4,5]=>{{1,2,3},{4,5,6}} [(1,4),(2,5),(3,6)]=>[4,5,6,1,2,3]=>[3,2,1,6,5,4]=>{{1,3},{2},{4,6},{5}} [(1,3),(2,5),(4,6)]=>[3,5,1,6,2,4]=>[4,2,6,1,5,3]=>{{1,4},{2},{3,6},{5}} [(1,2),(3,5),(4,6)]=>[2,1,5,6,3,4]=>[4,3,6,5,1,2]=>{{1,4,5},{2,3,6}} [(1,2),(3,6),(4,5)]=>[2,1,6,5,4,3]=>[3,4,5,6,1,2]=>{{1,3,5},{2,4,6}} [(1,3),(2,6),(4,5)]=>[3,6,1,5,4,2]=>[2,4,5,1,6,3]=>{{1,2,4},{3,5,6}} [(1,4),(2,6),(3,5)]=>[4,6,5,1,3,2]=>[2,3,1,5,6,4]=>{{1,2,3},{4,5,6}} [(1,5),(2,6),(3,4)]=>[5,6,4,3,1,2]=>[2,1,3,4,6,5]=>{{1,2},{3},{4},{5,6}} [(1,6),(2,5),(3,4)]=>[6,5,4,3,2,1]=>[1,2,3,4,5,6]=>{{1},{2},{3},{4},{5},{6}} [(1,2),(3,4),(5,6),(7,8)]=>[2,1,4,3,6,5,8,7]=>[7,8,5,6,3,4,1,2]=>{{1,7},{2,8},{3,5},{4,6}} [(1,3),(2,4),(5,6),(7,8)]=>[3,4,1,2,6,5,8,7]=>[7,8,5,6,2,1,4,3]=>{{1,4,6,7},{2,3,5,8}} [(1,4),(2,3),(5,6),(7,8)]=>[4,3,2,1,6,5,8,7]=>[7,8,5,6,1,2,3,4]=>{{1,3,5,7},{2,4,6,8}} [(1,5),(2,3),(4,6),(7,8)]=>[5,3,2,6,1,4,8,7]=>[7,8,4,1,6,2,3,5]=>{{1,3,4,7},{2,5,6,8}} [(1,6),(2,3),(4,5),(7,8)]=>[6,3,2,5,4,1,8,7]=>[7,8,1,4,5,2,3,6]=>{{1,3,7},{2,6,8},{4},{5}} [(1,7),(2,3),(4,5),(6,8)]=>[7,3,2,5,4,8,1,6]=>[6,1,8,4,5,2,3,7]=>{{1,2,6},{3,7,8},{4},{5}} [(1,8),(2,3),(4,5),(6,7)]=>[8,3,2,5,4,7,6,1]=>[1,6,7,4,5,2,3,8]=>{{1},{2,6},{3,7},{4},{5},{8}} [(1,8),(2,4),(3,5),(6,7)]=>[8,4,5,2,3,7,6,1]=>[1,6,7,3,2,5,4,8]=>{{1},{2,5,6},{3,4,7},{8}} [(1,7),(2,4),(3,5),(6,8)]=>[7,4,5,2,3,8,1,6]=>[6,1,8,3,2,5,4,7]=>{{1,2,5,6},{3,4,7,8}} [(1,6),(2,4),(3,5),(7,8)]=>[6,4,5,2,3,1,8,7]=>[7,8,1,3,2,5,4,6]=>{{1,3,4,7},{2,5,6,8}} [(1,5),(2,4),(3,6),(7,8)]=>[5,4,6,2,1,3,8,7]=>[7,8,3,1,2,6,4,5]=>{{1,4,7},{2,5,8},{3},{6}} [(1,4),(2,5),(3,6),(7,8)]=>[4,5,6,1,2,3,8,7]=>[7,8,3,2,1,6,5,4]=>{{1,5,7},{2,4,8},{3},{6}} [(1,3),(2,5),(4,6),(7,8)]=>[3,5,1,6,2,4,8,7]=>[7,8,4,2,6,1,5,3]=>{{1,5,6,7},{2,3,4,8}} [(1,2),(3,5),(4,6),(7,8)]=>[2,1,5,6,3,4,8,7]=>[7,8,4,3,6,5,1,2]=>{{1,7},{2,8},{3,4},{5,6}} [(1,2),(3,6),(4,5),(7,8)]=>[2,1,6,5,4,3,8,7]=>[7,8,3,4,5,6,1,2]=>{{1,7},{2,8},{3},{4},{5},{6}} [(1,3),(2,6),(4,5),(7,8)]=>[3,6,1,5,4,2,8,7]=>[7,8,2,4,5,1,6,3]=>{{1,6,7},{2,3,8},{4},{5}} [(1,4),(2,6),(3,5),(7,8)]=>[4,6,5,1,3,2,8,7]=>[7,8,2,3,1,5,6,4]=>{{1,5,6,7},{2,3,4,8}} [(1,5),(2,6),(3,4),(7,8)]=>[5,6,4,3,1,2,8,7]=>[7,8,2,1,3,4,6,5]=>{{1,4,6,7},{2,3,5,8}} [(1,6),(2,5),(3,4),(7,8)]=>[6,5,4,3,2,1,8,7]=>[7,8,1,2,3,4,5,6]=>{{1,3,5,7},{2,4,6,8}} [(1,7),(2,5),(3,4),(6,8)]=>[7,5,4,3,2,8,1,6]=>[6,1,8,2,3,4,5,7]=>{{1,2,4,6},{3,5,7,8}} [(1,8),(2,5),(3,4),(6,7)]=>[8,5,4,3,2,7,6,1]=>[1,6,7,2,3,4,5,8]=>{{1},{2,4,6},{3,5,7},{8}} [(1,8),(2,6),(3,4),(5,7)]=>[8,6,4,3,7,2,5,1]=>[1,5,2,7,3,4,6,8]=>{{1},{2,3,5},{4,6,7},{8}} [(1,7),(2,6),(3,4),(5,8)]=>[7,6,4,3,8,2,1,5]=>[5,1,2,8,3,4,6,7]=>{{1,2,3,5},{4,6,7,8}} [(1,6),(2,7),(3,4),(5,8)]=>[6,7,4,3,8,1,2,5]=>[5,2,1,8,3,4,7,6]=>{{1,3,5},{2},{4,6,8},{7}} [(1,5),(2,7),(3,4),(6,8)]=>[5,7,4,3,1,8,2,6]=>[6,2,8,1,3,4,7,5]=>{{1,4,6},{2},{3,5,8},{7}} [(1,4),(2,7),(3,5),(6,8)]=>[4,7,5,1,3,8,2,6]=>[6,2,8,3,1,5,7,4]=>{{1,5,6},{2},{3,4,8},{7}} [(1,3),(2,7),(4,5),(6,8)]=>[3,7,1,5,4,8,2,6]=>[6,2,8,4,5,1,7,3]=>{{1,6},{2},{3,8},{4},{5},{7}} [(1,2),(3,7),(4,5),(6,8)]=>[2,1,7,5,4,8,3,6]=>[6,3,8,4,5,7,1,2]=>{{1,6,7},{2,3,8},{4},{5}} [(1,2),(3,8),(4,5),(6,7)]=>[2,1,8,5,4,7,6,3]=>[3,6,7,4,5,8,1,2]=>{{1,3,7},{2,6,8},{4},{5}} [(1,3),(2,8),(4,5),(6,7)]=>[3,8,1,5,4,7,6,2]=>[2,6,7,4,5,1,8,3]=>{{1,2,6},{3,7,8},{4},{5}} [(1,4),(2,8),(3,5),(6,7)]=>[4,8,5,1,3,7,6,2]=>[2,6,7,3,1,5,8,4]=>{{1,2,5,6},{3,4,7,8}} [(1,5),(2,8),(3,4),(6,7)]=>[5,8,4,3,1,7,6,2]=>[2,6,7,1,3,4,8,5]=>{{1,2,4,6},{3,5,7,8}} [(1,6),(2,8),(3,4),(5,7)]=>[6,8,4,3,7,1,5,2]=>[2,5,1,7,3,4,8,6]=>{{1,2,3,5},{4,6,7,8}} [(1,7),(2,8),(3,4),(5,6)]=>[7,8,4,3,6,5,1,2]=>[2,1,5,6,3,4,8,7]=>{{1,2},{3,5},{4,6},{7,8}} [(1,8),(2,7),(3,4),(5,6)]=>[8,7,4,3,6,5,2,1]=>[1,2,5,6,3,4,7,8]=>{{1},{2},{3,5},{4,6},{7},{8}} [(1,8),(2,7),(3,5),(4,6)]=>[8,7,5,6,3,4,2,1]=>[1,2,4,3,6,5,7,8]=>{{1},{2},{3,4},{5,6},{7},{8}} [(1,7),(2,8),(3,5),(4,6)]=>[7,8,5,6,3,4,1,2]=>[2,1,4,3,6,5,8,7]=>{{1,2},{3,4},{5,6},{7,8}} [(1,6),(2,8),(3,5),(4,7)]=>[6,8,5,7,3,1,4,2]=>[2,4,1,3,7,5,8,6]=>{{1,2,3,4},{5,6,7,8}} [(1,5),(2,8),(3,6),(4,7)]=>[5,8,6,7,1,3,4,2]=>[2,4,3,1,7,6,8,5]=>{{1,2,4},{3},{5,7,8},{6}} [(1,4),(2,8),(3,6),(5,7)]=>[4,8,6,1,7,3,5,2]=>[2,5,3,7,1,6,8,4]=>{{1,2,5},{3},{4,7,8},{6}} [(1,3),(2,8),(4,6),(5,7)]=>[3,8,1,6,7,4,5,2]=>[2,5,4,7,6,1,8,3]=>{{1,2,5,6},{3,4,7,8}} [(1,2),(3,8),(4,6),(5,7)]=>[2,1,8,6,7,4,5,3]=>[3,5,4,7,6,8,1,2]=>{{1,3,4,7},{2,5,6,8}} [(1,2),(3,7),(4,6),(5,8)]=>[2,1,7,6,8,4,3,5]=>[5,3,4,8,6,7,1,2]=>{{1,5,6,7},{2,3,4,8}} [(1,3),(2,7),(4,6),(5,8)]=>[3,7,1,6,8,4,2,5]=>[5,2,4,8,6,1,7,3]=>{{1,5,6},{2},{3,4,8},{7}} [(1,4),(2,7),(3,6),(5,8)]=>[4,7,6,1,8,3,2,5]=>[5,2,3,8,1,6,7,4]=>{{1,5},{2},{3},{4,8},{6},{7}} [(1,5),(2,7),(3,6),(4,8)]=>[5,7,6,8,1,3,2,4]=>[4,2,3,1,8,6,7,5]=>{{1,4},{2},{3},{5,8},{6},{7}} [(1,6),(2,7),(3,5),(4,8)]=>[6,7,5,8,3,1,2,4]=>[4,2,1,3,8,5,7,6]=>{{1,3,4},{2},{5,6,8},{7}} [(1,7),(2,6),(3,5),(4,8)]=>[7,6,5,8,3,2,1,4]=>[4,1,2,3,8,5,6,7]=>{{1,2,3,4},{5,6,7,8}} [(1,8),(2,6),(3,5),(4,7)]=>[8,6,5,7,3,2,4,1]=>[1,4,2,3,7,5,6,8]=>{{1},{2,3,4},{5,6,7},{8}} [(1,8),(2,5),(3,6),(4,7)]=>[8,5,6,7,2,3,4,1]=>[1,4,3,2,7,6,5,8]=>{{1},{2,4},{3},{5,7},{6},{8}} [(1,7),(2,5),(3,6),(4,8)]=>[7,5,6,8,2,3,1,4]=>[4,1,3,2,8,6,5,7]=>{{1,2,4},{3},{5,7,8},{6}} [(1,6),(2,5),(3,7),(4,8)]=>[6,5,7,8,2,1,3,4]=>[4,3,1,2,8,7,5,6]=>{{1,2,3,4},{5,6,7,8}} [(1,5),(2,6),(3,7),(4,8)]=>[5,6,7,8,1,2,3,4]=>[4,3,2,1,8,7,6,5]=>{{1,4},{2,3},{5,8},{6,7}} [(1,4),(2,6),(3,7),(5,8)]=>[4,6,7,1,8,2,3,5]=>[5,3,2,8,1,7,6,4]=>{{1,5},{2,3},{4,8},{6,7}} [(1,3),(2,6),(4,7),(5,8)]=>[3,6,1,7,8,2,4,5]=>[5,4,2,8,7,1,6,3]=>{{1,5,6,7},{2,3,4,8}} [(1,2),(3,6),(4,7),(5,8)]=>[2,1,6,7,8,3,4,5]=>[5,4,3,8,7,6,1,2]=>{{1,5,7},{2,4,8},{3},{6}} [(1,2),(3,5),(4,7),(6,8)]=>[2,1,5,7,3,8,4,6]=>[6,4,8,3,7,5,1,2]=>{{1,5,6,7},{2,3,4,8}} [(1,3),(2,5),(4,7),(6,8)]=>[3,5,1,7,2,8,4,6]=>[6,4,8,2,7,1,5,3]=>{{1,6},{2,4},{3,8},{5,7}} [(1,4),(2,5),(3,7),(6,8)]=>[4,5,7,1,2,8,3,6]=>[6,3,8,2,1,7,5,4]=>{{1,5,6,7},{2,3,4,8}} [(1,5),(2,4),(3,7),(6,8)]=>[5,4,7,2,1,8,3,6]=>[6,3,8,1,2,7,4,5]=>{{1,4,6,7},{2,3,5,8}} [(1,6),(2,4),(3,7),(5,8)]=>[6,4,7,2,8,1,3,5]=>[5,3,1,8,2,7,4,6]=>{{1,2,3,5},{4,6,7,8}} [(1,7),(2,4),(3,6),(5,8)]=>[7,4,6,2,8,3,1,5]=>[5,1,3,8,2,6,4,7]=>{{1,2,5},{3},{4,7,8},{6}} [(1,8),(2,4),(3,6),(5,7)]=>[8,4,6,2,7,3,5,1]=>[1,5,3,7,2,6,4,8]=>{{1},{2,5},{3},{4,7},{6},{8}} [(1,8),(2,3),(4,6),(5,7)]=>[8,3,2,6,7,4,5,1]=>[1,5,4,7,6,2,3,8]=>{{1},{2,5,6},{3,4,7},{8}} [(1,7),(2,3),(4,6),(5,8)]=>[7,3,2,6,8,4,1,5]=>[5,1,4,8,6,2,3,7]=>{{1,2,5,6},{3,4,7,8}} [(1,6),(2,3),(4,7),(5,8)]=>[6,3,2,7,8,1,4,5]=>[5,4,1,8,7,2,3,6]=>{{1,3,5,7},{2,4,6,8}} [(1,5),(2,3),(4,7),(6,8)]=>[5,3,2,7,1,8,4,6]=>[6,4,8,1,7,2,3,5]=>{{1,2,4,6},{3,5,7,8}} [(1,4),(2,3),(5,7),(6,8)]=>[4,3,2,1,7,8,5,6]=>[6,5,8,7,1,2,3,4]=>{{1,2,5,6},{3,4,7,8}} [(1,3),(2,4),(5,7),(6,8)]=>[3,4,1,2,7,8,5,6]=>[6,5,8,7,2,1,4,3]=>{{1,6},{2,5},{3,8},{4,7}} [(1,2),(3,4),(5,7),(6,8)]=>[2,1,4,3,7,8,5,6]=>[6,5,8,7,3,4,1,2]=>{{1,4,6,7},{2,3,5,8}} [(1,2),(3,4),(5,8),(6,7)]=>[2,1,4,3,8,7,6,5]=>[5,6,7,8,3,4,1,2]=>{{1,3,5,7},{2,4,6,8}} [(1,3),(2,4),(5,8),(6,7)]=>[3,4,1,2,8,7,6,5]=>[5,6,7,8,2,1,4,3]=>{{1,2,5,6},{3,4,7,8}} [(1,4),(2,3),(5,8),(6,7)]=>[4,3,2,1,8,7,6,5]=>[5,6,7,8,1,2,3,4]=>{{1,5},{2,6},{3,7},{4,8}} [(1,5),(2,3),(4,8),(6,7)]=>[5,3,2,8,1,7,6,4]=>[4,6,7,1,8,2,3,5]=>{{1,4},{2,6},{3,7},{5,8}} [(1,6),(2,3),(4,8),(5,7)]=>[6,3,2,8,7,1,5,4]=>[4,5,1,7,8,2,3,6]=>{{1,3,4,7},{2,5,6,8}} [(1,7),(2,3),(4,8),(5,6)]=>[7,3,2,8,6,5,1,4]=>[4,1,5,6,8,2,3,7]=>{{1,2,4,6},{3,5,7,8}} [(1,8),(2,3),(4,7),(5,6)]=>[8,3,2,7,6,5,4,1]=>[1,4,5,6,7,2,3,8]=>{{1},{2,4,6},{3,5,7},{8}} [(1,8),(2,4),(3,7),(5,6)]=>[8,4,7,2,6,5,3,1]=>[1,3,5,6,2,7,4,8]=>{{1},{2,3,5},{4,6,7},{8}} [(1,7),(2,4),(3,8),(5,6)]=>[7,4,8,2,6,5,1,3]=>[3,1,5,6,2,8,4,7]=>{{1,2,3,5},{4,6,7,8}} [(1,6),(2,4),(3,8),(5,7)]=>[6,4,8,2,7,1,5,3]=>[3,5,1,7,2,8,4,6]=>{{1,3},{2,5},{4,7},{6,8}} [(1,5),(2,4),(3,8),(6,7)]=>[5,4,8,2,1,7,6,3]=>[3,6,7,1,2,8,4,5]=>{{1,3,4,7},{2,5,6,8}} [(1,4),(2,5),(3,8),(6,7)]=>[4,5,8,1,2,7,6,3]=>[3,6,7,2,1,8,5,4]=>{{1,3,5,7},{2,4,6,8}} [(1,3),(2,5),(4,8),(6,7)]=>[3,5,1,8,2,7,6,4]=>[4,6,7,2,8,1,5,3]=>{{1,2,4,6},{3,5,7,8}} [(1,2),(3,5),(4,8),(6,7)]=>[2,1,5,8,3,7,6,4]=>[4,6,7,3,8,5,1,2]=>{{1,3,4,7},{2,5,6,8}} [(1,2),(3,6),(4,8),(5,7)]=>[2,1,6,8,7,3,5,4]=>[4,5,3,7,8,6,1,2]=>{{1,4,7},{2,5,8},{3},{6}} [(1,3),(2,6),(4,8),(5,7)]=>[3,6,1,8,7,2,5,4]=>[4,5,2,7,8,1,6,3]=>{{1,4,6,7},{2,3,5,8}} [(1,4),(2,6),(3,8),(5,7)]=>[4,6,8,1,7,2,5,3]=>[3,5,2,7,1,8,6,4]=>{{1,2,3,5},{4,6,7,8}} [(1,5),(2,6),(3,8),(4,7)]=>[5,6,8,7,1,2,4,3]=>[3,4,2,1,7,8,6,5]=>{{1,2,3,4},{5,6,7,8}} [(1,6),(2,5),(3,8),(4,7)]=>[6,5,8,7,2,1,4,3]=>[3,4,1,2,7,8,5,6]=>{{1,3},{2,4},{5,7},{6,8}} [(1,7),(2,5),(3,8),(4,6)]=>[7,5,8,6,2,4,1,3]=>[3,1,4,2,6,8,5,7]=>{{1,2,3,4},{5,6,7,8}} [(1,8),(2,5),(3,7),(4,6)]=>[8,5,7,6,2,4,3,1]=>[1,3,4,2,6,7,5,8]=>{{1},{2,3,4},{5,6,7},{8}} [(1,8),(2,6),(3,7),(4,5)]=>[8,6,7,5,4,2,3,1]=>[1,3,2,4,5,7,6,8]=>{{1},{2,3},{4},{5},{6,7},{8}} [(1,7),(2,6),(3,8),(4,5)]=>[7,6,8,5,4,2,1,3]=>[3,1,2,4,5,8,6,7]=>{{1,2,3},{4},{5},{6,7,8}} [(1,6),(2,7),(3,8),(4,5)]=>[6,7,8,5,4,1,2,3]=>[3,2,1,4,5,8,7,6]=>{{1,3},{2},{4},{5},{6,8},{7}} [(1,5),(2,7),(3,8),(4,6)]=>[5,7,8,6,1,4,2,3]=>[3,2,4,1,6,8,7,5]=>{{1,3,4},{2},{5,6,8},{7}} [(1,4),(2,7),(3,8),(5,6)]=>[4,7,8,1,6,5,2,3]=>[3,2,5,6,1,8,7,4]=>{{1,3,5},{2},{4,6,8},{7}} [(1,3),(2,7),(4,8),(5,6)]=>[3,7,1,8,6,5,2,4]=>[4,2,5,6,8,1,7,3]=>{{1,4,6},{2},{3,5,8},{7}} [(1,2),(3,7),(4,8),(5,6)]=>[2,1,7,8,6,5,3,4]=>[4,3,5,6,8,7,1,2]=>{{1,4,6,7},{2,3,5,8}} [(1,2),(3,8),(4,7),(5,6)]=>[2,1,8,7,6,5,4,3]=>[3,4,5,6,7,8,1,2]=>{{1,3,5,7},{2,4,6,8}} [(1,3),(2,8),(4,7),(5,6)]=>[3,8,1,7,6,5,4,2]=>[2,4,5,6,7,1,8,3]=>{{1,2,4,6},{3,5,7,8}} [(1,4),(2,8),(3,7),(5,6)]=>[4,8,7,1,6,5,3,2]=>[2,3,5,6,1,7,8,4]=>{{1,2,3,5},{4,6,7,8}} [(1,5),(2,8),(3,7),(4,6)]=>[5,8,7,6,1,4,3,2]=>[2,3,4,1,6,7,8,5]=>{{1,2,3,4},{5,6,7,8}} [(1,6),(2,8),(3,7),(4,5)]=>[6,8,7,5,4,1,3,2]=>[2,3,1,4,5,7,8,6]=>{{1,2,3},{4},{5},{6,7,8}} [(1,7),(2,8),(3,6),(4,5)]=>[7,8,6,5,4,3,1,2]=>[2,1,3,4,5,6,8,7]=>{{1,2},{3},{4},{5},{6},{7,8}} [(1,8),(2,7),(3,6),(4,5)]=>[8,7,6,5,4,3,2,1]=>[1,2,3,4,5,6,7,8]=>{{1},{2},{3},{4},{5},{6},{7},{8}}
Map
to permutation
Description
Returns the fixed point free involution whose transpositions are the pairs in the perfect matching.
Map
reverse
Description
Sends a permutation to its reverse.
The reverse of a permutation $\sigma$ of length $n$ is given by $\tau$ with $\tau(i) = \sigma(n+1-i)$.
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].