Set Partitions Upto Rotation

initialisations

nice[a_]:=If[a==={},a,Partition[Join[a, Table[Graphics[Point[{0, 0}]], 
     { Mod[Ceiling[Sqrt[#]]-Mod[#,(Ceiling[Sqrt[#]] )],Ceiling[Sqrt[#]]] & @ Length[a]}]], Ceiling[Sqrt[Length[a]]] ] ]

Set Partitions :: counted by the Bell numbers

Compare to  http://mathforum.org/advanced/robertd/bell.html

Show[GraphicsArray[nice[plotbell /@ SetPartitions[4]], GraphicsSpacing -> 0.25]]

The function 'shrink'  generates all rotations of each set partition, sorts them internally and overall, and picks the first of these rotated states as representative. Finally, it takes the union of these representatives.

The count of these Set Partitions upto rotation is:

the plots for n=1 up to n=7  are :