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(Definition: for a set $S$, and a partition $P=\{P_1,P_2,\ldots,P_k\}$ of $S$, we say that the length of $P$ is $k$.)

I want to iterate over all partitions of $S$ whose length is $k$.

Motivation: I am testing a property for graphs which satisfy a certain coloring-property (which is not standard coloring). So I need to check, in turn, whether a partition in $1,2,\ldots,p$ parts will satisfy the restriction for this particular type of coloring, before I can proceed to the test I want to make.

If it helps, for traditional (proper) graph coloring there is an iterator that does precisely the type of thing I am looking for; it is called all_graph_colorings.