Enumerating submultisets of multisets

We consider the problem of enumerating the submultisets of a multiset, in which each element has equal multiplicity. The crucial property is that consecutive submultisets in this listing differ one in the cardinality of exactly one of the elements. This is a generalization to k-ary numbers of the so-called Gray Codes for binary numbers. The result is not new, but we think the proof is.

We give an example how these results were used during our research into switching classes. Finally, we indicate how the method can be generalized to arbitrary multisets.