Mutually describing multisets and integer partitions

To each finite multiset AA, with underlying set S(A)S(A), we associate a new multiset d(A)d(A), obtained by adjoining to S(A)S(A) the multiplicities of its elements in AA. We study the orbits of the map dd under iteration, and show that if AA consists of nonnegative integers, then its orbit under dd converges to a cycle. Moreover, we prove that all cycles of dd over ZZ are of length at most 33, and we completely determine them. This amounts to finding all systems of mutually describing multisets. In the process, we are led to introduce and study a related discrete dynamical system on the set of integer partitions of nn for each n≥1n≥1.