Mutually describing multisets and integer partitions

To each finite multiset A , with underlying set S ( A ) , we associate a new multiset d ( A ) , obtained by adjoining to S ( A ) the multiplicities of its elements in A . We study the orbits of the map d under iteration, and show that if A consists of nonnegative integers, then its orbit under d converges to a cycle. Moreover, we prove that all cycles of d over Z are of length at most 3 , 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 n for each n ≥ 1 .