Partitions of elements in a monoid and its applications to systems theory

The feedback class of a locally Brunovsky linear system is fully determined by the decomposition of state space as direct sum of system invariants [2]. In this paper we attack the problem of enumerating all feedback classes of locally Brunovsky systems over an n-dimensional state space and translate to the combinatorial problem of enumerating all the partitions of integer n   in some abelian semigroup. The problem of computing the number ν(n,k)ν(n,k) of all the partitions of integer n into k different summands is pointed out.