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

TL;DR

This paper explores the enumeration of feedback classes of locally Brunovsky systems by translating the problem into partitioning integers within an abelian semigroup, linking systems theory with combinatorial mathematics.

Contribution

It introduces a novel combinatorial approach to classify feedback systems by connecting system invariants with integer partitions in semigroups.

Findings

Enumeration method for feedback classes of systems

Connection between system invariants and integer partitions

Analysis of the number of partitions into distinct summands

Abstract

The feedback class of a locally Brunovsky linear system is fully determined by the decomposition of state space as direct sum of system invariants [4]. In this paper we attack the problem of enumerating all feedback classes of locally Brunovsky systems over a $n$-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 $\nu(n,k)$ of all the partitions of integer $n$ into $k$ different summands is pointed out.