Occupation Patterns and Parikh Images in Markov Support Dynamics

Abstract

The directed support graph of a discrete-time, time-homogeneous Markov chain naturally defines a regular language whose words are the admissible trajectories from a fixed initial state. Applying the Parikh map associates with each trajectory its occupation vector, recording how many times each state is visited while disregarding the chronological order of the visits. Equivalently, each trajectory determines a monomial whose exponents are its occupation numbers, yielding a natural commutative representation of occupation patterns. For each trajectory length n, we define the occupation ideal generated by the corresponding Parikh monomials. The minimal generators of this ideal are in one-to-one correspondence with the distinct occupation patterns realized at that length, providing an algebraic encoding of the combinatorial structure of admissible trajectories. This construction naturally gives rise to three complementary measures of support complexity describing reachability growth, trajectory growth, and occupation-pattern growth. The resulting framework connects Markov support graphs, regular languages, Parikh images, and monomial ideals, establishing occupation ideals as a new algebraic object associated with Markov support dynamics. It provides a natural interface between stochastic processes, formal language theory, and combinatorial commutative algebra, and suggests new algebraic and geometric approaches to the study of support dynamics. Examples illustrate how occupation patterns reflect branching, recurrence, transience, and local oscillation in the underlying support graph.