Rationality of the Möbius function of a composition poset

We consider the zeta and Möbius functions of a partial order on integer compositions first studied by Bergeron, Bousquet-Mélou, and Dulucq. The Möbius function of this poset was determined by Sagan and Vatter. We prove rationality of various formal power series in noncommuting variables whose coefficients are evaluations of the zeta function, ζ, and the Möbius function, μ. The proofs are either directly from the definitions or by constructing finite-state automata.We also obtain explicit expressions for generating functions obtained by specializing the variables to commutative ones. We reprove Sagan and Vatter's formula for μ using this machinery. These results are closely related to those of Björner and Reutenauer about subword order, and we discuss a common generalization.