  We consider the zeta and M\"obius functions of a partial order on integer
compositions first studied by Bergeron, Bousquet-M\'elou, and Dulucq. The
M\"obius 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 and the M\"obius function.
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 the M\"obius function using this machinery. These results are
closely related to those of Bj\"orner and Reutenauer about subword order, and
we discuss a common generalization.
