The Mobius function of a composition poset

We determine the Mobius function of a poset of compositions of an integer. In fact, we give two proofs of this formula, one using an involution and one involving discrete Morse theory. This composition poset turns out to be intimately connected with subword order, whose Mobius function was determined by Bjorner. We show that, using a generalization of subword order, we can obtain both Bjorner's results and our own as special cases.