The Mِbius function of generalized factor order

We use discrete Morse theory to determine the Möbius function of generalized factor order. Ordinary factor order on the Kleene closure A ∗ of a set A is the partial order defined by letting u ≤ w if w contains u as a subsequence of consecutive letters. Generalized factor order takes into account a partial order P A on the alphabet A , that is, u ≤ w whenever w contains a subsequence w ( i + 1 ) ⋯ w ( i + | u | ) such that for each j , u ( j ) ≤ w ( i + j ) in A . Using Babson and Hersh’s application of Robin Forman’s discrete Morse theory to poset order complexes, we are able to give a recursive formula for the Möbius function in the case where each element of A covers a unique letter in P A .