TITLE: What is the Möbius function of substrings?
QUESTION [6 upvotes]: Define a poset on the set of all finite binary strings, defined by $a \le b$ whenever $b = uav$ for (possibly empty) binary strings $u, v$.
What is the Möbius function of this poset?

REPLY [9 votes]: This problem was solved by Anders Björner, The Möbius function of factor order, Theoretical Computer Science 117 (1993), 91-98. In particular, the Möbius function assumes only the values $-1,0,1$. It is also Exercise 3.134(b) in Enumerative Combinatorics, vol. 1, second edition.