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](https://en.wikipedia.org/wiki/Incidence_algebra#Special_elements) of this poset?



What is the Möbius function of substrings?
This problem was solved by [Anders Björner, The Möbius function of factor order, *Theoretical Computer Science* **117** (1993), 91-98](https://doi.org/10.1016/0304-3975(93)90305-D). 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.


