The Möbius function of factor order

Abstract

Intervals in the factor ordering of a free monoid are investigated. It was shown by Farmer (1982) that such intervals (β, α) are contractible or homotopy spheres in case β is the empty word. We observe here that the same is true in general. This implies that the Möbius function of factor order takes values in {0, + 1, −1}. A recursive rule for this Möbius function is given, which allows efficient computation via the Knuth—Morris—Pratt algorithm. The Möbius function of subword order was studied in Björner (1990). We give here a simpler proof (a parity-changing involution) for its combinatorial interpretation.