Discrete Morse theory and the consecutive pattern poset

We use discrete Morse theory to provide another proof of Bernini, Ferrari, and Steingrímsson’s formula for the M?bius function of the consecutive pattern poset. In addition, we are able to determine the homotopy type of this poset. Earlier, Bj?rner determined the M?bius function and homotopy type of factor order and the results are remarkably similar to those in the pattern case. In his thesis, Willenbring used discrete Morse theory to give an illuminating proof of Bj?rner’s result. Since our proof parallels Willenbring’s, we also consider the relationship between the two posets. In particular, we show that some of their intervals are isomorphic, and also that there is a sequence of posets interpolating between the two all of whom have essentially the same M?bius function.