Digital Commons Network^™

The consecutive pattern poset is the infinite partially ordered set of all permutations where sigma < tau if \tau has a subsequence of adjacent entries in the same relative order as the entries of sigma.  We study the structure of the intervals in this poset from topological, poset-theoretic, and enumerative perspectives. In particular, we prove that all intervals are rank-unimodal and strongly Sperner, and we characterize disconnected and shellable intervals. We also show that most intervals are not shellable and have Möbius function equal to zero.