The structure of the consecutive pattern poset
International Mathematics Research Notices
The consecutive pattern poset is the infinite partially ordered set of all permutations where σ ≤ τ if τ has a subsequence of adjacent entries in the same relative order as the entries of σ. 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.
Elizalde, Sergi and McNamara, Peter R. W.. "The structure of the consecutive pattern poset." International Mathematics Research Notices (2018) : 2099-2134.