Abstract
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 Mo ̈bius function equal to zero.
Highlights
Consecutive patterns in permutations generalize well-studied notions such as descents, ascents, peaks, valleys and runs
A permutation σ is said to be contained in another one τ as a consecutive pattern if τ has a subsequence of adjacent entries in the same relative order as the entries of σ
An early inspiration for research from this viewpoint was a question of Wilf [Wil02] asking for the Mobius function of intervals in the poset defined by classical pattern containment
Summary
Consecutive patterns in permutations generalize well-studied notions such as descents, ascents, peaks, valleys and runs. An early inspiration for research from this viewpoint was a question of Wilf [Wil02] asking for the Mobius function of intervals in the poset defined by classical pattern containment. The Mobius function for intervals in P, the consecutive pattern poset, has been determined by Bernini, Ferrari and Steingrımsson [BFS11] and by Sagan and Willenbring [SW12] This already gives an indication that the consecutive pattern case is more tractable for certain types of questions than the classical case. The precursor in the classical case to the present work is [MS15], where the focus is on classifying disconnected open intervals and showing that certain special intervals are shellable We successfully address these same topological questions of disconnectivity and shellability for the consecutive pattern poset P. The proofs that have been omitted in this extended abstract due to lack of space can be found in the full paper version [EM15]
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