Picture of athlete cycling

Open Access research with a real impact on health...

The Strathprints institutional repository is a digital archive of University of Strathclyde's Open Access research outputs. Strathprints provides access to thousands of Open Access research papers by Strathclyde researchers, including by researchers from the Physical Activity for Health Group based within the School of Psychological Sciences & Health. Research here seeks to better understand how and why physical activity improves health, gain a better understanding of the amount, intensity, and type of physical activity needed for health benefits, and evaluate the effect of interventions to promote physical activity.

Explore open research content by Physical Activity for Health...

On the topology of the permutation pattern poset

McNamara, Peter R. W. and Steingrímsson, Einar (2015) On the topology of the permutation pattern poset. Journal of Combinatorial Theory Series A, 134. pp. 1-35. ISSN 0097-3165

Text (McNamara-Steingrimsson-JCTSA2015-topology-of-the-permutation-pattern-poset)
McNamara_Steingrimsson_JCTSA2015_topology_of_the_permutation_pattern_poset.pdf - Accepted Author Manuscript
License: Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 logo

Download (322kB) | Preview


The set of all permutations, ordered by pattern containment, forms a poset. This paper presents the first explicit major results on the topology of intervals in this poset. We show that almost all (open) intervals in this poset have a disconnected subinterval and are thus not shellable. Nevertheless, there seem to be large classes of intervals that are shellable and thus have the homotopy type of a wedge of spheres. We prove this to be the case for all intervals of layered permutations that have no disconnected subintervals of rank 3 or more. We also characterize in a simple way those intervals of layered permutations that are disconnected. These results carry over to the poset of generalized subword order when the ordering on the underlying alphabet is a rooted forest. We conjecture that the same applies to intervals of separable permutations, that is, that such an interval is shellable if and only if it has no disconnected subinterval of rank 3 or more. We also present a simplified version of the recursive formula for the Möbius function of decomposable permutations given by Burstein et al. [9].