Abstract
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].
Original language  English 

Pages (fromto)  135 
Number of pages  35 
Journal  Journal of Combinatorial Theory Series A 
Volume  134 
Early online date  20 Mar 2015 
DOIs  
Publication status  Published  31 Aug 2015 
Keywords
 pattern poset
 shellable
 disconnected
 layered permutations
 generalized subword order
 Möbius function
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Einar Steingrimsson
 SICSA
 Mathematics And Statistics  Visiting Professor
Person: Academic, Visiting Professor