On the topology of the permutation pattern poset

JOURNAL OF COMBINATORIAL THEORY SERIES A

卷:134

Article

McNamara, Peter R. W.¹  Steingrimsson, Einar² 

[1] Bucknell Univ, Dept Math, Lewisburg, PA 17837 USA

[2] Univ Strathclyde, Dept Comp & Informat Sci, Glasgow G1 1XH, Lanark, Scotland

关键词: Pattern poset;    Shellable;    Disconnected;    Layered permutations;    Generalized subword order;    Mobius function;   

DOI  :  10.1016/j.jcta.2015.02.009

来源: Elsevier

【 摘 要 】

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 Mobius function of decomposable permutations given by Burstein et al. [9]. (C) 2015 Elsevier Inc. All rights reserved.

【 授权许可】

Free