# On the topology of the permutation pattern poset

Peter R. W. McNamara, Einar Steingrímsson

Research output: Contribution to journalArticle

6 Citations (Scopus)

### 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. .
Language English 1-35 35 Journal of Combinatorial Theory Series A 134 20 Mar 2015 10.1016/j.jcta.2015.02.009 Published - 31 Aug 2015

### Fingerprint

Poset
Permutation
Topology
Interval
Möbius Function
Subword
Open interval
Recursive Formula
Homotopy Type
Decomposable
Wedge
If and only if

### Keywords

• pattern poset
• shellable
• disconnected
• layered permutations
• generalized subword order
• Möbius function

### Cite this

@article{a88cf3105d9842a48d18d0c5c8dd88f5,
title = "On the topology of the permutation pattern poset",
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 M{\"o}bius function of decomposable permutations given by Burstein et al. .",
keywords = "pattern poset, shellable, disconnected, layered permutations, generalized subword order, M{\"o}bius function",
author = "McNamara, {Peter R. W.} and Einar Steingr{\'i}msson",
year = "2015",
month = "8",
day = "31",
doi = "10.1016/j.jcta.2015.02.009",
language = "English",
volume = "134",
pages = "1--35",
journal = "Journal of Combinatorial Theory Series A",
issn = "0097-3165",

}

On the topology of the permutation pattern poset. / McNamara, Peter R. W.; Steingrímsson, Einar.

In: Journal of Combinatorial Theory Series A , Vol. 134, 31.08.2015, p. 1-35.

Research output: Contribution to journalArticle

TY - JOUR

T1 - On the topology of the permutation pattern poset

AU - McNamara, Peter R. W.

AU - Steingrímsson, Einar

PY - 2015/8/31

Y1 - 2015/8/31

N2 - 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. .

AB - 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. .

KW - pattern poset

KW - shellable

KW - disconnected

KW - layered permutations

KW - generalized subword order

KW - Möbius function

UR - http://www.sciencedirect.com/science/article/pii/S009731651500028X

UR - http://www.sciencedirect.com/science/journal/00973165

U2 - 10.1016/j.jcta.2015.02.009

DO - 10.1016/j.jcta.2015.02.009

M3 - Article

VL - 134

SP - 1

EP - 35

JO - Journal of Combinatorial Theory Series A

T2 - Journal of Combinatorial Theory Series A

JF - Journal of Combinatorial Theory Series A

SN - 0097-3165

ER -