1,989 research outputs found
On the topology of the permutation pattern poset
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\"obius function of decomposable
permutations given by Burstein et al.Comment: 33 pages, 4 figures. Incorporates changes suggested by the referees;
new open problems in Subsection 9.4. To appear in JCT(A
Some open problems on permutation patterns
This is a brief survey of some open problems on permutation patterns, with an
emphasis on subjects not covered in the recent book by Kitaev, \emph{Patterns
in Permutations and words}. I first survey recent developments on the
enumeration and asymptotics of the pattern 1324, the last pattern of length 4
whose asymptotic growth is unknown, and related issues such as upper bounds for
the number of avoiders of any pattern of length for any given . Other
subjects treated are the M\"obius function, topological properties and other
algebraic aspects of the poset of permutations, ordered by containment, and
also the study of growth rates of permutation classes, which are containment
closed subsets of this poset.Comment: 20 pages. Related to upcoming talk at the British Combinatorial
Conference 2013. To appear in London Mathematical Society Lecture Note Serie
The Möbius function of the permutation pattern Poset
A permutation \tau contains another permutation \sigma as a pattern if \tau has a subsequence whose elements are in the same order with respect to size as the elements in \sigma. This defines a partial order on the set of all permutations, and gives a graded poset P. We give a large class of pairs of permutations whose intervals in P have Mobius function 0. Also, we give a solution to the problem when \sigma occurs precisely once in \tau, and \sigma and \tau satisfy certain further conditions, in which case the Mobius function is shown to be either -1, 0 or 1. We conjecture that for intervals [\sigma,\tau] consisting of permutations avoiding the pattern 132, the magnitude of the Mobius function is bounded by the number of occurrences of \sigma in \tau. We also conjecture that the Mobius function of the interval [1,\tau] is -1, 0 or 1
The structure of the consecutive pattern poset
The consecutive pattern poset is the infinite partially ordered set of all
permutations where if has a subsequence of adjacent
entries in the same relative order as the entries of . 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 M\"obius function equal to zero.Comment: 29 pages, 7 figures. To appear in IMR
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