47 research outputs found
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
Statistics on ordered partitions of sets
We introduce several statistics on ordered partitions of sets, that is, set
partitions where the blocks are permuted arbitrarily. The distribution of these
statistics is closely related to the q-Stirling numbers of the second kind.
Some of the statistics are generalizations of known statistics on set
partitions, but others are entirely new. All the new ones are sums of two
statistics, inspired by statistics on permutations, where one of the two
statistics is based on a certain partial ordering of the blocks of a partition.Comment: Added a Prologue, as this paper is soon to be published in a journa
The Mƶbius 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
Generalized permutation patterns - a short survey
An occurrence of a classical pattern p in a permutation Ļ is a subsequence of Ļ whose letters are in the same relative order (of size) as those in p. In an occurrence of a generalized pattern, some letters of that subsequence may be required to be adjacent in the permutation. Subsets of permutations characterized by the avoidanceāor the prescribed number of occurrencesā of generalized patterns exhibit connections to an enormous variety of other combinatorial structures, some of them apparently deep. We give a short overview of the state of the art for generalized patterns
The Coloring Ideal and Coloring Complex of a Graph
Let be a simple graph on vertices. We define a monomial ideal in
the Stanley-Reisner ring of the order complex of the Boolean algebra on
atoms. The monomials in are in one-to-one correspondence with the proper
colorings of . In particular, the Hilbert polynomial of equals the
chromatic polynomial of .
The ideal is generated by square-free monomials, so is the
Stanley-Reisner ring of a simplicial complex . The -vector of is a
certain transformation of the tail of the chromatic polynomial
of . The combinatorial structure of the complex is described
explicitly and it is shown that the Euler characteristic of equals the
number of acyclic orientations of .Comment: 13 pages, 3 figure
Random Walks and Mixed Volumes of Hypersimplices
Below is a method for relating a mixed volume computation for polytopes
sharing many facet directions to a symmetric random walk. The example of
permutahedra and particularly hypersimplices is expanded upon.Comment: 6 page
Decreasing subsequences in permutations and Wilf equivalence for involutions
In a recent paper, Backelin, West and Xin describe a map that
recursively replaces all occurrences of the pattern in a permutation
by occurrences of the pattern . The resulting
permutation contains no decreasing subsequence of length .
We prove that, rather unexpectedly, the map commutes with taking the
inverse of a permutation. In the BWX paper, the definition of is
actually extended to full rook placements on a Ferrers board (the permutations
correspond to square boards), and the construction of the map is the
key step in proving the following result. Let be a set of patterns starting
with the prefix . Let be the set of patterns obtained by
replacing this prefix by in every pattern of . Then for all ,
the number of permutations of the symmetric group \Sn_n that avoid equals
the number of permutations of \Sn_n that avoid . Our commutation result,
generalized to Ferrers boards, implies that the number of {\em involutions} of
\Sn_n that avoid is equal to the number of involutions of \Sn_n
avoiding , as recently conjectured by Jaggard
Generalized permutation patterns -- a short survey
An occurrence of a classical pattern p in a permutation \pi is a subsequence
of \pi whose letters are in the same relative order (of size) as those in p. In
an occurrence of a generalized pattern, some letters of that subsequence may be
required to be adjacent in the permutation. Subsets of permutations
characterized by the avoidance--or the prescribed number of occurrences--of
generalized patterns exhibit connections to an enormous variety of other
combinatorial structures, some of them apparently deep. We give a short
overview of the state of the art for generalized patterns.Comment: 11 pages. Added a section on asymptotics (Section 8), added more
examples of barred patterns equal to generalized patterns (Section 7) and
made a few other minor additions. To appear in ``Permutation Patterns, St
Andrews 2007'', S.A. Linton, N. Ruskuc, V. Vatter (eds.), LMS Lecture Note
Series, Cambridge University Pres
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