45 research outputs found
Upper bounds for the Stanley-Wilf limit of 1324 and other layered patterns
We prove that the Stanley-Wilf limit of any layered permutation pattern of
length is at most , and that the Stanley-Wilf limit of the
pattern 1324 is at most 16. These bounds follow from a more general result
showing that a permutation avoiding a pattern of a special form is a merge of
two permutations, each of which avoids a smaller pattern. If the conjecture is
true that the maximum Stanley-Wilf limit for patterns of length is
attained by a layered pattern then this implies an upper bound of for
the Stanley-Wilf limit of any pattern of length .
We also conjecture that, for any , the set of 1324-avoiding
permutations with inversions contains at least as many permutations of
length as those of length . We show that if this is true then the
Stanley-Wilf limit for 1324 is at most
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
Staircases, dominoes, and the growth rate of 1324-avoiders
We establish a lower bound of 10.271 for the growth rate of the permutations avoiding 1324, and an upper bound of 13.5. This is done by first finding the precise growth rate of a subclass whose enumeration is related to West-2-stack-sortable permutations, and then combining copies of this subclass in particular ways
On the Wilf-Stanley limit of 4231-avoiding permutations and a conjecture of Arratia
We construct a sequence of finite automata that accept subclasses of the
class of 4231-avoiding permutations. We thereby show that the Wilf-Stanley
limit for the class of 4231-avoiding permutations is bounded below by 9.35.
This bound shows that this class has the largest such limit among all classes
of permutations avoiding a single permutation of length 4 and refutes the
conjecture that the Wilf-Stanley limit of a class of permutations avoiding a
single permutation of length k cannot exceed (k-1)^2.Comment: Submitted to Advances in Applied Mathematic
Pattern-Avoiding Involutions: Exact and Asymptotic Enumeration
We consider the enumeration of pattern-avoiding involutions, focusing in
particular on sets defined by avoiding a single pattern of length 4. As we
demonstrate, the numerical data for these problems demonstrates some surprising
behavior. This strange behavior even provides some very unexpected data related
to the number of 1324-avoiding permutations