6,477 research outputs found
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
Avoiding vincular patterns on alternating words
A word is alternating if either
(when the word is up-down) or (when the word is
down-up). The study of alternating words avoiding classical permutation
patterns was initiated by the authors in~\cite{GKZ}, where, in particular, it
was shown that 123-avoiding up-down words of even length are counted by the
Narayana numbers.
However, not much was understood on the structure of 123-avoiding up-down
words. In this paper, we fill in this gap by introducing the notion of a
cut-pair that allows us to subdivide the set of words in question into
equivalence classes. We provide a combinatorial argument to show that the
number of equivalence classes is given by the Catalan numbers, which induces an
alternative (combinatorial) proof of the corresponding result in~\cite{GKZ}.
Further, we extend the enumerative results in~\cite{GKZ} to the case of
alternating words avoiding a vincular pattern of length 3. We show that it is
sufficient to enumerate up-down words of even length avoiding the consecutive
pattern and up-down words of odd length avoiding the
consecutive pattern to answer all of our enumerative
questions. The former of the two key cases is enumerated by the Stirling
numbers of the second kind.Comment: 25 pages; To appear in Discrete Mathematic
Counting occurrences of some subword patterns
We find generating functions the number of strings (words) containing a
specified number of occurrences of certain types of order-isomorphic classes of
substrings called subword patterns. In particular, we find generating functions
for the number of strings containing a specified number of occurrences of a
given 3-letter subword pattern.Comment: 9 page
Pattern avoidance in labelled trees
We discuss a new notion of pattern avoidance motivated by the operad theory:
pattern avoidance in planar labelled trees. It is a generalisation of various
types of consecutive pattern avoidance studied before: consecutive patterns in
words, permutations, coloured permutations etc. The notion of Wilf equivalence
for patterns in permutations admits a straightforward generalisation for (sets
of) tree patterns; we describe classes for trees with small numbers of leaves,
and give several bijections between trees avoiding pattern sets from the same
class. We also explain a few general results for tree pattern avoidance, both
for the exact and the asymptotic enumeration.Comment: 27 pages, corrected various misprints, added an appendix explaining
the operadic contex
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
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