315 research outputs found
Classical and consecutive pattern avoidance in rooted forests
Following Anders and Archer, we say that an unordered rooted labeled forest
avoids the pattern if in each tree, each sequence of
labels along the shortest path from the root to a vertex does not contain a
subsequence with the same relative order as . For each permutation
, we construct a bijection between -vertex
forests avoiding and
-vertex forests avoiding ,
giving a common generalization of results of West on permutations and
Anders--Archer on forests. We further define a new object, the forest-Young
diagram, which we use to extend the notion of shape-Wilf equivalence to
forests. In particular, this allows us to generalize the above result to a
bijection between forests avoiding and forests avoiding for . Furthermore, we give recurrences
enumerating the forests avoiding , , and other sets
of patterns. Finally, we extend the Goulden--Jackson cluster method to study
consecutive pattern avoidance in rooted trees as defined by Anders and Archer.
Using the generalized cluster method, we prove that if two length- patterns
are strong-c-forest-Wilf equivalent, then up to complementation, the two
patterns must start with the same number. We also prove the surprising result
that the patterns and are strong-c-forest-Wilf equivalent, even
though they are not c-Wilf equivalent with respect to permutations.Comment: 39 pages, 11 figure
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
Using homological duality in consecutive pattern avoidance
Using the approach suggested in [arXiv:1002.2761] we present below a
sufficient condition guaranteeing that two collections of patterns of
permutations have the same exponential generating functions for the number of
permutations avoiding elements of these collections as consecutive patterns. In
short, the coincidence of the latter generating functions is guaranteed by a
length-preserving bijection of patterns in these collections which is identical
on the overlappings of pairs of patterns where the overlappings are considered
as unordered sets. Our proof is based on a direct algorithm for the computation
of the inverse generating functions. As an application we present a large class
of patterns where this algorithm is fast and, in particular, allows to obtain a
linear ordinary differential equation with polynomial coefficients satisfied by
the inverse generating function.Comment: 12 pages, 1 figur
Computational Approaches to Consecutive Pattern Avoidance in Permutations
In recent years, there has been increasing interest in consecutive pattern
avoidance in permutations. In this paper, we introduce two approaches to
counting permutations that avoid a set of prescribed patterns consecutively.
These algoritms have been implemented in the accompanying Maple package CAV,
which can be downloaded from the author's website. As a byproduct of the first
algorithm, we have a theorem giving a sufficient condition for when two pattern
sets are strongly (consecutively) Wilf-Equivalent. For the implementation of
the second algorithm, we define the cluster tail generating function and show
that it always satisfies a certain functional equation. We also explain how the
CAV package can be used to approximate asymptotic constants for single pattern
avoidance.Comment: 12 page
Inversion Polynomials for Permutations Avoiding Consecutive Patterns
In 2012, Sagan and Savage introduced the notion of -Wilf equivalence for
a statistic and for sets of permutations that avoid particular permutation
patterns which can be extended to generalized permutation patterns. In this
paper we consider -Wilf equivalence on sets of two or more consecutive
permutation patterns. We say that two sets of generalized permutation patterns
and are -Wilf equivalent if the generating function for the
inversion statistic on the permutations that simultaneously avoid all elements
of is equal to the generating function for the inversion statistic on the
permutations that simultaneously avoid all elements of .
In 2013, Cameron and Killpatrick gave the inversion generating function for
Fibonacci tableaux which are in one-to-one correspondence with the set of
permutations that simultaneously avoid the consecutive patterns and
In this paper, we use the language of Fibonacci tableaux to study the
inversion generating functions for permutations that avoid where is
a set of five or fewer consecutive permutation patterns. In addition, we
introduce the more general notion of a strip tableaux which are a useful
combinatorial object for studying consecutive pattern avoidance. We go on to
give the inversion generating functions for all but one of the cases where
is a subset of three consecutive permutation patterns and we give several
results for a subset of two consecutive permutation patterns
Descent c-Wilf Equivalence
Let denote the symmetric group. For any , we let
denote the number of descents of ,
denote the number of inversions of , and
denote the number of left-to-right minima of .
For any sequence of statistics on
permutations, we say two permutations and in are
-c-Wilf equivalent if the generating
function of over all permutations which
have no consecutive occurrences of equals the generating function of
over all permutations which have no
consecutive occurrences of . We give many examples of pairs of
permutations and in which are -c-Wilf
equivalent, -c-Wilf equivalent, and
-c-Wilf equivalent. For example, we
will show that if and are minimally overlapping permutations
in which start with 1 and end with the same element and
and , then and are
-c-Wilf equivalent.Comment: arXiv admin note: text overlap with arXiv:1510.0431
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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