51,383 research outputs found
Transport of patterns by Burge transpose
We take the first steps in developing a theory of transport of patterns from
Fishburn permutations to (modified) ascent sequences. Given a set of pattern
avoiding Fishburn permutations, we provide an explicit construction for the
basis of the corresponding set of modified ascent sequences. Our approach is in
fact more general and can transport patterns between permutations and
equivalence classes of so called Cayley permutations. This transport of
patterns relies on a simple operation we call the Burge transpose. It operates
on certain biwords called Burge words. Moreover, using mesh patterns on Cayley
permutations, we present an alternative view of the transport of patterns as a
Wilf-equivalence between subsets of Cayley permutations. We also highlight a
connection with primitive ascent sequences.Comment: 24 pages, 4 figure
Combinatorial Optimization of Subsequence Patterns in Words
Packing patterns in words concerns finding a word with the maximum number of a prescribed pattern. The majority of the work done thus far is on packing patterns into permutations. In 2002, Albert, Atkinson, Handley, Holton and Stromquist showed that there always exists a layered permutation containing the maximum number of a layered pattern among all permutations of length n. Consequently, the packing density for all but two (up to equivalence) permutation patterns up to length 4 can be obtained. In this thesis we consider the analogous question for colored patterns and permutations. By introducing the concept of colored blocks we characterize the optimal permutations with the maximum number of a given colored pattern when it contains at most three colored blocks. As examples, we apply this characterization to find the optimal permutations of various colored patterns and subsequently obtain their corresponding packing densities
Place-difference-value patterns: A generalization of generalized permutation and word patterns
Motivated by study of Mahonian statistics, in 2000, Babson and Steingrimsson
introduced the notion of a "generalized permutation pattern" (GP) which
generalizes the concept of "classical" permutation pattern introduced by Knuth
in 1969. The invention of GPs led to a large number of publications related to
properties of these patterns in permutations and words. Since the work of
Babson and Steingrimsson, several further generalizations of permutation
patterns have appeared in the literature, each bringing a new set of
permutation or word pattern problems and often new connections with other
combinatorial objects and disciplines. For example, Bousquet-Melou et al.
introduced a new type of permutation pattern that allowed them to relate
permutation patterns theory to the theory of partially ordered sets.
In this paper we introduce yet another, more general definition of a pattern,
called place-difference-value patterns (PDVP) that covers all of the most
common definitions of permutation and/or word patterns that have occurred in
the literature. PDVPs provide many new ways to develop the theory of patterns
in permutations and words. We shall give several examples of PDVPs in both
permutations and words that cannot be described in terms of any other pattern
conditions that have been introduced previously. Finally, we raise several
bijective questions linking our patterns to other combinatorial objects.Comment: 18 pages, 2 figures, 1 tabl
Locally Convex Words and Permutations
We introduce some new classes of words and permutations characterized by the
second difference condition , which we
call the -convexity condition. We demonstrate that for any sized alphabet
and convexity parameter , we may find a generating function which counts
-convex words of length . We also determine a formula for the number of
0-convex words on any fixed-size alphabet for sufficiently large by
exhibiting a connection to integer partitions. For permutations, we give an
explicit solution in the case and show that the number of 1-convex and
2-convex permutations of length are and ,
respectively, and use the transfer matrix method to give tight bounds on the
constants and . We also providing generating functions similar to
the the continued fraction generating functions studied by Odlyzko and Wilf in
the "coins in a fountain" problem.Comment: 20 pages, 4 figure
Set Systems and Families of Permutations with Small Traces
We study the maximum size of a set system on elements whose trace on any
elements has size at most . We show that if for some the
shatter function of a set system satisfies then ; this generalizes Sauer's Lemma on the size of
set systems with bounded VC-dimension. We use this bound to delineate the main
growth rates for the same problem on families of permutations, where the trace
corresponds to the inclusion for permutations. This is related to a question of
Raz on families of permutations with bounded VC-dimension that generalizes the
Stanley-Wilf conjecture on permutations with excluded patterns
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
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