57,325 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
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
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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