10,044 research outputs found
Asymptotic distribution of fixed points of pattern-avoiding involutions
For a variety of pattern-avoiding classes, we describe the limiting
distribution for the number of fixed points for involutions chosen uniformly at
random from that class. In particular we consider monotone patterns of
arbitrary length as well as all patterns of length 3. For monotone patterns we
utilize the connection with standard Young tableaux with at most rows and
involutions avoiding a monotone pattern of length . For every pattern of
length 3 we give the bivariate generating function with respect to fixed points
for the involutions that avoid that pattern, and where applicable apply tools
from analytic combinatorics to extract information about the limiting
distribution from the generating function. Many well-known distributions
appear.Comment: 16 page
Distributions of several infinite families of mesh patterns
Br\"and\'en and Claesson introduced mesh patterns to provide explicit
expansions for certain permutation statistics as linear combinations of
(classical) permutation patterns. The first systematic study of avoidance of
mesh patterns was conducted by Hilmarsson et al., while the first systematic
study of the distribution of mesh patterns was conducted by the first two
authors.
In this paper, we provide far-reaching generalizations for 8 known
distribution results and 5 known avoidance results related to mesh patterns by
giving distribution or avoidance formulas for certain infinite families of mesh
patterns in terms of distribution or avoidance formulas for smaller patterns.
Moreover, as a corollary to a general result, we find the distribution of one
more mesh pattern of length 2.Comment: 27 page
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
Avoidance of Partitions of a Three-element Set
Klazar defined and studied a notion of pattern avoidance for set partitions,
which is an analogue of pattern avoidance for permutations. Sagan considered
partitions which avoid a single partition of three elements. We enumerate
partitions which avoid any family of partitions of a 3-element set as was done
by Simion and Schmidt for permutations. We also consider even and odd set
partitions. We provide enumerative results for set partitions restricted by
generalized set partition patterns, which are an analogue of the generalized
permutation patterns of Babson and Steingr{\'{\i}}msson. Finally, in the spirit
of work done by Babson and Steingr{'{\i}}msson, we will show how these
generalized partition patterns can be used to describe set partition
statistics.Comment: 23 pages, 2 tables, 1 figure, to appear in Advances in Applied
Mathematic
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