91 research outputs found
On the least exponential growth admitting uncountably many closed permutation classes
We show that the least exponential growth of counting functions which admits
uncountably many closed permutation classes lies between 2^n and
(2.33529...)^n.Comment: 13 page
Growth rates of permutation classes: categorization up to the uncountability threshold
In the antecedent paper to this it was established that there is an algebraic
number such that while there are uncountably many growth
rates of permutation classes arbitrarily close to , there are only
countably many less than . Here we provide a complete characterization of
the growth rates less than . In particular, this classification
establishes that is the least accumulation point from above of growth
rates and that all growth rates less than or equal to are achieved by
finitely based classes. A significant part of this classification is achieved
via a reconstruction result for sum indecomposable permutations. We conclude by
refuting a suggestion of Klazar, showing that is an accumulation point
from above of growth rates of finitely based permutation classes.Comment: To appear in Israel J. Mat
Small permutation classes
We establish a phase transition for permutation classes (downsets of
permutations under the permutation containment order): there is an algebraic
number , approximately 2.20557, for which there are only countably many
permutation classes of growth rate (Stanley-Wilf limit) less than but
uncountably many permutation classes of growth rate , answering a
question of Klazar. We go on to completely characterize the possible
sub- growth rates of permutation classes, answering a question of
Kaiser and Klazar. Central to our proofs are the concepts of generalized grid
classes (introduced herein), partial well-order, and atomicity (also known as
the joint embedding property)
Intervals of permutation class growth rates
We prove that the set of growth rates of permutation classes includes an
infinite sequence of intervals whose infimum is , and
that it also contains every value at least . These
results improve on a theorem of Vatter, who determined that there are
permutation classes of every growth rate at least .
Thus, we also refute his conjecture that the set of growth rates below
is nowhere dense. Our proof is based upon an analysis of expansions
of real numbers in non-integer bases, the study of which was initiated by
R\'enyi in the 1950s. In particular, we prove two generalisations of a result
of Pedicini concerning expansions in which the digits are drawn from sets of
allowed values.Comment: 20 pages, 10 figures, ancillary files containing computer-aided
calculations include
Intervals of permutation class growth rates
We prove that the set of growth rates of permutation classes includes an infinite sequence of intervals whose infimum is θB ≈ 2.35526, and that it also contains every value at least λB ≈ 2.35698. These results improve on a theorem of Vatter, who determined that there are permutation classes of every growth rate at least λA ≈ 2.48187. Thus, we also refute his conjecture that the set of growth rates below λA is nowhere dense. Our proof is based upon an analysis of expansions of real numbers in non-integer bases, the study of which was initiated by Rényi in the 1950s. In particular, we prove two generalisations of a result of Pedicini concerning expansions in which the digits are drawn from sets of allowed values
Infinite presentability of groups and condensation
We describe various classes of infinitely presented groups that are
condensation points in the space of marked groups. A well-known class of such
groups consists of finitely generated groups admitting an infinite minimal
presentation. We introduce here a larger class of condensation groups, called
infinitely independently presentable groups, and establish criteria which allow
one to infer that a group is infinitely independently presentable. In addition,
we construct examples of finitely generated groups with no minimal
presentation, among them infinitely presented groups with Cantor-Bendixson rank
1, and we prove that every infinitely presented metabelian group is a
condensation group.Comment: 32 pages, no figure. 1->2 Major changes (the 13-page first version,
authored by Y.C. and L.G., was entitled "On infinitely presented soluble
groups") 2->3 some changes including cuts in Section
Permutation classes
This is a survey on permutation classes for the upcoming book Handbook of
Enumerative Combinatorics
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