1,681 research outputs found
An Erd\H{o}s--Hajnal analogue for permutation classes
Let be a permutation class that does not contain all layered
permutations or all colayered permutations. We prove that there is a constant
such that every permutation in of length contains a
monotone subsequence of length
Finitely labeled generating trees and restricted permutations
Generating trees are a useful technique in the enumeration of various
combinatorial objects, particularly restricted permutations. Quite often the
generating tree for the set of permutations avoiding a set of patterns requires
infinitely many labels. Sometimes, however, this generating tree needs only
finitely many labels. We characterize the finite sets of patterns for which
this phenomenon occurs. We also present an algorithm - in fact, a special case
of an algorithm of Zeilberger - that is guaranteed to find such a generating
tree if it exists.Comment: Accepted by J. Symb. Comp.; 12 page
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)
Maximal independent sets and separating covers
In 1973, Katona raised the problem of determining the maximum number of
subsets in a separating cover on n elements. The answer to Katona's question
turns out to be the inverse to the answer to a much simpler question: what is
the largest integer which is the product of positive integers with sum n? We
give a combinatorial explanation for this relationship, via Moon and Moser's
answer to a question of Erdos: how many maximal independent sets can a graph on
n vertices have? We conclude by showing how Moon and Moser's solution also
sheds light on a problem of Mahler and Popken's about the complexity of
integers.Comment: To appear in the Monthl
On the effective and automatic enumeration of polynomial permutation classes
We describe an algorithm, implemented in Python, which can enumerate any
permutation class with polynomial enumeration from a structural description of
the class. In particular, this allows us to find formulas for the number of
permutations of length n which can be obtained by a finite number of block
sorting operations (e.g., reversals, block transpositions, cut-and-paste
moves)
Grid classes and the Fibonacci dichotomy for restricted permutations
We introduce and characterise grid classes, which are natural generalisations
of other well-studied permutation classes. This characterisation allows us to
give a new, short proof of the Fibonacci dichotomy: the number of permutations
of length n in a permutation class is either at least as large as the nth
Fibonacci number or is eventually polynomial
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