106 research outputs found
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
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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