328 research outputs found
An entropy argument for counting matroids
We show how a direct application of Shearers' Lemma gives an almost optimum
bound on the number of matroids on elements.Comment: Short note, 4 page
Counting matroids in minor-closed classes
A flat cover is a collection of flats identifying the non-bases of a matroid.
We introduce the notion of cover complexity, the minimal size of such a flat
cover, as a measure for the complexity of a matroid, and present bounds on the
number of matroids on elements whose cover complexity is bounded. We apply
cover complexity to show that the class of matroids without an -minor is
asymptotically small in case is one of the sparse paving matroids
, , , , or , thus confirming a few special
cases of a conjecture due to Mayhew, Newman, Welsh, and Whittle. On the other
hand, we show a lower bound on the number of matroids without -minor
which asymptoticaly matches the best known lower bound on the number of all
matroids, due to Knuth.Comment: 13 pages, 3 figure
Elementary bounds on Poincare and log-Sobolev constants for decomposable Markov chains
We consider finite-state Markov chains that can be naturally decomposed into
smaller ``projection'' and ``restriction'' chains. Possibly this decomposition
will be inductive, in that the restriction chains will be smaller copies of the
initial chain. We provide expressions for Poincare (resp. log-Sobolev)
constants of the initial Markov chain in terms of Poincare (resp. log-Sobolev)
constants of the projection and restriction chains, together with further a
parameter. In the case of the Poincare constant, our bound is always at least
as good as existing ones and, depending on the value of the extra parameter,
may be much better. There appears to be no previously published decomposition
result for the log-Sobolev constant. Our proofs are elementary and
self-contained.Comment: Published at http://dx.doi.org/10.1214/105051604000000639 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org
On the number of matroids compared to the number of sparse paving matroids
It has been conjectured that sparse paving matroids will eventually
predominate in any asymptotic enumeration of matroids, i.e. that
, where denotes the number of
matroids on elements, and the number of sparse paving matroids. In
this paper, we show that We prove this by arguing that each matroid on elements has a
faithful description consisting of a stable set of a Johnson graph together
with a (by comparison) vanishing amount of other information, and using that
stable sets in these Johnson graphs correspond one-to-one to sparse paving
matroids on elements.
As a consequence of our result, we find that for some ,
asymptotically almost all matroids on elements have rank in the range .Comment: 12 pages, 2 figure
Log-concavity, ultra-log-concavity, and a maximum entropy property of discrete compound Poisson measures
Sufficient conditions are developed, under which the compound Poisson
distribution has maximal entropy within a natural class of probability measures
on the nonnegative integers. Recently, one of the authors [O. Johnson, {\em
Stoch. Proc. Appl.}, 2007] used a semigroup approach to show that the Poisson
has maximal entropy among all ultra-log-concave distributions with fixed mean.
We show via a non-trivial extension of this semigroup approach that the natural
analog of the Poisson maximum entropy property remains valid if the compound
Poisson distributions under consideration are log-concave, but that it fails in
general. A parallel maximum entropy result is established for the family of
compound binomial measures. Sufficient conditions for compound distributions to
be log-concave are discussed and applications to combinatorics are examined;
new bounds are derived on the entropy of the cardinality of a random
independent set in a claw-free graph, and a connection is drawn to Mason's
conjecture for matroids. The present results are primarily motivated by the
desire to provide an information-theoretic foundation for compound Poisson
approximation and associated limit theorems, analogous to the corresponding
developments for the central limit theorem and for Poisson approximation. Our
results also demonstrate new links between some probabilistic methods and the
combinatorial notions of log-concavity and ultra-log-concavity, and they add to
the growing body of work exploring the applications of maximum entropy
characterizations to problems in discrete mathematics.Comment: 30 pages. This submission supersedes arXiv:0805.4112v1. Changes in
v2: Updated references, typos correcte
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