142 research outputs found
On the structure of the h-vector of a paving matroid.
We give two proofs that the h-vector of any paving matroid is a pure 0-sequence, thus answering in the affirmative a conjecture made by Stanley, for this particular class of matroids. We also investigate the problem of obtaining good lower bounds for the number of bases of a paving matroid given its rank and number of elements.The first author was supported by Conacyt of México Proyect8397
Matroids with nine elements
We describe the computation of a catalogue containing all matroids with up to
nine elements, and present some fundamental data arising from this cataogue.
Our computation confirms and extends the results obtained in the 1960s by
Blackburn, Crapo and Higgs. The matroids and associated data are stored in an
online database, and we give three short examples of the use of this database.Comment: 22 page
Generalized Laminar Matroids
Nested matroids were introduced by Crapo in 1965 and have appeared frequently
in the literature since then. A flat of a matroid is Hamiltonian if it has
a spanning circuit. A matroid is nested if and only if its Hamiltonian
flats form a chain under inclusion; is laminar if and only if, for every
-element independent set , the Hamiltonian flats of containing
form a chain under inclusion. We generalize these notions to define the classes
of -closure-laminar and -laminar matroids. This paper focuses on
structural properties of these classes noting that, while the second class is
always minor-closed, the first is if and only if . The main results
are excluded-minor characterizations for the classes of 2-laminar and
2-closure-laminar matroids.Comment: 12 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
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