597 research outputs found
-vectors of small matroid complexes
Stanley conjectured in 1977 that the -vector of a matroid simplicial
complex is a pure -sequence. We give simple constructive proofs that the
conjecture is true for matroids of rank less than or equal to 3, and corank 2.
We used computers to verify that Stanley's conjecture holds for all matroids on
at most nine elements.Comment: 11 pages, 1 figur
Generic and special constructions of pure O-sequences
It is shown that the h-vectors of Stanley-Reisner rings of three classes of
matroids are pure O-sequences. The classes are (a) matroids that are
truncations of other matroids, or more generally of Cohen-Macaulay complexes,
(b) matroids whose dual is (rank + 2)-partite, and (c) matroids of
Cohen-Macaulay type at most five. Consequences for the computational search for
a counterexample to a conjecture of Stanley are discussed.Comment: 16 pages, v2: various small improvements, accepted by Bulletin of the
London Math. Societ
Lexicographic shellability, matroids and pure order ideals
In 1977 Stanley conjectured that the -vector of a matroid independence
complex is a pure -sequence. In this paper we use lexicographic shellability
for matroids to motivate a combinatorial strengthening of Stanley's conjecture.
This suggests that a pure -sequence can be constructed from combinatorial
data arising from the shelling. We then prove that our conjecture holds for
matroids of rank at most four, settling the rank four case of Stanley's
conjecture. In general, we prove that if our conjecture holds for all rank
matroids on at most elements, then it holds for all matroids
Pure O-sequences and matroid h-vectors
We study Stanley's long-standing conjecture that the h-vectors of matroid
simplicial complexes are pure O-sequences. Our method consists of a new and
more abstract approach, which shifts the focus from working on constructing
suitable artinian level monomial ideals, as often done in the past, to the
study of properties of pure O-sequences. We propose a conjecture on pure
O-sequences and settle it in small socle degrees. This allows us to prove
Stanley's conjecture for all matroids of rank 3. At the end of the paper, using
our method, we discuss a first possible approach to Stanley's conjecture in
full generality. Our technical work on pure O-sequences also uses very recent
results of the third author and collaborators.Comment: Contains several changes/updates with respect to the previous
version. In particular, a discussion of a possible approach to the general
case is included at the end. 13 pages. To appear in the Annals of
Combinatoric
COMs: Complexes of Oriented Matroids
In his seminal 1983 paper, Jim Lawrence introduced lopsided sets and featured
them as asymmetric counterparts of oriented matroids, both sharing the key
property of strong elimination. Moreover, symmetry of faces holds in both
structures as well as in the so-called affine oriented matroids. These two
fundamental properties (formulated for covectors) together lead to the natural
notion of "conditional oriented matroid" (abbreviated COM). These novel
structures can be characterized in terms of three cocircuits axioms,
generalizing the familiar characterization for oriented matroids. We describe a
binary composition scheme by which every COM can successively be erected as a
certain complex of oriented matroids, in essentially the same way as a lopsided
set can be glued together from its maximal hypercube faces. A realizable COM is
represented by a hyperplane arrangement restricted to an open convex set. Among
these are the examples formed by linear extensions of ordered sets,
generalizing the oriented matroids corresponding to the permutohedra. Relaxing
realizability to local realizability, we capture a wider class of combinatorial
objects: we show that non-positively curved Coxeter zonotopal complexes give
rise to locally realizable COMs.Comment: 40 pages, 6 figures, (improved exposition
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