相关论文: The augmented external activity complex of a matro…
We prove that the external activity complex $\textrm{Act}_<(M)$ of a matroid is shellable. In fact, we show that every linear extension of LasVergnas's external/internal order $<_{ext/int}$ on $M$ provides a shelling of $\textrm{Act}_<(M)$.…
We introduce the Schubert variety of a pair of linear subspaces in $\mathbf{C}^n$ and the external activity complex of a pair of not necessarily realizable matroids. Both of these generalize constructions of Ardila et al., which occur when…
We study the augmented Bergman complex of a closure operator on a finite set, which interpolates between the order complex of proper flats and the independence complex of the operator. In 2020, Braden, Huh, Matherne, Proudfoot, and Wang…
The augmented Bergman complex of a matroid is a simplicial complex introduced recently in work of Braden, Huh, Matherne, Proudfoot and Wang. It may be viewed as a hybrid of two well-studied pure shellable simplicial complexes associated to…
Shelling orders are a ubiquitous tool used to understand invariants of cell complexes. Significant effort has been made to develop techniques to decide when a given complex is shellable. However, empirical evidence shows that some shelling…
We say that a pure $d$-dimensional simplicial complex $\Delta$ on $n$ vertices is \emph{shelling completable} if $\Delta$ can be realized as the initial sequence of some shelling of $\Delta_{n-1}^{(d)}$, the $d$-skeleton of the…
Given a matroid $M=(E,{\cal I})$, and a total ordering over the elements $E$, a broken circuit is a circuit where the smallest element is removed and an NBC independent set is an independent set in ${\cal I}$ with no broken circuit. The set…
We introduce the separability complex, a one-complex associated to a finite regular cover of the rose and show that it is connected if and only if the fundamental group of the associated cover is generated by its intersection with the set…
Chordal graph shelling antimatroids have received little attention with regard to their combinatorial properties and related optimization problems, as compared to the case of poset shelling antimatroids. Here we consider a special case of…
One of the major open questions in matroid theory asks whether the $h$-vector $(h_0,h_1,\ldots,h_s)$ of the broken circuit complex of a matroid $M$ satisfies the following inequalities: $$ h_0\leq h_1\leq \cdots\leq h_{\lfloor s/2\rfloor}…
We extend the splitting operation from binary matroids (Raghunathan et al., 1998) to $p$- matroids, where $p$-matroids refer to matroids representable over $GF(p).$ We also characterize circuits, bases, and independent sets of the resulting…
In 1977 Stanley conjectured that the $h$-vector of a matroid independence complex is a pure $O$-sequence. In this paper we use lexicographic shellability for matroids to motivate a combinatorial strengthening of Stanley's conjecture. This…
If a pure simplicial complex is partitionable, then its $h$-vector has a combinatorial interpretation in terms of any partitioning of the complex. Given a non-partitionable complex $\Delta$, we construct a complex $\Gamma \supseteq \Delta$…
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…
Let L be a lattice admitting a left-modular chain of length r, not necessarily maximal. We show that if either L is graded or the chain is modular, then the (r-2)-skeleton of L is vertex-decomposable (hence shellable). This proves a…
We study structural and topological properties of nested set complexes of matroids with arbitrary building sets, proving that these complexes are vertex decomposable and admit convex ear decompositions. These results unify and generalize…
We explore a combinatorial theory of linear dependency in complex space, "complex matroids", with foundations analogous to those for oriented matroids. We give multiple equivalent axiomatizations of complex matroids, showing that this…
We study the behavior of $h$-vectors associated to matroid complexes under weak maps, or inclusions of matroid polytopes. Specifically, we show that the $h$-vector of the order complex of the lattice of flats of a matroid is component-wise…
A matroid base polytope is a polytope in which each vertex has 0,1 coordinates and each edge is parallel to a difference of two coordinate vectors. Matroid base polytopes are described combinatorially by integral submodular functions on a…
We give inductive constructions of independent graphs that contain implied nonedges but do not contain any non-trivial rigid subgraphs, or \emph{nucleations}: some of the constructions and proofs apply to 3-dimensional abstract rigidity…