Related papers: The $cd$-index of base polytopes for connected spl…
In this paper, we study flag structures of matroid base polytopes. We describe faces of matroid base polytopes in terms of matroid data, and give conditions for hyperplane splits of matroid base polytopes. Also, we show how the cd-index of…
This paper initiates the explicit study of face numbers of matroid polytopes and their computation. We prove that, for the large class of split matroid polytopes, their face numbers depend solely on the number of cyclic flats of each rank…
The flag vector contains all the face incidence data of a polytope, and in the poset setting, the chain enumerative data. It is a classical result due to Bayer and Klapper that for face lattices of polytopes, and more generally, Eulerian…
A well-known conjecture of McMullen, proved by Billera, Lee and Stanley, describes the face numbers of simple polytopes. The necessary and sufficient condition is that the toric g-vector of the polytope is an M-vector, that is, the vector…
A positroid is a special case of a realizable matroid that arose from the study of the totally nonnegative part of the Grassmannian by Postnikov. In this paper, we study the facets of its matroid polytope and the independent set polytope.…
For a lattice polytope $P$, the rank of $P$ is defined by $F-(\dim P+1)$, where $F$ is the number of facets of $P$. In this paper, we study matroid polytopes with small rank. More precisely, we characterize matroid independence polytopes…
Let $P(M)$ be the matroid base polytope of a matroid $M$. A {\em matroid base polytope decomposition} of $P(M)$ is a decomposition of the form $P(M) = \bigcup\limits_{i=1}^t P(M_{i})$ where each $P(M_i)$ is also a matroid base polytope for…
We show that the $\gamma$-vector of the order complex of any polytope is the f-vector of a balanced simplicial complex. This is done by proving this statement for a subclass of Stanley's S-shellable spheres which includes all polytopes. The…
Polymatroids can be considered as "fractional matroid" where the rank function is not required to be integer valued. Many, but not every notion in matroid terminology translates naturally to polymatroids. Defining cyclic flats of a…
This is a continuation of the early paper concerning matroid base polytope decomposition. Here, we will present sufficient conditions on $M$ so its base matroid polytope $P(M)$ has a {\em sequence} of hyperplane splits. The latter yields to…
Over a finite field $\mathbb{F}_{q^m}$, the evaluation of skew polynomials is intimately related to the evaluation of linearized polynomials. This connection allows one to relate the concept of polynomial independence defined for skew…
A flag positroid of ranks $\boldsymbol{r}:=(r_1<\dots <r_k)$ on $[n]$ is a flag matroid that can be realized by a real $r_k \times n$ matrix $A$ such that the $r_i \times r_i$ minors of $A$ involving rows $1,2,\dots,r_i$ are nonnegative for…
In this sequel to "Foundations of matroids - Part 1", we establish several presentations of the foundation of a matroid in terms of small building blocks. For example, we show that the foundation of a matroid M is the colimit of the…
We specify what is meant for a polytope to be reconstructible from its graph or dual graph. And we introduce the problem of class reconstructibility, i.e., the face lattice of the polytope can be determined from the (dual) graph within a…
It is possible to write the indicator function of any matroid polytope as an integer combination of indicator functions of Schubert matroid polytopes. In this way, every matroid on $n$ elements of rank $r$ can be thought of as a lattice…
In this manuscript we study the subdivisions of the permutahedron $\Pi_n$ into two subpolytopes corresponding to flags of positroids, which are in particular flags of lattice path matroids (LPFMs). A subpolytope $P_{[u,v]}$ of $\Pi_n$ is a…
Motivated by entanglement-assisted quantum error-correcting codes, where the hull dimension determines the number of required pre-shared entangled pairs, we study hulls of two families of $\mathbb{F}_q$-linear codes defined by…
Let $M$ be a matroid defined on a finite set $E$ and $L\subset E$. $L$ is locked in $M$ if $M|L$ and $M^*|(E\backslash L)$ are 2-connected, and $min\{r(L), r^*(E\backslash L)\} \geq 2$. Locked subsets characterize nontrivial facets of the…
We derive formulas for the cd-index and the toric h-vector of a convex polytope P from a sweeping by a hyperplane. These arise from interpreting the corresponding S-shelling of the dual of P. We describe a partition of the faces of the…
A partitioned matroid $(M, \{X_1,X_2,\dots,X_n\})$ consists of a matroid $M$ and a partition $\{X_1,X_2,\dots,X_n\}$ of its ground set. As such structures arise frequently in structural matroid theory, this paper introduces a general…