Related papers: Faces of Generalized Permutohedra
The concept of typical and weighted typical spherical faces for tessellations of the $d$-dimensional unit sphere, generated by $n$ independent random great hyperspheres distributed according to a non-degenerate directional distribution, is…
We give the facet description of the deformation cones of graph associahedra and nestohedra, generalizing the classical parametrization of the family of deformed permutahedra by the cone of submodular functions. When the underlying building…
We consider redundant analogues of the f- and h-vectors of simplicial complexes and present bases of R^{m+1} related to these ``long'' f- and h-vectors describing the face systems from 2^{1,...,m}; we list the corresponding change of basis…
McMullen's g-vector is important for simple convex polytopes. This paper postulates axioms for its extension to general convex polytopes. It also conjectures that, for each dimension d, a stated finite calculation gives the formula for the…
The family of Buchsbaum simplicial posets generalizes the family of simplicial cell manifolds. The $h'$-vector of a simplicial complex or simplicial poset encodes the combinatorial and topological data of its face numbers and the reduced…
We introduce a higher-uniformity analogue of graphic zonotopes and permutohedra. Specifically, given a $(d+1)$-uniform hypergraph $H$, we define its hypergraphic zonotope $\mathcal{Z}_H$, and when $H$ is the complete $(d+1)$-uniform…
Let $G$ be a finite group. For a fixed element $g$ in $G$ and a given subgroup $H$ of $G$, the relative $g$-noncommuting graph of $G$ is a simple undirected graph whose vertex set is $G$ and two vertices $x$ and $y$ are adjacent if $x \in…
Kupavskii, Volostnov, and Yarovikov have recently shown that any set of $n$ points in general position in the plane has at least as many (partial) triangulations as the convex $n$-gon. We generalize this in two directions: we show that…
When the number of non-triangular faces adjacent to a vertex $v$ is less than or equal to three, the vertex $v$ will be called (\emph{combinatorially}) \emph{rigid}. We study the number of rigid vertices and suggest a conjecture on a…
We present two algorithms that compute the Newton polytope of a polynomial defining a hypersurface H in C^n using numerical computation. The first algorithm assumes that we may only compute values of f - this may occur if f is given as a…
We find an explicit formula for the gamma vector in terms of the input polynomial in a way that extends it to arbitrary polynomials. More specifically, we find explicit linear combination in terms of coefficients of the input polynomial…
There is a well-established dictionary between zonotopes, hyperplane arrangements, and their (oriented) matroids. Arguably one of the most famous examples is the class of graphical zonotopes, also called acyclotopes, which encode…
This paper introduces a colored generalization of the Eulerian polynomials, denoted the $\alpha$-colored Eulerian polynomials. We first compute these polynomials by taking the $h$-vector of the $\alpha$-colored permutohedron, a colored…
We compare various viewpoints on down-sets (simplicial complexes), illustrating how the combinatorial inclusion-exclusion principle may serve as an alternative to more advanced methods of studying their face numbers.
Several recent papers have examined a rational polyhedron $P_m$ whose integer points are in bijection with the numerical semigroups (cofinite, additively closed subsets of the non-negative integers) containing $m$. A combinatorial…
We prove that if a pure simplicial complex of dimension d with n facets has the least possible number of (d-1)-dimensional faces among all complexes with n faces of dimension d, then it is vertex decomposable. This answers a question of J.…
We characterize all permutations which realize as the $z$-monodromies of faces in connected simple finite graphs embedded in surfaces whose duals are also simple.
Generalized permutohedra are deformations of regular permutohedra, and arise in many different fields of mathematics. One important characterization of generalized permutohedra is the Submodular Theorem, which is related to the deformation…
The $f$-vector of a polytope consists of the numbers of its $i$-dimensional faces. An open field of study is the characterization of all possible $f$-vectors. It has been solved in three dimensions by Steinitz in the early 19th century. We…
We connect generalized permutahedra with Schubert calculus. Thereby, we give sufficient vanishing criteria for Schubert intersection numbers of the flag variety. Our argument utilizes recent developments in the study of Schubitopes, which…