Related papers: On posets and polytopes attached to arbors
Let $P\subset\mathbb R^n$ be a convex polytope and let $\ell$ be a linear functional which is nonconstant on every edge of $P$. The induced acyclic orientation determines positive and negative Bia{\l}ynicki-Birula type partitions of $P$…
Ehrhart theory measures a polytope P discretely by counting the lattice points inside its dilates P, 2P, 3P, .... We compute the Ehrhart quasipolynomials of the standard Coxeter permutahedra for the classical Coxeter groups, expressing them…
We describe a family of shellings for the canonical triangulation of the order polytope of the zig-zag poset. This gives a new combinatorial interpretation for the coefficients in the numerator of the Ehrhart series of this order polytopein…
Motivated by the graph associahedron KG, a polytope whose face poset is based on connected subgraphs of G, we consider the notion of associativity and tubes on posets. This leads to a new family of simple convex polytopes obtained by…
The Interval poset of a permutation is an effective way of capturing all the intervals of the permutation and the inclusions between them and was introduced recently by Tenner. Thi paper explores the geometric interpretation of interval…
We give an algorithm for computing the Teichm\"uller polynomial for a certain class of fibered alternating links associated to trees. Furthermore, we exhibit a mutant pair of such links distinguished by the Teichm\"uller polynomial.
The type A_n full root polytope is the convex hull in R^{n+1} of the origin and the points e_i-e_j for 1<= i<j <= n+1. Given a tree T on the vertex set [n+1], the associated root polytope P(T) is the intersection of the full root polytope…
We prove that the acyclic reorientation poset of a directed acyclic graph $D$ is a lattice if and only if the transitive reduction of any induced subgraph of $D$ is a forest. We then show that the acyclic reorientation lattice is always…
We prove the unimodality of the Ehrhart $\delta$-polynomial of the chain polytope of the zig-zag poset, which was conjectured by Kirillov. First, based on a result due to Stanley, we show that this polynomial coincides with the…
We study restricted chain-order polytopes associated to Young diagrams using combinatorial mutations. These polytopes are obtained by intersecting chain-order polytopes with certain hyperplanes. The family of chain-order polytopes…
Given a functor from any category into the category of topological spaces, one obtains a linear representation of the category by post-composing the given functor with a homology functor with field coefficients. This construction is…
Generalizing the decomposition of a connected planar graph into a tree and a dual tree, we prove a combinatorial analog of the classic Helmholz-Hodge decomposition of a smooth vector field. Specifically, we show that for every polyhedral…
To most mathematicians and computer scientists the word ``tree'' conjures up, in addition to the usual image, the image of a connected graph with no circuits. In the last few years various types of trees have been the subject of much…
We describe in this note a torsor structure arising on the affine scheme defined by a system of rationnal algebraic relations between polyzetas at roots of unity (values of hyperlogarithmic functions on a fixed finite group of complex roots…
Our input is a directed, rooted graph $G = (V \cup \{r\},E)$ where each vertex in $V$ has a partial order preference over its incoming edges. The preferences of a vertex extend naturally to preferences over arborescences rooted at $r$. We…
To every poset P, Stanley (1986) associated two polytopes, the order polytope and the chain polytope, whose geometric properties reflect the combinatorial qualities of P. This construction allows for deep insights into combinatorics by way…
The homological information about a filtered simplicial complex over the poset of positive real numbers is often presented by a barcode which depicts the evolution of the associated Betti numbers. However, there is a wonderfully complex…
For a discrete poset $\mathcal X$ McCord proved that the natural map $|{\mathcal X}|\to {\mathcal X}$ from the order complex to the poset equipped with the Up topology is a weak homotopy equivalence. Much later, Zivaljevi\'{c} defined the…
We adapt here the computation of characters on incidence Hopf algebras introduced by W. Schmitt in the 1990s to a family mixing bounded and unbounded posets. We then apply our results to the family of hypertree posets and partition posets.…
The tree complex is a simplicial complex defined in recent work of Belk, Lanier, Margalit, and Winarski with natural applications to mapping class groups and complex dynamics. In this article, we connect this setting with the study of…