Related papers: On posets and polytopes attached to arbors
Preorder polytopes, defined from preorders on finite sets, are introduced and studied from a lattice point enumeration point of view. They naturally generalize arbor polytopes, recently introduced and studied by the second named author.…
This paper proves four conjectured generating series, due to Chapoton, which concern invariants of posets and polytopes associated with a specific sequence of arbors. Two of these conjectures provide closed-form formulas for the generating…
We compute the characteristic polynomials of the posets of hypertrees. We show that the generating series of the polynomials can be expressed using cyclic hypertrees. We also propose a conjecture on the action of the symmetric groups on the…
In the last decade, the order polytope of the zigzag poset has been thoroughly studied. A related poset, called \emph{crown poset}, obtained by adding an extra relation between the endpoints of an even zigzag poset, is not so well…
Combinatorial Hopf algebras of trees exemplify the connections between operads and bialgebras. Painted trees were introduced recently as examples of how graded Hopf operads can bequeath Hopf structures upon compositions of coalgebras. We…
We introduce a new matroid (graph) invariant, the arboricity polynomial. Given a matroid, the arboricity polynomial enumerates the number of covers of the ground set by disjoint independent sets. We establish the polynomiality of the…
This is an elementary presentation of the arithmetic of trees. We show how it is related to the Tamari poset. In the last part we investigate various ways of realizing this poset as a polytope (associahedron), including one inferred from…
We investigate a trivariate polynomial associated with rooted trees. It generalises a bivariate polynomial for rooted trees that was recently introduced by Liu. We show that this polynomial satisfies a deletion-contraction recursion and can…
We study generating functions of strict and non-strict order polynomials of series-parallel posets, called order series. These order series are closely related to Ehrhart series and h*-polynomials of the associated order polytopes. We…
Motivated by representation theory and geometry, we introduce and develop an equivariant generalization of Ehrhart theory, the study of lattice points in dilations of lattice polytopes. We prove representation-theoretic analogues of…
Tutte's dichromate T(x,y) is a well known graph invariant. Using the original definition in terms of internal and external activities as our point of departure, we generalize the valuations T(x,1) and T(1,y) to hypergraphs. In the…
We study two different objects attached to an arbitrary quadrangulation of a regular polygon. The first one is a poset, closely related to the Stokes polytopes introduced by Baryshnikov. The second one is a set of some paths configurations…
We investigate linear and additive codes in partially ordered Hamming-like spaces that satisfy the extension property, meaning that automorphisms of ideals extend to automorphisms of the poset. The codes are naturally described in terms of…
The proper parts of face lattices of convex polytopes are shown to satisfy a strong form of the Cohen--Macaulay property, namely that removing from their Hasse diagram all edges in any closed interval results in a Cohen--Macaulay poset of…
Trees or rooted trees have been generously studied in the literature. A forest is a set of trees or rooted trees. Here we give recurrence relations between the number of some kind of rooted forest with $k$ roots and that with $k+1$ roots on…
To any poset or quasi-poset is attached a lattice polytope, whose Ehrhart polynomial we study from a Hopf-algebraic point of view. We use for this two interacting bialgebras on quasi-posets. The Ehrhart polynomial defines a Hopf algebra…
In this paper, we provide an overview of Ehrhart polynomials associated with order polytopes of finite posets, a concept first introduced by Stanley. We focus on their combinatorial interpretations for many sequences listed on the OEIS. We…
We discuss generalizations of some results on lattice polygons to certain piecewise linear loops which may have a self-intersection but have vertices in the lattice $\mathbb{Z}^2$. We first prove a formula on the rotation number of a…
Given a tree $T$, its path polytope is the convex hull of the edge indicator vectors for the paths between any two distinct leaves in $T$. These polytopes arise naturally in polyhedral geometry and applications, such as phylogenetics,…
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…