We present a common generalization of counting lattice points in rational polytopes and the enumeration of proper graph colorings, nowhere-zero flows on graphs, magic squares and graphs, antimagic squares and graphs, compositions of an integer whose parts are partially distinct, and generalized latin squares. Our method is to generalize Ehrhart's theory of lattice-point counting to a convex polytope dissected by a hyperplane arrangement. We particularly develop the applications to graph and signed-graph coloring, compositions of an integer, and antimagic labellings.
Generalized permutahedra are a class of polytopes with many interesting combinatorial subclasses. We introduce pruned inside-out polytopes, a generalization of inside-out polytopes introduced by Beck--Zaslavsky (2006), which have many applications such as recovering the famous reciprocity result for graph colorings by Stanley. We study the integer point count of pruned inside-out polytopes by applying classical Ehrhart polynomials and Ehrhart-Macdonald reciprocity. This yields a geometric perspective on and a generalization of a combinatorial reciprocity theorem for generalized permutahedra by…
The goal of this article is to obtain bounds on the coefficients of modular and integral flow and tension polynomials of graphs. To this end we make use of the fact that these polynomials can be realized as Ehrhart polynomials of inside-out polytopes. Inside-out polytopes come with an associated relative polytopal complex and, for a wide class of inside-out polytopes, we show that this complex has a convex ear decomposition. This leads to the desired bounds on the coefficients of these polynomials.
We show that, if the interior of a lattice d-polytope P contains at least one lattice point, then it contains a lattice point whose coefficient of asymmetry with respect to P is at most b for some number b depending on d only. As an application, we obtain new upper bounds on the volume of a lattice polytope given the number of lattice points in its interior.
There is a simple formula for the Ehrhart polynomial of a cyclic polytope. The purpose of this paper is to show that the same formula holds for a more general class of polytopes, lattice-face polytopes. We develop a way of decomposing any d-dimensional simplex in general position into d! signed sets, each of which corresponds to a permutation in the symmetric group, and reduce the problem of counting lattice points in a polytope in general position to that of counting lattice points in these special signed sets. Applying this decomposition to a lattice-face simplex, we obtain signed sets with …
We describe how to construct and enumerate Magic squares, Franklin squares, Magic cubes, and Magic graphs as lattice points inside polyhedral cones using techniques from Algebraic Combinatorics. The main tools of our methods are the Hilbert Poincare series to enumerate lattice points and the Hilbert bases to generate lattice points. We define polytopes of magic labelings of graphs and digraphs, and give a description of the faces of the Birkhoff polytope as polytopes of magic labelings of digraphs.
Can one build an arbitrary polytope from any polytope inside by iteratively stacking pyramids onto facets, without losing the convexity throughout the process? We prove that this is indeed possible for (i) 3-polytopes, (ii) 4-polytopes under a certain infinitesimal quasi-pyramidal relaxation, and (iii) all dimensions asymptotically. The motivation partly comes from our study of K-theory of monoid rings and of certain posets of discrete-convex objects.
We initiate the study of a class of polytopes, which we coin polypositroids, defined to be those polytopes that are simultaneously generalized permutohedra (or polymatroids) and alcoved polytopes. Whereas positroids are the matroids arising from the totally nonnegative Grassmannian, polypositroids are "positive" polymatroids. We parametrize polypositroids using Coxeter necklaces and balanced graphs, and describe the cone of polypositroids by extremal rays and facet inequalities. We introduce a notion of (W,c)-polypositroid for a finite Weyl group W and a choice of Coxeter element c. We connect…
We generalize Ehrhart's idea of counting lattice points in dilated rational polytopes: Given a rational simplex, that is, an n-dimensional polytope with n+1 rational vertices, we use its description as the intersection of n+1 halfspaces, which determine the facets of the simplex. Instead of just a single dilation factor, we allow different dilation factors for each of these facets. We give an elementary proof that the lattice point counts in the interior and closure of such a "vector-dilated" simplex are quasipolynomials satisfying an Ehrhart-type reciprocity law. This generalizes the classica…
We show that lattice polytopes cut out by root systems of classical type are normal and Koszul, generalizing a well-known result of Bruns, Gubeladze, and Trung in type A. We prove similar results for Cayley sums of collections of polytopes whose Minkowski sums are cut out by root systems. The proofs are based on a combinatorial characterization of diagonally split toric varieties.
We provide an elementary proof of a formula for the number of northeast lattice paths that lie in a certain region of the plane. Equivalently, this formula counts the lattice points inside the Pitman--Stanley polytope of an n-tuple.
We describe a provably complete algorithm for the generation of a tight, possibly exact superset of all combinatorially distinct simple n-facet polytopes in R^d, along with their graphs, f-vectors, and face lattices. The technique applies repeated cutting planes and planar sweeps to a d-simplex. Our generator has implications for several outstanding problems in polytope theory, including conjectures about the number of distinct polytopes, the edge expansion of polytopal graphs, and the d-step conjecture.
Sturmfels-Sullivant conjectured that the cut polytope of a graph is normal if and only if the graph has no K_5 minor. In the present paper, it is proved that the normality of cut polytopes of graphs is a minor closed property. By using this result, we have large classes of normal cut polytopes. Moreover, it turns out that, in order to study the conjecture, it is enough to consider 4-connected plane triangulations.
Understanding the face structure of the balanced minimal evolution (BME) polytope, especially its top-dimensional facets, is crucially important to phylogenetic applications. We show that BME polytope has a sub-lattice of its poset of faces which is isomorphic to a quotient of the well-studied permutoassociahedron. This sub-lattice corresponds to compatible sets of splits displayed by phylogenetic trees, and extends the lattice of faces of the BME polytope found by Hodge, Haws, and Yoshida. Each of the maximal elements in our new poset of faces corresponds to a single split of the leaves. Near…
We introduce revlex-initial 0/1-polytopes as the convex hulls of reverse-lexicographically initial subsets of 0/1-vectors. These polytopes are special knapsack-polytopes. It turns out that they have remarkable extremal properties. In particular, we use these polytopes in order to prove that the minimum numbers f(d, n) of facets and the minimum average degree a(d, n) of the graph of a d-dimensional 0/1-polytope with n vertices satisfy f(d, n) <= 3d and a(d, n) <= d + 4. We furthermore show that, despite the sparsity of their graphs, revlex-initial 0/1-polytopes satisfy a conjecture due to Mihai…
This note studies the geometric structure of monotone moment polytopes (the duals of smooth Fano polytopes) using probes. The latter are line segments that enter the polytope at an interior point of a facet and whose direction is integrally transverse to this facet. A point inside the polytope is displaceable by a probe if it lies less than half way along it. Using a construction due to Fukaya-Oh-Ohta-Ono, we show that every rational polytope has a central point that is not displaceable by probes. In the monotone (or more generally, the reflexive) case, this central point is its unique interio…
Given a (finite) simplicial complex, we define its $i$-th Laplacian polytope as the convex hull of the columns of its $i$-th Laplacian matrix. This extends Laplacian simplices of finite simple graphs, as introduced by Braun and Meyer. After studying basic properties of these polytopes, we focus on the $d$-th Laplacian polytope of the boundary of a $(d+1)$-simplex $\partial(σ_{d+1})$. If $d$ is odd, then as for graphs, the $d$-th Laplacian polytope turns out to be a $(d+1)$-simplex in this case. If $d$ is even, we show that the $d$-th Laplacian polytope of $\partial(σ_{d+1})$ is combinatorially…
We completely solve the problem of enumerating combinatorially inequivalent $d$-dimensional polytopes with $d+3$ vertices. A first solution of this problem, by Lloyd, was published in 1970. But the obtained counting formula was not correct, as pointed out in the new edition of Grünbaum's book. We both correct the mistake of Lloyd and propose a more detailed and self-contained solution, relying on similar preliminaries but using then a different enumeration method involving automata. In addition, we introduce and solve the problem of counting oriented and achiral (i.e. stable under reflection) …
Adjacency polytopes appear naturally in the study of nonlinear emergent phenomena in complex networks. The "PQ-type" adjacency polytope, denoted $\nabla^{\mathrm{PQ}}_G$ and which is the focus of this work, encodes rich combinatorial information about power-flow solutions in sparse power networks that are studied in electric engineering. Of particular importance is the normalized volume of such an adjacency polytope, which provides an upper bound on the number of distinct power-flow solutions. In this article we show that the problem of computing normalized volumes for $\nabla^{\mathrm{PQ}}_G$…
Recently, a combinatorial interpretation of Baldoni and Vergne's generalized Lidskii formula for the volume of a flow polytope was developed by Benedetti et al.. This converts the problem of computing Kostant partition functions into a problem of enumerating a set of objects called unified diagrams. We devise an enhanced version of this combinatorial model to compute the volumes of flow polytopes defined on a family of graphs called the k-caracol graphs, resulting in the first application of the model to non-planar graphs. At k=1 and k=n-1, we recover results for the classical caracol graph an…
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 in terms of the Lambert W function. A central tool is a description of the Ehrhart theory of a rational translate of an integer zonotope.
We give a simple construction of an orthogonal basis for the space of m by n matrices with row and column sums equal to zero. This vector space corresponds to the affine space naturally associated with the Birkhoff polytope, contingency tables and Latin squares. We also provide orthogonal bases for the spaces underlying magic squares and Sudoku boards. Our construction combines the outer (i.e., tensor or dyadic) product on vectors with certain rooted, vector-labeled, binary trees. Our bases naturally respect the decomposition of a vector space into centrosymmetric and skew-centrosymmetric piec…
A well-known theorem of Blind and Mani says that every simple polytope is uniquely determined by its graph. Kalai gave a very short and elegant proof of this result using the concept of acyclic orientations. As it turns out, Kalai's proof can be suitably generalized without much effort. We apply our results to a special class of cubical polytopes.
We introduce a graph structure on Euclidean polytopes. The vertices of this graph are the $d$-dimensional polytopes contained in $\mathbb{R}^d$ and its edges connect any two polytopes that can be obtained from one another by either inserting or deleting a vertex, while keeping their vertex sets otherwise unaffected. We prove several results on the connectivity of this graph, and on a number of its subgraphs. We are especially interested in several families of subgraphs induced by lattice polytopes, such as the subgraphs induced by the lattice polytopes with $n$ or $n+1$ vertices, that turn out…
Lattice polytopes which possess the integer decomposition property (IDP for short) turn up in many fields of mathematics. It is known that if the Cayley sum of lattice polytopes possesses IDP, then so does their Minkowski sum. In this paper, the Cayley sum of the order polytope of a finite poset and the stable set polytope of a finite simple graph is studied. We show that the Cayley sum of an order polytope and the stable set polytope of a perfect graph possesses a regular unimodular triangulation and IDP, and hence so does their Minkowski sum. Moreover, it turns out that, for an order polytop…
We investigate the complexity of counting the number of integer points in tropical polytopes, and the complexity of calculating their volume. We study the tropical analogue of the outer parallel body and establish bounds for its volume. We deduce that there is no approximation algorithm of factor $α=2^{\text{poly}(m,n)}$ for the volume of a tropical polytope given by $n$ vertices in a space of dimension $m$, unless P$=$NP. Neither is there such an approximation algorithm for counting the number of integer points in tropical polytopes described by vertices. If follows that approximating these v…
In this paper we motivate some new directions of research regarding the lattice width of convex bodies. We show that convex bodies of sufficiently large width contain a unimodular copy of a standard simplex. This implies that every lattice polytope contains a minimal generating set of the affine lattice spanned by its lattice points such that the number of generators is bounded by a constant which only depends on the dimension. We also discuss relations to recent results on spanning lattice polytopes and how our results could be viewed as the beginning of the study of generalized flatness cons…
We use the notions of reflexivity and of reflexive dimensions in order to introduce probability measures for lattice polytopes and initiate the investigation of their statistical properties. Examples of applications to discrete geometry include a study of randomness of self-duality of reflexive polytopes and implications for expectation values of the numbers of such polytopes in higher dimensions. We also discuss enumeration problems and related algorithms and point out interesting open problems. In this context we define the notion of IP-confined polytopes. Our new results include the list of…
Given a configuration $A$ of $n$ points in $\mathbb{R}^{d-1}$, we introduce the higher secondary polytopes $Σ_{A,1},\dots, Σ_{A,n-d}$, which have the property that $Σ_{A,1}$ agrees with the secondary polytope of Gelfand--Kapranov--Zelevinsky, while the Minkowski sum of these polytopes agrees with Billera--Sturmfels' fiber zonotope associated with (a lift of) $A$. In a special case when $d=3$, we refer to our polytopes as higher associahedra. They turn out to be related to the theory of total positivity, specifically, to certain combinatorial objects called plabic graphs, introduced by the seco…
The stable set polytope of a graph $G$, denoted as STAB($G$), is the convex hull of all the incidence vectors of stable sets of $G$. To describe a linear system which defines STAB($G$) seems to be a difficult task in the general case. In this paper we present a complete description of the stable set polytope of ($P_6$,triangle)-free graphs (and more generally of ($P_6$,paw)-free graphs). For that we combine different tools, in the context of a well known result of Chvátal \cite{Chvatal1975} which allows to focus just on prime facet-inducing graphs, with particular reference to a structure resu…
This article concerns the computational problem of counting the lattice points inside convex polytopes, when each point must be counted with a weight associated to it. We describe an efficient algorithm for computing the highest degree coefficients of the weighted Ehrhart quasi-polynomial for a rational simple polytope in varying dimension, when the weights of the lattice points are given by a polynomial function h. Our technique is based on a refinement of an algorithm of A. Barvinok [Computing the Ehrhart quasi-polynomial of a rational simplex, Math. Comp. 75 (2006), pp. 1449--1466] in the u…
We describe a new sequence of polytopes which characterize A_infinity maps from a topological monoid to an A_infinity space. Therefore each of these polytopes is a quotient of the corresponding multiplihedron. Later term(s) in our sequence of polytopes are demonstrated not to be combinatorially equivalent to the associahedron, as was previously assumed. They are given the new collective name composihedra. We point out how these polytopes are used to parameterize compositions in the formulation of the theories of enriched bicategories and pseudomonoids in a monoidal bicategory. We present a sim…
