The combinatorics of interval-vector polytopes

An \emph{interval vector} is a $(0,1)$-vector in $\mathbb{R}^n$ for which all the 1's appear consecutively, and an \emph{interval-vector polytope} is the convex hull of a set of interval vectors in $\mathbb{R}^n$. We study three particular classes of interval vector polytopes which exhibit interesting geometric-combinatorial structures; e.g., one class has volumes equal to the Catalan numbers, whereas another class has face numbers given by the Pascal 3-triangle.Comment: 10 pages, 3 figure

The EtE_t-Construction for Lattices, Spheres and Polytopes

We describe and analyze a new construction that produces new Eulerian lattices from old ones. It specializes to a construction that produces new strongly regular cellular spheres (whose face lattices are Eulerian). The construction does not always specialize to convex polytopes; however, in a number of cases where we can realize it, it produces interesting classes of polytopes. Thus we produce an infinite family of rational 2-simplicial 2-simple 4-polytopes, as requested by Eppstein, Kuperberg and Ziegler. We also construct for each $d\ge3$ an infinite family of $(d-2)$-simplicial 2-simple $d$-polytopes, thus solving a problem of Gr\"unbaum.Comment: 21 pages, many figure

Combinatorics and Geometry of Transportation Polytopes: An Update

A transportation polytope consists of all multidimensional arrays or tables of non-negative real numbers that satisfy certain sum conditions on subsets of the entries. They arise naturally in optimization and statistics, and also have interest for discrete mathematics because permutation matrices, latin squares, and magic squares appear naturally as lattice points of these polytopes. In this paper we survey advances on the understanding of the combinatorics and geometry of these polyhedra and include some recent unpublished results on the diameter of graphs of these polytopes. In particular, this is a thirty-year update on the status of a list of open questions last visited in the 1984 book by Yemelichev, Kovalev and Kravtsov and the 1986 survey paper of Vlach.Comment: 35 pages, 13 figure

Posets arising as 1-skeleta of simple polytopes, the nonrevisiting path conjecture, and poset topology

Given any polytope $P$ and any generic linear functional ${\bf c}$, one obtains a directed graph $G(P,{\bf c})$ by taking the 1-skeleton of $P$ and orienting each edge $e(u,v)$ from $u$ to $v$ for ${\bf c} (u) < {\bf c} ( v)$. This paper raises the question of finding sufficient conditions on a polytope $P$ and generic cost vector ${\bf c}$ so that the graph $G(P, {\bf c} )$ will not have any directed paths which revisit any face of $P$ after departing from that face. This is in a sense equivalent to the question of finding conditions on $P$ and ${\bf c}$ under which the simplex method for linear programming will be efficient under all choices of pivot rules. Conditions on $P$ and ${\bf c}$ are given which provably yield a corollary of the desired face nonrevisiting property and which are conjectured to give the desired property itself. This conjecture is proven for 3-polytopes and for spindles having the two distinguished vertices as source and sink; this shows that known counterexamples to the Hirsch Conjecture will not provide counterexamples to this conjecture. A part of the proposed set of conditions is that $G(P, {\bf c} )$ be the Hasse diagram of a partially ordered set, which is equivalent to requiring non revisiting of 1-dimensional faces. This opens the door to the usage of poset-theoretic techniques. This work also leads to a result for simple polytopes in which $G(P, {\bf c})$ is the Hasse diagram of a lattice L that the order complex of each open interval in L is homotopy equivalent to a ball or a sphere of some dimension. Applications are given to the weak Bruhat order, the Tamari lattice, and more generally to the Cambrian lattices, using realizations of the Hasse diagrams of these posets as 1-skeleta of permutahedra, associahedra, and generalized associahedra.Comment: new results for 3-polytopes and spindles added; exposition substantially improved throughou

Bier spheres and posets

In 1992 Thomas Bier presented a strikingly simple method to produce a huge number of simplicial (n-2)-spheres on 2n vertices as deleted joins of a simplicial complex on n vertices with its combinatorial Alexander dual. Here we interpret his construction as giving the poset of all the intervals in a boolean algebra that "cut across an ideal." Thus we arrive at a substantial generalization of Bier's construction: the Bier posets Bier(P,I) of an arbitrary bounded poset P of finite length. In the case of face posets of PL spheres this yields cellular "generalized Bier spheres." In the case of Eulerian or Cohen-Macaulay posets P we show that the Bier posets Bier(P,I) inherit these properties. In the boolean case originally considered by Bier, we show that all the spheres produced by his construction are shellable, which yields "many shellable spheres", most of which lack convex realization. Finally, we present simple explicit formulas for the g-vectors of these simplicial spheres and verify that they satisfy a strong form of the g-conjecture for spheres.Comment: 15 pages. Revised and slightly extended version; last section rewritte