Face numbers of triangulations of simplicial complexes were studied by Stanley by use of his concept of a local $h$-vector. It is shown that a parallel theory exists for cubical subdivisions of cubical complexes, in which the role of the $h$-vector of a simplicial complex is played by the (short or long) cubical $h$-vector of a cubical complex, defined by Adin, and the role of the local $h$-vector of a triangulation of a simplex is played by the (short or long) cubical local $h$-vector of a cubical subdivision of a cube. The cubical local $h$-vectors are defined in this paper and are shown to share many of the properties of their simplicial counterparts. Generalizations to subdivisions of locally Eulerian posets are also discussed.
The cubical barycentric subdivision sd_c(K) of a cubical complex K is introduced as an analogue of the barycentric subdivision of a simplicial complex. Explicit formulas for the short and long cubical h-vector of sd_c(K) are given, in terms of those of K. It is deduced that symmetry and nonnegativity of these h-vectors, as well as real rootedness of the short cubical h-polynomial, are preserved under cubical barycentric subdivision. The asymptotic behavior of the short and long cubical h-vectors of successive cubical barycentric subdivisions of K is also determined.
The $h$-polynomial of the barycentric subdivision of any $n$-dimensional cubical complex with nonnegative cubical $h$-vector is shown to have only real roots and to be interlaced by the Eulerian polynomial of type $B_n$. This result applies to barycentric subdivisions of shellable cubical complexes and, in particular, to barycentric subdivisions of cubical convex polytopes and answers affirmatively a question of Brenti, Mohammadi and Welker.
In geometric, algebraic, and topological combinatorics, the unimodality of combinatorial generating polynomials is frequently studied. Unimodality follows when the polynomial is (real) stable, a property often deduced via the theory of interlacing polynomials. Many of the open questions on stability and unimodality of polynomials pertain to the enumeration of faces of cell complexes. In this paper, we relate the theory of interlacing polynomials to the shellability of cell complexes. We first derive a sufficient condition for stability of the $h$-polynomial of a subdivision of a shellable comp…
What is the minimal closed cone containing all $f$-vectors of cubical $d$-polytopes? We construct cubical polytopes showing that this cone, expressed in the cubical $g$-vector coordinates, contains the nonnegative $g$-orthant, thus verifying one direction of the Cubical Generalized Lower Bound Conjecture of Babson, Billera and Chan. Our polytopes also show that a natural cubical analogue of the simplicial Generalized Lower Bound Theorem does not hold.
This paper introduces a new and simple statistic on noncrossing partitions that expresses each coordinate of the toric $h$-vector of a cubical complex, written in the basis of the Adin $h$-vector entries, as the total weight of all noncrossing partitions. The same model may also be used to obtain a very simple combinatorial interpretation of the contribution of a cubical shelling component to the toric $h$-vector. In this model, a strengthening of the symmetry expressed by the Dehn-Sommerville equations may be derived from the self-duality of the noncrossing partition lattice, exhibited by the…
Although the Unimodality Conjecture holds for some certain classes of cubical polytopes (e.g. cubes, capped cubical polytopes, neighborly cubical polytopes), it fails for cubical polytopes in general. A 12-dimensional cubical polytope with non-unimodal face vector is constructed by using capping operations over a neighborly cubical polytope with 2 to the power 131 vertices. For cubical polytopes, the Unimodality Conjecture is proved for dimensions less than 11. The first one-third of the face vector of a cubical polytope is increasing and its last one-third is decreasing in any dimension.
Several recent papers have addressed the problem of characterizing the $f$-vectors of cubical polytopes. This is largely motivated by the complete characterization of the $f$-vectors of simplicial polytopes given by Stanley, Billera, and Lee in 1980. Along these lines Blind and Blind have shown that unlike in the simplicial case, there are parity restrictions on the $f$-vectors of cubical polytopes. In particular, except for polygons, all even dimensional cubical polytopes must have an even number of vertices. Here this result is extended to a class of zonotopal complexes which includes simply…
In this paper, we answer two questions on local $h$-vectors, which were asked by Athanasiadis. First, we characterize all possible local $h$-vectors of quasi-geometric subdivisions of a simplex. Second, we prove that the local $γ$-vector of the barycentric subdivision of any CW-regular subdivision of a simplex is nonnegative. Along the way, we derive a new recurrence formula for the derangement polynomials.
S. Ekhad and D. Zeilberger recently proved that the multivariate generating function for the number of simple singular vector tuples of a generic $m_1 \times \cdots \times m_d$ tensor has an elegant rational form involving elementary symmetric functions, and provided a partial conjecture for the asymptotic behavior of the cubical case $m_1 = \cdots = m_d$. We prove this conjecture and further identify completely the dominant asymptotic term, including the multiplicative constant. Finally, we use the method of differential approximants to conjecture that the subdominant connective constant effe…
The enumerative theory of simplicial subdivisions (triangulations) of simplicial complexes was developed by Stanley in order to understand the effect of such subdivisions on the $h$-vector of a simplicial complex. A key role there is played by the concept of a local $h$-vector. This paper surveys some of the highlights of this theory and some recent developments, concerning subdivisions of flag homology spheres and their $γ$-vectors. Several interesting examples and open problems are discussed.
For a poset $P$ and an integer $r\geq 1$, let $P_r$ be a collection of all $r$-multichains in $P$. Corresponding to each strictly increasing map $ı:[r]\rightarrow [2r]$, there is an order $\preceq_ı$ on $P_r$. Let $\D(G_ı(P_r))$ be the clique complex of the graph $G_ı$ associated to $P_r$ and $ı$. In a recent paper \cite{NW}, it is shown that $\D(G_ı(P_r))$ is a subdivision of $P$ for a class of strictly increasing maps. In this paper, we show that all these subdivisions have the same $f$-vector. We give an explicit description of the transformation matrices from the $f$- and $h$-vectors of $Δ…
The cluster complex $Δ(Φ)$ is an abstract simplicial complex, introduced by Fomin and Zelevinsky for a finite root system $Φ$. The positive part of $Δ(Φ)$ naturally defines a simplicial subdivision of the simplex on the vertex set of simple roots of $Φ$. The local $h$-vector of this subdivision, in the sense of Stanley, is computed and the corresponding $γ$-vector is shown to be nonnegative. Combinatorial interpretations to the entries of the local $h$-vector and the corresponding $γ$-vector are provided for the classical root systems, in terms of noncrossing partitions of types $A$ and $B$. A…
The lth partial barycentric subdivision is defined for a (d-1)-dimensional simplicial complex Δand studied along with its combinatorial, geometric and algebraic aspects. We analyze the behavior of the f- and h-vector under the lth partial barycentric subdivision extending previous work of Brenti and Welker on the standard barycentric subdivision -- the case l = 1. We discuss and provide properties of the transformation matrices sending the f- and h-vector of Δto the f- and h-vector of its lth partial barycentric subdivision. We conclude with open problems.
The $γ$-vector is an important enumerative invariant of a flag simplicial homology sphere. It has been conjectured by Gal that this vector is nonnegative for every such sphere $Δ$ and by Reiner, Postnikov and Williams that it increases when $Δ$ is replaced by any flag simplicial homology sphere which geometrically subdivides $Δ$. Using the nonnegativity of the $γ$-vector in dimension 3, proved by Davis and Okun, as well as Stanley's theory of simplicial subdivisions and local $h$-vectors, the latter conjecture is confirmed in this paper in dimensions 3 and 4.
For any flag simplicial complex $Θ$ obtained by stellar subdividing the boundary of the cross polytope in edges, we define a flag simplicial complex $Γ(Θ)$ (dependent on the sequence of subdivisions) whose $f$-vector is the $γ$-vector of $Θ$. This proves that the $γ$-vector of any such simplicial complex satisfies the Frankl-Füredi-Kalai inequalities, partially solving a conjecture by Nevo and Petersen \cite{np}. We show that when $Θ$ is the dual simplicial complex to a nestohedron, and the sequence of subdivisions corresponds to a flag ordering as defined in \cite{ai}, that $Γ(Θ)$ is equal to…
Shmuel Friedland and Giorgio Ottaviani's beautiful constant term expression for the number of singular vector tuples of generic tensors is used to derive a rational generating function for these numbers, that in turn, is used to obtain an asymptotic formula for the number of such tuples for n by n by n three-dimensional tensors, and to conjecture an asymptotic formula for the general d-dimensional case. A donation of 100 dollars, in honor of the first prover, will be made to the On-line Encyclopedia of Integer Sequences.
Given a finite CAT(0) cubical complex, we define a flag simplicial complex associated to it, called the crossing complex. We show that the crossing complex holds much of the combinatorial information of the original cubical complex: for example, hyperplanes in the cubical complex correspond to vertex links in the crossing complex, and the crossing complex is balanced if and only if the cubical complex is cubically balanced. The most significant result is that the sets of $f$-vectors of CAT(0) cubical complexes and flag simplicial complexes are equal, up to an invertible linear transformation.
We study nestohedra $P_B$ corresponding to building sets $B$. It is shown that every flag nestohedron can be obtained from a cube by successive shavings faces of codimension 2. We receive new Delzant geometric realization of flag nestohedra. The main result of the paper is that Gal's conjecture holds for every flag nestohedron. Moreover, we get the exact estimation of $γ$-vectors of $n$-dimensional flag nestohedra: $0\leqγ_i(P_B)\leqγ_i(Pe^n)$.
We prove that the $γ$-vector of the barycentric subdivision of a simplicial sphere is the $f$-vector of a balanced simplicial complex. The combinatorial basis for this work is the study of certain refinements of Eulerian numbers used by Brenti and Welker to describe the $h$-vector of the barycentric subdivision of a boolean complex.
This thesis aims to give the reader an introduction and overview of the cd-index of a poset, as well as establish some new results. We give a combinatorial proof of Ehrenborg and Karu's cd-index subdivision decomposition for Gorenstein* complexes and extend it to a wider class of subdivisions. In doing so, we define a local cd-index that behaves analogously to the well studied local h-vector. We examine known cd-index and h-vector bounds, and then use the local cd-index to bound a particular class of polytopes with the cd-index of a stacked polytope. We conclude by investigating the h-vector a…
The interval subdivision Int$(Δ)$ of a simplicial complex $Δ$ was introduced by Walker. We give the complete combinatorial description of the entries of the transformation matrices from the $f$- and $h$-vectors of $Δ$ to the $f$- and $h$-vectors of Int$(Δ)$. We show that if $Δ$ has non-negative $h$-vector then the $h$-polynomial of its interval subdivision has only real roots. As a consequence, we prove the Charney-Davis conjecture for Int$(Δ)$, if $Δ$ has non-negative reciprocal $h$-vector.
For a simplicial complex or more generally Boolean cell complex $Δ$ we study the behavior of the $f$- and $h$-vector under barycentric subdivision. We show that if $Δ$ has a non-negative $h$-vector then the $h$-polynomial of its barycentric subdivision has only simple and real zeros. As a consequence this implies a strong version of the Charney-Davis conjecture for spheres that are the subdivision of a Boolean cell complex. For a general $(d-1)$-dimensional simplicial complex $Δ$ the $h$-polynomial of its $n$-th iterated subdivision shows convergent behavior. More precisely, we show that among…
Through tropical normal idempotent matrices, we introduce isocanted alcoved polytopes, computing their $f$--vectors and checking the validity of the following five conjectures: Bárány, unimodality, $3^d$, flag and cubical lower bound (CLBC). Isocanted alcoved polytopes are centrally symmetric, almost simple cubical polytopes. They are zonotopes. We show that, for each dimension, there is a unique combinatorial type. In dimension $d$, an isocanted alcoved polytope has $2^{d+1}-2$ vertices, its face lattice is the lattice of proper subsets of $[d+1]$ and its diameter is $d+1$. They are realizati…
We show that an 'almost strong Lefschetz' property holds for the barycentric subdivision of a shellable complex. From this we conclude that for the barycentric subdivision of a Cohen-Macaulay complex, the $h$-vector is unimodal, peaks in its middle degree (one of them if the dimension of the complex is even), and that its $g$-vector is an $M$-sequence. In particular, the (combinatorial) $g$-conjecture is verified for barycentric subdivisions of homology spheres. In addition, using the above algebraic result, we derive new inequalities on a refinement of the Eulerian statistics on permutations,…
Brenti and Welker have shown that for any simplicial complex X, the face vectors of successive barycentric subdivisions of X have roots which converge to fixed values depending only on the dimension of X. We improve and generalize this result here. We begin with an alternative proof based on geometric intuition. We then prove an interesting symmetry of these roots about the real number -2. This symmetry can be seen via a nice algebraic realization of barycentric subdivision as a simple map on formal power series in two variables. Finally, we use this algebraic machinery with some geometric mot…
Let $ρ:(D^2)^m\to I^m$ be the orbit map for the diagonal action of the torus $T^m$ on the unit poly-disk $(D^2)^m$, $I^m=[0,1]^m$ is the unit cube. Let $C$ be a cubical subcomplex in $I^m$. The moment-angle complex $\ma(C)$ is a $T^m$-invariant bigraded cellular decomposition of the subset $ρ^{-1}(C)\subset(D^2)^m$ with cells corresponding to the faces of $C$. Different combinatorial problems concerning cubical complexes and related combinatorial objects can be treated by studying the equivariant topology of corresponding moment-angle complexes. Here we consider moment-angle complexes defined …
We express the generating function for lattice points in a rational polyhedral cone with a simplicial subdivision in terms of multivariate analogues of the h-polynomials of the subdivision and "local contributions" of the links of its nonunimodular faces. We also compute new examples of nonunimodal h^*-vectors of reflexive polytopes.
This article introduces Gamma-triangles, which are closely related to and more fundamental than F-triangles and H-triangles that have been used in the combinatorics of cluster complexes. It is proved that Gamma-triangles can be expressed as sums of the local gamma-vectors introduced by Athanasiadis, which themselves refine the local h-vectors attached by Stanley to simplicial subdivisions. Gamma triangles of all cluster complexes are then explicitly computed.
Sphere-bases are constructed for the $\mathbb{Z}_2$ vector space formed by the $k$-dimensional subcomplexes, of $n$-simplex (or $n$-cube), for which every $(k{-}1)$-face is contained in a positive even number of $k$-cells; addition is symmetric difference of the corresponding sets of $k$-cells. The bases consist of the boundaries of an algorithmically-specified family of $k{+}1$-simplexes or $k{+}1$-cubes. Geometric properties of these bases are investigated.
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 with the cone generated by the vectors e_i-e_j, where (i, j) is an edge of T, i<j. The reduced forms of a certain monomial m[T] in commuting variables x_{ij} under the reduction x_{ij}x_{jk} --> x_{ik}x_{ij}+x_{jk}x_{ik}+βx_{ik}, can be interpreted as triangulations of P(T). Using these triangulations, the volume and Ehrhart polynomial of P(T) are obtained. I…
There are natural polynomial invariants of polytopes and lattice polytopes coming from enumerative combinatorics and Ehrhart theory, namely the $h$- and $h^*$-polynomials, respectively. In this paper, we study their generalization to subdivisions and lattice subdivisions of polytopes. By abstracting constructions in mixed Hodge theory, we introduce multivariable polynomials which specialize to the $h$-, $h^*$- polynomials. These polynomials, the mixed $h$-polynomial and the (refined) limit mixed $h^*$-polynomial have rich symmetry, non-negativity, and unimodality properties, which both refine …
The type C_n full root polytope is the convex hull in R^n of the origin and the points e_i-e_j, e_i+e_j, 2e_k for 1 <= i < j <= n, k \in [n]. Given a graph G, with edges labeled positive or negative, associate to each edge e of G a vector v(e) which is e_i-e_j if e=(i, j), i < j, is labeled negative and e_i+e_j if it is labeled positive. For such a signed graph G, the associated root polytope P(G) is the intersection of the full root polytope with the cone generated by the vectors v(e), for edges e in G. The reduced forms of a certain monomial m[G] in commuting variables x_{ij}, y_{ij}, z_k un…
The paper surveys some new results and open problems connected with such fundamental combinatorial concepts as polytopes, simplicial complexes, cubical complexes, and subspace arrangements. Particular attention is paid to the case of simplicial and cubical subdivisions of manifolds and, especially, spheres. We describe important constructions which allow to study all these combinatorial objects by means of methods of commutative and homological algebra. The proposed approach to combinatorial problems relies on the theory of moment-angle complexes, currently being developed by the authors. The …
The antiprism triangulation provides a natural way to subdivide a simplicial complex $Δ$, similar to barycentric subdivision, which appeared independently in combinatorial algebraic topology and computer science. It can be defined as the simplicial complex of chains of multi-pointed faces of $Δ$, from a combinatorial point of view, and by successively applying the antiprism construction, or balanced stellar subdivisions, on the faces of $Δ$, from a geometric point of view. This paper studies enumerative invariants associated to this triangulation, such as the transformation of the $h$-vector o…
An h-tiling on a finite simplicial complex is a partition of its geometric realization by maximal simplices deprived of several codimension one faces together with possibly their remaining face of highest codimension. In this last case, the tiles are said to be critical. An h-tiling thus induces a partitioning of its face poset by closed or semi-open intervals. We prove the existence of h-tilings on every finite simplicial complex after finitely many stellar subdivisions at maximal simplices. These tilings are moreover shellable. We also prove that the number of tiles of each type used by a ti…
A triangulation of a simplicial complex $Δ$ is called uniform if the $f$-vector of its restriction to a face of $Δ$ depends only on the dimension of that face. This paper proves that the entries of the $h$-vector of a uniform triangulation of $Δ$ can be expressed as nonnegative integer linear combinations of those of the $h$-vector of $Δ$, where the coefficients depend only on the dimension of $Δ$ and the $f$-vectors of the restrictions of the triangulation to simplices of various dimensions. Moreover, it provides information about these coefficients, including formulas, recurrence relations a…
We give new interpretations of the $ν$-Tamari lattice of Préville-Ratelle and Viennot. First, we describe it as a rotation lattice of $ν$-trees, which uncovers the relation with known combinatorial objects such as tree-like tableaux and north-east fillings. Then, using a formulation in terms of bracket vectors of $ν$-trees and componentwise order, we provide a simple description of the lattice property. We also show that the $ν$-Tamari lattice is isomorphic to the increasing-flip poset of a suitably chosen subword complex, and settle a special case of Rubey's lattice conjecture concerning the …
In this paper we study general tropical linear spaces locally: For any basis B of the matroid underlying a tropical linear space L, we define the local tropical linear space L_B to be the subcomplex of L consisting of all vectors v that make B a basis of maximal v-weight. The tropical linear space L can then be expressed as the union of all its local tropical linear spaces, which we prove are homeomorphic to Euclidean space. Local tropical linear spaces have a simple description in terms of polyhedral matroid subdivisions, and we prove that they are dual to mixed subdivisions of Minkowski sums…
Given a finite set $V=\{v^1, \dots, v^n\} \subset \mathbb R^d$ with dim conv $(V)=d$, a triangulation $T$ of $V$ is a collection of distinct subsets $\{T_1, \dots, T_m\}$ where $T_i \subseteq V$ is the vertex set of a $d$-simplex, $\mathrm{conv} (V)=\bigcup_{i=1}^m \mathrm{conv} (T_i)$, and $T_i \cap T_j$ is a common (possibly empty) face of both $T_i$ and $T_j$. Associated with each triangulation $T$ of $V$ is the GKZ-vector $ϕ(T)=(z_1, \dots, z_n)$ where $z_i$ is the sum of the volumes of all $d$-simplices of $T$ having $v^i \in V$ as a vertex. It is clear that given $V$ and a triangulation …
We recently defined a property of Morse shellability (and tileability) of finite simplicial complexes which extends the classical one and its relations with discrete Morse theory. We now prove that the product of two Morse tileable or shellable simplicial complexes carries Morse tileable or shellable triangulations under some tameness condition, and that any tiling or shelling becomes tame after one barycentric subdivision. We deduce that any finite product of closed manifolds of dimensions less than four carries Morse shellable triangulations whose critical and h-vectors are palindromic. We a…
We present extremal constructions connected with the property of simplicial collapsibility. (1) For each $d \ge 2$, there are collapsible (and shellable) simplicial $d$-complexes with only one free face. Also, there are non-evasive $d$-complexes with only two free faces. (Both results are optimal in all dimensions.) (2) Optimal discrete Morse vectors need not be unique. We explicitly construct a contractible, but non-collapsible $3$-dimensional simplicial complex with face vector $f=(106,596,1064,573)$ that admits two distinct optimal discrete Morse vectors, $(1,1,1,0)$ and $(1,0,1,1)$. Indeed…
