We show that the $γ$-vector of the order complex of any polytope is the f-vector of a balanced simplicial complex. This is done by proving this statement for a subclass of Stanley's S-shellable spheres which includes all polytopes. The proof shows that certain parts of the cd-index, when specializing $c=1$ and considering the resulted polynomial in $d$, are the f-polynomials of simplicial complexes that can be colored with "few" colors. We conjecture that the cd-index of a regular CW-sphere is itself the flag f-vector of a colored simplicial complex in a certain sense.
The construction of the Bier sphere Bier(K) for a simplicial complex K is due to Bier. Björner, Paffenholz, Sjöstrand and Ziegler generalize this construction to obtain a Bier poset Bier(P,I) from any bounded poset P and any proper ideal I of P. They show shellability of Bier(P,I) for the case where P is the boolean lattice, and obtain thereby 'many shellable spheres' in the sense of Kalai. We put the Bier construction into the general framework of the theory of nested set complexes of Feichtner and Kozlov. We obtain 'more shellable spheres' by proving the general statement that combinatorial …
We prove that if a simplicial complex is shellable, then the intersection lattice for the corresponding diagonal arrangement is homotopy equivalent to a wedge of spheres. Furthermore, we describe precisely the spheres in the wedge, based on the data of shelling. Also, we give some examples of diagonal arrangements where the complement is K(π,1), coming from rank 3 matroids.
We consider the poset of vector partitions of $[n]$ into $s$ components, denoted $Π_{n,s}$, which was first defined by Stanley in 1978. In 1986, Sagan showed that this poset is CL-shellable, and hence has the homotopy type of a wedge of spheres of dimension $(n-2)$. We extend on this result to show that $Π_{n,s}$ is edge-lexicographic shellable. We then use this edge-labeling to find a recursive expression for the number of spheres, and show that when $s=1$ the number of spheres is equal to the number of complete non-ambiguous trees, first defined in 2014 by Aval, Boussicault, Bouvel and Silim…
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 Eule…
In 1992, Thomas Bier introduced a surprisingly simple way to construct a large number of simplicial spheres. He proved that, for any simplicial complex $Δ$ on the vertex set $V$ with $Δ\ne 2^V$, the deleted join of $Δ$ with its Alexander dual $Δ^\vee$ is a combinatorial sphere. In this paper, we extend Bier's construction to multicomplexes, and study their combinatorial and algebraic properties. We show that all these spheres are shellable and edge decomposable, which yields a new class of many shellable edge decomposable spheres that are not realizable as polytopes. It is also shown that thes…
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,…
We introduce topological prismatoids, a combinatorial abstraction of the (geometric) prismatoids recently introduced by the second author to construct counter-examples to the Hirsch conjecture. We show that the `strong $d$-step Theorem' that allows to construct such large-diameter polytopes from `non-$d$-step' prismatoids still works at this combinatorial level. Then, using metaheuristic methods on the flip graph, we construct four combinatorially different non-$d$-step $4$-dimensional topological prismatoids with $14$ vertices. This implies the existence of $8$-dimensional spheres with $18$ v…
We give a new proof of an old identity of Dixon (1865-1936) that uses tools from topological combinatorics. Dixon's identity is re-established by constructing an infinite family of non-pure simplicial complexes $Δ(n)$, indexed by the positive integers, such that the alternating sum of the numbers of faces of $Δ(n)$ of each dimension is the left-hand side of the identity. We show that $Δ(n)$ is shellable for all $n$. Then, using the fact that a shellable simplicial complex is homotopy equivalent to a wedge of spheres, we compute the Betti numbers of $Δ(n)$ by counting (via a generating function…
A well-known conjecture of McMullen, proved by Billera, Lee and Stanley, describes the face numbers of simple polytopes. The necessary and sufficient condition is that the toric g-vector of the polytope is an M-vector, that is, the vector of dimensions of graded pieces of a standard graded algebra A. Recent work by Murai, Nevo and Yanagawa suggests a similar condition for the coefficients of the cd-index of a poset P. The coefficients of the cd-index are conjectured to be the dimensions of graded pieces in a standard multigraded algebra A. We prove the conjecture for simplicial spheres and we …
The open intervals in the Bruhat order on twisted involutions in a Coxeter group are shown to be PL spheres. This implies results conjectured by F. Incitti and sharpens the known fact that these posets are Gorenstein* over Z_2. We also introduce a Boolean cell complex which is an analogue for twisted involutions of the Coxeter complex. Several classical Coxeter complex properties are shared by our complex. When the group is finite, it is a shellable sphere, shelling orders being given by the linear extensions of the weak order on twisted involutions. Furthermore, the h-polynomial of the comple…
We exhibit a canonical connection between maximal (0,1)-fillings of a moon polyomino avoiding north-east chains of a given length and reduced pipe dreams of a certain permutation. Following this approach we show that the simplicial complex of such maximal fillings is a vertex-decomposable, and thus shellable, sphere. In particular, this implies a positivity result for Schubert polynomials. For Ferrers shapes, we moreover construct a bijection to maximal fillings avoiding south-east chains of the same length which specializes to a bijection between k-triangulations of the n-gon and k-fans of Dy…
Durhuus and Jonsson (1995) introduced the class of "locally constructible" (LC) 3-spheres and showed that there are only exponentially-many combinatorial types of simplicial LC 3-spheres. Such upper bounds are crucial for the convergence of models for 3D quantum gravity. We characterize the LC property for d-spheres ("the sphere minus a facet collapses to a (d-2)-complex") and for d-balls. In particular, we link it to the classical notions of collapsibility, shellability and constructibility, and obtain hierarchies of such properties for simplicial balls and spheres. The main corollaries from …
For any Coxeter system $(W,S)$ of rank $n$, we introduce an abstract boolean complex (simplicial poset) of dimension $2n-1$ that contains the Coxeter complex as a relative subcomplex. Faces are indexed by triples $(I,w,J)$, where $I$ and $J$ are subsets of the set $S$ of simple generators, and $w$ is a minimal length representative for the parabolic double coset $W_I w W_J$. There is exactly one maximal face for each element of the group $W$. The complex is shellable and thin, which implies the complex is a sphere for the finite Coxeter groups. In this case, a natural refinement of the $h$-pol…
Shellability of a simplicial complex has many useful structural implications. In particular, it was shown by Danaraj and Klee that every shellable pseudo-manifold is a PL-sphere. The purpose of this paper is to prove the shellability of the quasi-arc complex of the Möbius strip. Along the way we provide elementary proofs of the shellability of the arc complex of the $n$-gon and the cylinder. In turn, applying the result of Danaraj and Klee, we obtain the sphericity of all of these complexes.
Gamma-positivity is an elementary property that polynomials with symmetric coefficients may have, which directly implies their unimodality. The idea behind it stems from work of Foata, Schützenberger and Strehl on the Eulerian polynomials; it was revived independently by Brändén and Gal in the course of their study of poset Eulerian polynomials and face enumeration of flag simplicial spheres, respectively, and has found numerous applications since then. This paper surveys some of the main results and open problems on gamma-positivity, appearing in various combinatorial or geometric contexts, a…
