The space Td,n of n tropically collinear points in a fixed tropical projective space TP is equivalent to the tropicalization of the determinantal variety of matrices of rank at most 2, which consists of real d × n matrices of tropical or Kapranov rank at most 2, modulo projective equivalence of columns. We show that it is equal to the image of the moduli space M0,n(TP , 1) of n-marked tropical ...

In this paper we study topological properties of the poset of injective words and the lattice of classical non-crossing partitions. Specifically, it is shown that after the removal of the bottom and top elements (if existent) these posets are doubly Cohen-Macaulay. This extends the well-known result that those posets are shellable. Both results rely on a new poset fiber theorem, for doubly homo...

Hochschild lattices are specific intervals in the dexter meet-semilattices recently introduced by Chapoton. A natural geometric realization of these leads to some cell complexes Saneblidze, called polytopes. We obtain several geometrical properties lattices, namely we give cubic realizations, establish that EL-shellable, and show they constructible interval doubling. also prove combinatorial as...

All lattices are assumed to be finite. Björner [2] has shown that a dismantlable (see Rival, [5]) lattice L is Cohen-Macaulay (see [6] for definition) if and only if L is ranked and interval-connected. A lattice is planar if its Hasse diagram can be drawn in the plane with no edges crossing. Baker, Fishburn and Roberts have shown that planar lattices are dismantlable, see [1]. Lexicographically...

Let n,k,k and n,k,h , h < k, denote the intersection lattices of the k-equal subspace arrangement of type Dn and the k, h-equal subspace arrangement of type Bn respectively. Denote by SB n the group of signed permutations. We show that ( n,k,k )/SB n is collapsible. For ( n,k,h )/S B n , h < k, we show the following. If n ≡ 0 (mod k), then it is homotopy equivalent to a sphere of dimension 2n k...

Each positive rational number x > 0 can be written uniquely as x = a/(b− a) for coprime positive integers 0 < a < b. We will identify x with the pair (a, b). In this paper we define for each positive rational x > 0 a simplicial complex Ass(x) = Ass(a, b) called the rational associahedron. It is a pure simplicial complex of dimension a − 2, and its maximal faces are counted by the rational Catal...

Suppose that each edge of a connected graph G of order n is independently operational with probability p; the reliability of G is the probability that the operational edges form a spanning connected subgraph. A useful expansion of the reliability is as pn-1 5dj=0 Hi(l — p ) i , and the Ball-Provan method for bounding reliability relies on Stanley's combinatorial bounds for the H-vectors of shel...

We investigate the conjectured sufficiency of a condition for h-vectors (1, h1, h2, . . . , hd, 0) of regular d-dimensional triangulations. (The condition is already shown to be necessary in [2]). We first prove that the condition is sufficient when h1 ≥ h2 ≥ · · · ≥ hd. We then derive some new shellings of squeezed spheres and use them to prove that the condition is sufficient when d = 3. Fina...

Let Cn = {compositions of n}, C = ∪Cn. We define a partial order making C into a ranked poset having 1 as its bottom element and Cn as its (n− 1)-st rank level. Let α = a1 + · · · + ak ∈ Cn. The interval [1, α] is shown to have the following properties: • The number of maximal chains in [1, α] equals the number of permutations of [n] with descent set {a1, a1 + a2, . . .}. • The interval [1, α] ...