Related papers: A Lattice Path Interpretation of the Diamond Produ…
Let $P$ be a finite partially ordered set. In a recent series of works, Proudfoot introduced the notion of $Z$-polynomials associated with $P$-kernels, providing a unified framework for various intersection cohomology Poincar\'e polynomials…
In this paper, we study algorithmic questions concerning products of matrices and their consequences for recognition algorithms for polyhedra. The 1-product of matrices $S_1$, $S_2$ is a matrix whose columns are the concatenation of each…
Macdonald polynomials are orthogonal polynomials associated to root systems, and in the type A case, the symmetric kind is a common generalization of Schur functions, Macdonald spherical functions, and Jack polynomials. We use the…
The monomial basis for polynomials in N variables is labeled by compositions. To each composition there is associated a hook-length product, which is a product of linear functions of a parameter. The zeroes of this product are related to…
For a directed polytope, we construct a colored operad whose Poincare-Hilbert series encodes certain operations on the cellular complex of the polytope. We conjecture that for a class of short polytopes the constructed operads are Koszul…
The goal of this paper is to study convex lattice sets by the discrete Legendre transform. The definition of the polar of convex lattice sets in $\mathbb{Z}^n$ is provided. It is worth mentioning that the polar of convex lattice sets have…
We use Fourier methods to prove that if $n > 1$ translates of sublattices of $Z^d$ tile $Z^d$, and all the sublattices are Cartesian products of arithmetic progressions, then two of the tiles must be translates of each other. This is a…
The M\"obius polynomial is an invariant of ranked posets, closely related to the M\"obius function. In this paper, we study the M\"obius polynomial of face posets of convex polytopes. We present formulas for computing the M\"obius…
We present a formula for the degree of the discriminant of a smooth projective toric variety associated to a lattice polytope P, in terms of the number of integral points in the interior of dilates of faces of dimension greater or equal…
A T\"oplitz determinant whose entries are described by a q-analogue of the Narayana polynomials is evaluated by means of Laurent biorthogonal polynomials which allow of a combinatorial interpretation in terms of Schr\"oder paths. As an…
Dowling and Rhodes defined different lattices on the set of triples (Subset, Partition, Cross Section) over a fixed finite group G. Although the Rhodes lattice is not a geometric lattice, it defines a matroid in the sense of the theory of…
Given a directed graph $D$ with transitive closure $\operatorname{tc}(D)$ and path hypergraph $\mathbb{P}(D)$, we study the connections between the (acyclic) reorientation poset of $\operatorname{tc}(D)$, the (acyclic) sourcing poset of…
Regular triangulations of products of lattice polytopes are constructed with the additional property that the dual graphs of the triangulations are bipartite. The (weighted) size difference of this bipartition is a lower bound for the…
We introduce the wedge product of two polytopes. The wedge product is described in terms of inequality systems, in terms of vertex coordinates as well as purely combinatorially, from the corresponding data of its constituents. The wedge…
This paper is concerned with the linkedness of Cartesian products of complete graphs. A graph with at least $2k$ vertices is {\it $k$-linked} if, for every set of $2k$ distinct vertices organised in arbitrary $k$ pairs of vertices, there…
We study lattice point visibility along polynomial lines of sight and prove the Visibility Density Conjecture of Chaubey and Pandey for a large class of polynomials.
In this note we give the definition of the "doubling operation" for simple polytopes, find the formula for the h-polynomial of new polytope.As an application of this operation we establish the relationship between moment-angle manifolds and…
Define an expansion poset to be the poset of monomials of a cluster variable attached to an arc in a polygon, where each monomial is represented by the corresponding combinatorial object from some fixed combinatorial cluster expansion…
We introduce and study, for a process P delivering edges on the Cartesian product of the vertex sets of a given set of graphs, the P-product of these graphs, thereby generalizing many types of product graph. Analogous to the notion of a…
We consider a product of fundamental crystals of type $C_n$ in monomial realization, where the product means a natural product of Laurent monomials, not a tensor product. Then we shall show that the product still holds a crystal structure…