Mixed-integer mathematical programs are among the most commonly used models for a wide set of problems in Operations Research and related fields. However, there is still very little known about what can be expressed by small mixed-integer programs. In particular, prior to this work, it was open whether some classical problems, like the minimum odd-cut problem, can be expressed by a compact mixe...

The weak order polytope P n WO is the polytope in R n(n?1) whose ver-tices correspond to the members of the family of reeexive weak orders on f1; 2 : : : ; ng when each coordinate of R n(n?1) is associated with a diierent ordered pair (i; j) of distinct points in f1; 2; : : : ; ng. The vertex w that corresponds to weak order W has w (i;j) = 1 if i-j in W and has w (i;j) = 0 otherwise, with P n ...

The algebraic technique of Gr obner bases is applied to study triangulations of the second hypersimplex (2; n). We present a quadratic Gr obner basis for the associated toric ideal I(K n ). The simplices in the resulting triangulation of (2; n) have unit volume, and they are indexed by subgraphs which are linear thrackles [28] with respect to a circular embedding of K n . For n 6 the number o...

For a word v in variables x and y, Chebikin and Ehrenborg found that the number of faces of the descent polytope DPv equals the number of factorizations of v using subfactors of the form xy and yx with some additional constraints. They also showed the number of faces of DPv equals the number of alternating subwords of v and raised the problem of finding a bijective proof between these two enume...

The Monotone Upper Bound Problem asks for the maximal number M(d, n) of vertices on a strictly-increasing edge-path on a simple d-polytope with n facets. More specifically, it asks whether the upper bound M(d, n) ≤ Mubt(d, n) provided by McMullen’s (1970) Upper Bound Theorem is tight, where Mubt(d, n) is the number of vertices of a dual-to-cyclic d-polytope with n facets. It was recently shown ...

The concept of perfection of a polytope was introduced by S. A. Robertson. Intuitively speaking, a polytope P is perfect if and only if it cannot be deformed to a polytope of different shape without changing the action of its symmetry group G(P ) on its face-lattice F (P ). By Rostami’s conjecture, the perfect 4-polytopes form a particular set of Wythoffian polytopes. In the present paper first...

Body centered structures are used as seeds for a variety of structures of rank 3 and higher. Propellane based structures are introduced and their design and topological properties are detailed.

Let OTd(n) be the smallest integer N such that every N -element point sequence in R d in general position contains an order-type homogeneous subset of size n, where a set is order-type homogeneous if all (d + 1)-tuples from this set have the same orientation. It is known that a point sequence in R that is order-type homogeneous, forms the vertex set of a convex polytope that is combinatorially ...

My research project involves investigations in the mathematical field of combinatorics. The research study will be based on the results of Professors Steven Edwards and William Griffiths, who recently found a new formula for the cross-polytope numbers. My topic will be focused on ”Generalizations of cross-polytope numbers”. It will include the proofs of the combinatorics results in Dr. Edwards ...