arXiv subjects
Publications and source records attributed to Andrey O. Matveev.
A three-tope committee K* for a simple oriented matroid M is a 3-subset of its maximal covectors such that every positive halfspace of M contains at least two topes from K*. We consider three-tope committees as the vertex sets of triangles in graphs associated with the topes and enumerate them making use of the properties of the poset of convex subsets of the ground set of M.
A tope committee K* for a simple oriented matroid M is a subset of its maximal covectors such that every positive halfspace of M contains more than half of the covectors from K*. The structures of the family of all committees for M, and of the family of its committees that contain no pairs of opposites, are described. A Farey subsequence associated with the elements of the m-th layer of the Boolean lattice of rank 2m is explored.
We establish monotone bijections between subsequences of the Farey sequences and the halfsequences of Farey subsequences associated with elements of the Boolean lattices.
We establish monotone bijections between the Farey sequences of order m and the halfsequences of Farey subsequences associated with the rank m elements of the Boolean lattice of subsets of a 2m-set. We also present a few related combinatorial identities.
Antichains of a finite bounded poset are assigned antichains playing a role analogous to that played by blockers in the Boolean lattice of all subsets of a finite set. Some properties of lattices of generalized blockers are discussed.
We consider redundant analogues of the f- and h-vectors of simplicial complexes and present bases of R^{m+1} related to these ``long'' f- and h-vectors describing the face systems from 2^{1,...,m}; we list the corresponding change of basis matrices. The representations of the long f- and h-vectors of a face system with respect to various bases are expressed based on partitions of the system into Boolean intervals.
We review several linear algebraic aspects of the Dehn-Sommerville relations and relate redundant analogues of the f- and h-vectors describing the subsets of a simplex 2^{1,...,m} that satisfy Dehn-Sommerville type relations to integer points contained in some rational polytopes.
An order-theoretic generalization of Seymour relations describing the connection between the set-theoretic blocker, deletion, and contraction maps on clutters, is presented.