2013/06/07 by Richard P. Stanley, Stanley, Richard P.
Engineering · Mathematics · #52C35 #Advanced Combinatorial Mathematics #Combinatorics (math.CO) #FOS: Mathematics #Mathematics and Applications #graph theory and CDMA systems #math.CO #msc:52C35
paper · pdf · doi:10.48550/arxiv.1306.1838
15 pages, 2 figures
arxiv created 2013/06/07 · openalex publication_date 2013/06/07 · arxiv updated 2013/06/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Given a real finite hyperplane arrangement A and a point p not on any of the hyperplanes, we define an arrangement vo(A,p), called the *valid order arrangement*, whose regions correspond to the different orders in which a line through p can cross the hyperplanes in A. If A is the set of affine spans of the facets of a convex polytope P and p lies in the interior of P, then the valid orderings with respect to p are just the line shellings of p where the shelling line contains p. When p is sufficiently generic, the intersection lattice of vo(A,p) is the *Dilworth truncation* of the semicone of A. Various applications and examples are given. For instance, we determine the maximum number of line shellings of a d-polytope with m facets when the shelling line contains a fixed point p. If P is the order polytope of a poset, then the sets of facets visible from a point involve a generalization of chromatic polynomials related to list colorings.