On Counting the k-face Cells of Cyclic Arrangements

A projective d-arrangement of n hyperplanes H(d, n) is a finite collection of hyperplanes in the real projective space Pd such that no point belongs to every hyperplane of H(d, n). Any arrangement H(d, n) decomposes Pd into a d-dimensional cell complex K . We may call cells of H(d, n) the d-cells of K , and facets of H(d, n) the (d − 1)-cells of K . Clearly any cell of H(d, n) has at least (resp. at most) d + 1 (resp. n) facets. We shall denote by f p[H(d, n)] the number of d-cells of H(d, n) having exactly p facets, d + 1 ≤ p ≤ n. The cyclic polytope of dimension d with n vertices Cd(t1, . . . , tn) was discovered by Carathéodory [3, 4] and has been rediscovered many times; it is usually defined as the convex hull in the Euclidean space Rd , d ≥ 2, of n, n ≥ d + 1, different points x(t1), . . . , x(tn) of the moment curve x : R → Rd , t → (t, t2, . . . , td). Cyclic polytopes, and simplicial neighbourly polytopes, in general, play an important role in the combinatorial convex geometry due to their connection with certain extremal problems. For example, the upper bound theorem established by McMullen [7, 8], says that the number of j-dimensional faces of d-polytope with n vertices is maximized by Cd(t1, . . . , tn). Here, we focus our attention to cyclic arrangements, A(d, n), defined as the dual to cyclic polytopes Cd(t1, . . . , tn). As for cyclic polytopes, cyclic arrangements also have extremal properties. For instance, Shannon [12] has introduced cyclic arrangements A(d, n) as examples of projective arrangements with a minimum number of cells with (d + 1)-facets. In this paper, we give an explicit formula to compute f p[A(d, n)] for each d + 1 ≤ p ≤ n.