Deletion-restriction is a fundamental tool in the theory of hyperplane arrangements. Various important results in this field have been proved using deletion-restriction. In this paper we use deletion-restriction to identify a class of toric arrangements for which the cohomology algebra of the complement is generated in degree 1. We also show that for these arrangements the complement is formal in the sense of Sullivan.

We define arrangements of codimension-1 submanifolds in a smooth manifold which generalize arrangements of hyperplanes. When these submanifolds are removed the manifold breaks up into regions, each of which is homeomorphic to an open disc. The aim of this paper is to derive formulas that count the number of regions formed by such an arrangement. We achieve this aim by generalizing Zaslavsky's theorem to this setting. We show that this number is determined by the combinatorics of the intersections of these submanifolds.2010 Mathematics Subject Classification. 52C35, 52B45, 05E45, 05A99, 57Q15.

For r ≥ 1, the r-independence complex of a graph G is a simplicial complex whose faces are subset I ⊆ V (G) such that each component of the induced subgraph G[I] has at most r vertices. In this article, we determine the homotopy type of r-independence complexes of certain families of graphs including complete s-partite graphs, fully whiskered graphs, cycle graphs and perfect m-ary trees. In each case, these complexes are either homotopic to a wedge of equi-dimensional spheres or are contractible. We also give a closed form formula for their homotopy types.

A. A hyperplane arrangement in R n is a finite collection of affine hyperplanes. The regions are the connected components of the complement of these hyperplanes. By a theorem of Zaslavsky, the number of regions of a hyperplane arrangement is the sum of coefficients of its characteristic polynomial. Arrangements that contain hyperplanes parallel to subspaces whose defining equations are x i − x j = 0 form an important class called the deformations of the braid arrangement. In a recent work, Bernardi showed that regions of certain deformations are in one-to-one correspondence with certain labeled trees. In this article, we define a statistic on these trees such that the distribution is given by the coefficients of the characteristic polynomial. In particular, our statistic applies to well-studied families like extended Catalan, Shi, Linial and semiorder.

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