A finite hyperplane arrangement A is a finite set of affine hyperplanes in a vector space. The hyperplanes in the arrangement divide the space into regions, which are the connected components of their complement. Some regions are bounded, while others extend to infinity. The intersections of the hyperplanes form a partially ordered set $L({\mathcal A})$, which, together with the Möbius function, is used to define the characteristic polynomial $\chi_{\mathcal A}(t)$. This polynomial contains important information about the geometric and combinatorial properties of the arrangement. An important theorem of Thomas Zaslavsky from 1975 uses the values of the characteristic polynomial at $−1$ and $1$ to determine the number of regions and bounded regions, respectively.
A special class of arrangements is given by graphic arrangements, for which the characteristic polynomial is equal to the chromatic polynomial of the corresponding graph. This connection also makes it possible to determine the number of acyclic orientations of a graph using the theory of hyperplane arrangements.
The characteristic polynomial of an arbitrary arrangement can be determined directly from the partially ordered set of intersections and its Möbius function, while other methods can be used in special cases. The finite field method determines the characteristic polynomial by counting points in finite vector spaces. For exponential sequences of arrangements, the characteristic polynomial can also be determined from the number of their regions using a generating function.
Some special arrangements are closely connected to other important combinatorial objects. The number of regions of the braid arrangement is equal to the number of permutations, the number of regions of the Shi arrangement is equal to the number of parking functions, and the number of regions of the Catalan arrangement is expressed in terms of Catalan numbers.
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