Combinatorics

Abstract

The intersection poset L(A) is an important combinatorial invariant of the arrangement A. We study its properties in this chapter. In Section 2.1 we give L(A) a partial order by reverse inclusion and show that it is a geometric lattice when A is a central arrangement. We construct the face poset of a real arrangement and show its connection with oriented matroids. We also define supersolvable arrangements here, and a generalization called arrangements with a nice partition. In Section 2.2 we define the Möbius function and study its properties. We also present notes on the interesting history of this function dating back to Euler. In Section 2.3 we define the Poincaré polynomial π(A, t), which is related to another combinatorial function called the characteristic polynomial. A fundamental technical tool in this book is the method of deletion and restriction, which allows induction on the number of hyperplanes in the arrangement. It uses the triple (A, A′, A″) of Definition 1.14. The Deletion-Restriction Theorem states:

$$ \pi \left( {A,t} \right) = \pi \left( {A',t} \right) + t\pi \left( {A'',t} \right) $$