Summary: PACIFIC JOURNAL OF MATHEMATICS
Vol. , No. ,
COMPUTING THE TUTTE POLYNOMIAL
OF A HYPERPLANE ARRAGEMENT
We define and study the Tutte polynomial of a hyperplane arrangement.
We introduce a method for computing the Tutte polynomial by solving a
related enumerative problem. As a consequence, we obtain new formulas
for the generating functions enumerating alternating trees, labelled trees,
semiorders and Dyck paths.
Much work has been devoted in recent years to studying hyperplane arrangements
and, in particular, their characteristic polynomials. The polynomial (q) is a very
powerful invariant of the arrangement ; it arises very naturally in many different
contexts. Two of the many important and beautiful results about the characteristic
polynomial of an arrangement are the following.
Theorem 1.1 [Zaslavsky 1975]. Let be a hyperplane arrangement in n. The
number of regions into which dissects n is equal to (-1)n (-1). The number
of regions which are relatively bounded is equal to (-1)n (1).
Theorem 1.2 [Orlik and Solomon 1980]. Let be a hyperplane arrangement in