Computing the Tutte polynomial of a hyperplane arrangement
نویسنده
چکیده
We define and study the Tutte polynomial of a hyperplane arrangement. We introduce a method for computing it by solving an enumerative problem in a finite field. For specific arrangements, the computation of Tutte polynomials is then reduced to certain related enumerative questions. As a consequence, we obtain new formulas for the generating functions enumerating alternating trees, labelled trees, semiorders and Dyck paths.
منابع مشابه
Tutte polynomials of hyperplane arrangements and the finite field method
The Tutte polynomial is a fundamental invariant associated to a graph, matroid, vector arrangement, or hyperplane arrangement, which answers a wide variety of questions about its underlying object. This short survey focuses on some of the most important results on Tutte polynomials of hyperplane arrangements. We show that many enumerative, algebraic, geometric, and topological invariants of a h...
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Theorem 1.1 [Zaslavsky 1975]. Let A be a hyperplane arrangement in Rn . The number of regions into which A dissects Rn is equal to (−1)χA(−1). The number of regions which are relatively bounded is equal to (−1)χA(1). Theorem 1.2 [Orlik and Solomon 1980]. Let A be a hyperplane arrangement in Cn , and let MA =Cn− ⋃ H∈A H be its complement. Then the Poincaré polynomial of the cohomology ring of MA...
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