[Paper Review] Computing the Tutte polynomial of a hyperplane arrangement

Federico Ardila|ArXiv.org|Sep 13, 2004

Advanced Combinatorial Mathematics26 references8 citations

TL;DR

This paper introduces the Tutte polynomial for hyperplane arrangements using a finite field method based on the coboundary polynomial, enabling computation via enumerative problems over finite fields. It derives new formulas for generating functions of combinatorial objects like labeled trees, Dyck paths, alternating trees, and semiorders by linking Tutte polynomials of specific arrangements—such as the braid, Shi, semiorder, and Catalan arrangements—to known enumerative sequences.

ABSTRACT

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.

Motivation & Objective

• To define and compute the Tutte polynomial for hyperplane arrangements, extending its applicability beyond graphs and matroids.
• To establish a finite field method for computing the Tutte polynomial through the coboundary polynomial, reducing the problem to enumerative questions over finite fields.
• To connect the Tutte polynomial of specific arrangements—braid, Shi, semiorder, and Catalan—to classical combinatorial enumeration problems.
• To derive new generating function formulas for combinatorial objects such as labeled trees, Dyck paths, alternating trees, and semiorders.
• To unify and generalize known results on characteristic polynomials of arrangements by embedding them within the broader framework of the Tutte polynomial.

Proposed method

• Define the Tutte polynomial of a hyperplane arrangement using the coboundary polynomial, a transformation of the Tutte polynomial.
• Use a finite field method: compute the Tutte polynomial by counting solutions to linear equations over finite fields, particularly via surjective functions with restricted differences.
• Apply the method to arrangements like the braid, Shi, semiorder, and Catalan arrangements by analyzing the number of surjective functions with forbidden differences in values.
• Derive generating functions for characteristic polynomials by taking limits of ratios of auxiliary polynomials $ A_r(x) $, which encode the number of valid functions for each $ r $.
• Leverage known bijections (e.g., between Shi arrangement regions and parking functions) to validate and interpret the results.
• Use exponential generating functions and limits as $ r \to \infty $ to extract the Tutte polynomial generating functions from the finite field enumeration.

Experimental results

Research questions

• RQ1 How can the Tutte polynomial be generalized to hyperplane arrangements, given its success in graphs and matroids?
• RQ2 What finite field method enables the computation of the Tutte polynomial of a hyperplane arrangement through enumerative problems?
• RQ3 How do the Tutte polynomials of specific arrangements (e.g., braid, Shi, semiorder, Catalan) relate to classical combinatorial sequences?
• RQ4 Can the generating functions for labeled trees, Dyck paths, alternating trees, and semiorders be derived from the Tutte polynomial of associated arrangements?
• RQ5 What is the connection between the limit of ratios $ A_r(x)/A_{r-1}(x) $ and the generating functions of characteristic polynomials of arrangements?

Key findings

• The Tutte polynomial of the braid arrangement $ A_n $ is computed via the finite field method, yielding a generating function related to labeled trees.
• For the Shi arrangement $ {/mathcal{S}}_n $, the generating function $ \sum_{n\geq 0} \chi_{{\mathcal{S}}_n}(q) \frac{x^n}{n!} $ is expressed as $ \left( \lim_{r\to\infty} \frac{A_r(x)}{A_{r-1}(x)} \right)^q $, where $ A_r(x) = \sum_{n=0}^r (r-n)^n \frac{x^n}{n!} $.
• The number of semiorders on $[n]$ is given by $ i_n $, and the exponential generating function $ \sum_{n\geq 0} (-1)^n i_n \frac{x^n}{n!} $ equals $ \lim_{r\to\infty} \frac{A_{r-1}(x)}{A_r(x)} $, with $ A_r(x) $ derived from the semiorder arrangement.
• For the Catalan arrangement $ C_n $, the generating function $ \sum_{n\geq 0} \chi_{C_n}(q) \frac{x^n}{n!} $ is $ \left( \lim_{r\to\infty} \frac{A_r(x)}{A_{r-1}(x)} \right)^q $, where $ A_r(x) = \sum_{n=0}^{\lfloor (r+1)/2 \rfloor} \binom{r-n+1}{n} x^n $.
• The limit $ \lim_{r\to\infty} \frac{F_{r-1}}{F_r} = \frac{\sqrt{5}-1}{2} $ is obtained by formally substituting $ x=1 $ into the generating function for Catalan arrangements, yielding a heuristic proof of Fibonacci growth rate.
• The method successfully recovers known results: the number of regions of the Shi arrangement equals the number of parking functions, and the number of regions of the Catalan arrangement is $ n! C_n $, where $ C_n $ is the $ n $-th Catalan number.