2025/04/07 by Michael Cuntz, Cuntz, Michael, Hung Manh Tran +5
Mathematics · #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Algebraic structures and combinatorial models #Combinatorics (math.CO) #FOS: Mathematics #Primary 05C50 #Secondary 52C35
paper · pdf · doi:10.48550/arxiv.2504.05423
openalex publication_date 2025/04/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Given a type A root system Φ of rank n, we introduce the concept of a signature for each subset S of Φ consisting of n+1 positive roots. For a subset S represented by a tuple (β1, …, βn+1), the signature of S is defined as an unordered pair \a, b\, where a and b denote the numbers of 1s and -1s, respectively, among the cofactors (-1)k det(S ∖ \βk\) for 1 ≤ k ≤ n+1. We prove that the number of tuples with a given signature can be expressed in terms of classical Eulerian numbers. The study of these signatures is motivated by their connections to the arithmetic and combinatorial properties of cones over deformed arrangements defined by Φ, including the Shi, Catalan, Linial, and Ish arrangements. We apply our main result to compute two important invariants of these arrangements: The minimum period of the characteristic quasi-polynomial, and the evaluation of the classical and arithmetic Tutte polynomials at (1, 1).