Signatures of Type $A$ Root Systems

Given a type $A$ root system $\Phi$ of rank $n$, we introduce the concept of a signature for each subset $S$ of $\Phi$ consisting of $n+1$ positive roots. For a subset $S$ represented by a tuple $(\beta_1, \ldots, \beta_{n+1})$, the signature of $S$ is defined as an unordered pair $\{a, b\}$, where $a$ and $b$ denote the numbers of $1$s and $-1$s, respectively, among the cofactors $(-1)^k \det(S \setminus \{\beta_k\})$ for $1 \le k \le n+1$. 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 $\Phi$, 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 $(1, 1)$.