Summary: Let \((M,g)\) be a compact Riemannian surface with volume 1. Let \(A=(a_{ij})_{n\times n}\) be an invertible matrix and \(A^{-1}=(a_{ij})_{n\times n}\) be the inverse of \(A\). In this paper, we consider the generalized Liouville system
\[ \Delta_gu_i+\sum^n_{j=1}a_{ij}\rho_j\left(\frac{h_je^{u_j}}{\int h_je^{u_j}}-1\right)=0\quad\text{in }M,\tag{1} \]
where \(0<h_j\in C^1(M)\) and \(\rho_j\in \mathbb R^+\), and prove that, under the assumptions of \((H_1)\) and \((H_2)\) (see the Introduction), the Leray-Schauder degree of (1) is equal to
\[ \frac{(-\chi(M)+1)\cdots(\chi(M)+N)}{N!} \]
if \(\rho=(\rho_1,\dots,\rho_n)\) satisfies
\[ 8\pi N\sum^n_{i=1}\rho_i<\sum_{1\leq i,j\leq n}a_{ij}\rho_i\rho_j<8\pi(N+1)\sum^n_{i=1}\rho_i. \]
Equation (1) is a natural generalization of the classic Liouville equation and is the Euler-Lagrangian equation of the nonlinear function \(\Phi_\rho\):
\[ \Phi_\rho(u)=\tfrac12\int_Ma^{ij}\nabla_gu_i\cdot\nabla_gu_j+\sum^n_{i=1}\int_M\rho_iu_i-\sum^n_{i=1}\rho_i\log\int_Mh_ie^{u_i}. \]
The Liouville system (1) has arisen in many different research areas in mathematics and physics. Our counting formulas are the first result in degree theory for Liouville systems.