Let A be an $n\times n$ matrix and let ${\vee }^{k}A$ be its k-th symmetric tensor power. We express the normalized trace of ${\vee }^{k}A$ as an integral of the k-th powers of the numerical values of A over the unit sphere ${\mathbb{S}}^n$ of ${\mathbb{C}}^n$ with respect to the rotation-invariant probability measure. Equivalently, this expression in turn can be interpreted as an integral representation for the (normalized) complete symmetric polynomials over ${\mathbb{C}}^n$. As applications, we present a new proof for the MacMahon Master Theorem in enumerative combinatorics. Then, our next application deals with a generalization of the work of Cuttler et al. in [Cuttler A, Greene C, Skandera M. Inequalities for symmetric means. Eur J Comb. 2011;32(6):745–761] concerning the monotonicity of products of complete symmetric polynomials. Finally, we give a solution to an open problem that was raised by Rovena and Temereanca in [Roventa I, Temereanca LE. A note on the positivity of the even degree complete homogeneous symmetric polynomials. Mediterr J Math. 2019;16(1):1–16].

让A成为$n\times n$矩阵并让${\vee }^{k}A$是它的第 k个对称张量幂。我们表达了归一化的踪迹${\vee }^{k}A$作为单位球面上A的数值的k次方的积分${\mathbb{小号}}^n$的${\mathbb{C}}^n$关于旋转不变概率测度。等价地，这个表达式又可以解释为（归一化的）完全对称多项式的积分表示${\mathbb{C}}^n$. 作为应用，我们在枚举组合学中提出了 MacMahon 主定理的新证明。然后，我们的下一个应用程序处理 Cuttler 等人的工作的概括。在 [Cuttler A, Greene C, Skandera M. 对称均值的不等式。Eur J 梳子。2011;32(6):745–761] 关于完全对称多项式乘积的单调性。最后，我们为 Rovena 和 Temereanca 在 [Roventa I, Temereanca LE. 关于偶数次完全齐次对称多项式正性的注释。Mediterr J 数学。2019;16(1):1-16]。