We use techniques in the shuffle algebra to present a formula for the partition function of a one-dimensional log-gas comprised of particles of (possibly) different integer charges at certain inverse temperature $\beta$ in terms of the Berezin integral of an associated non-homogeneous alternating tensor. This generalizes previously known results by removing the restriction on the number of species of odd charge. Our methods provide a unified framework extending the de Bruijn integral identities from classical $\beta$-ensembles ($\beta =1,2,4$) to multicomponent ensembles, as well as to iterated integrals of more general determinantal integrands.

中文翻译：

---

多奇电荷对数气体的配分函数

我们使用混洗代数中的技术来给出一维对数气体的配分函数公式，该对数气体由（可能）不同整数电荷的粒子在某个逆温度下组成$\beta$根据相关的非齐次交替张量的 Berezin 积分。这通过消除对奇数电荷种类数量的限制来概括先前已知的结果。我们的方法提供了一个统一的框架，将 de Bruijn 积分恒等式从经典$\beta$- 合奏（$\beta =1,2,4$) 到多分量系综，以及更一般的行列式被积函数的迭代积分。