The Partition Function of Log-Gases with Multiple Odd Charges

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.