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 nonhomogeneous 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.