Refined Counting of Necklaces in One-loop $\mathcal{N}=4$ SYM

We compute the grand partition function of $\mathcal{N}=4$ SYM at one-loop in the $SU(2)$ sector with general chemical potentials, extending the results of P\'olya's theorem. We make use of finite group theory, applicable to all orders of $1/N_c$ expansion. We show that only the planar terms contribute to the grand partition function, which is therefore equal to the grand partition function of an ensemble of XXX$_\frac12$ spin chains. We discuss how Hagedorn temperature changes on the complex plane of chemical potentials.