Refined counting of necklaces in one-loop N=4 SYM

We compute the grand partition function of N = 4 SYM at one-loop in the SU(2) sector with general chemical potentials, extending the results of Polya's theorem. We make use of finite group theory, applicable to all orders of perturbative 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 XXX1/2 spin chains. We discuss how Hagedorn temperature changes on the complex plane of chemical potentials.