  We study weakly coupled SU(N) N = 4 super Yang-Mills theory on R x S^3 at
infinite N, which has interesting thermodynamics, including a Hagedorn
transition, even at zero Yang-Mills coupling. We calculate the exact one-loop
partition function below the Hagedorn temperature. Our calculation employs the
representation of the one-loop dilatation operator as a spin chain Hamiltonian
acting on neighboring sites and a generalization of Polya's counting of
`necklaces' (gauge-invariant operators) to include necklaces with a `pendant'
(an operator which acts on neighboring beads). We find that the one-loop
correction to the Hagedorn temperature is delta ln T_H = + lambda/8 pi^2.
