  The proposal of Dijkgraaf, Verlinde and Verlinde for the emergence of smooth
strings from a supersymmetric U(N) Yang-Mills theory assumes that conjugacy
classes in the symmetric group with a few long cycles dominate the dynamics
about the infrared fixed point. It is shown that the average number of cycles
in a conjugacy class of $S_{N}$ is bounded below by $const.\sqrt N\ln\sqrt N,$
implying that some physical mechanism is needed to ensure the assumed
dominance. It is shown that if individual cycles have positive energies that
depend very weakly on their lengths, then long cycles dominate the dynamics at
low temperatures.
