  We examine Bose-Einstein condensation (BEC) for particles trapped in a
harmonic potential by considering it as a transition in the length of
permutation cycles that arise from wave-function symmetry. This ``loop-gas''
approach was originally developed by Feynman in his path-integral study of BEC
for an homogeneous gas in a box. For the harmonic oscillator potential it is
possible to treat the ideal gas exactly so that one can easily see how standard
approximations become more accurate in the thermodynamic limit (TDL). One
clearly sees that the condensate is made up of very long permutation loops
whose length fluctuates ever more wildly as the number of particles increases.
In the TDL, the WKB approximation, equivalent to the standard approach to BEC,
becomes precise for the non-condensate; however, this approximation neglects
completely the long cycles that make up the condensate. We examine the exact
form for the density matrix for the system and show how it describes the
condensate and behaves in the TDL.
