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We study statistical properties of $N$ non-interacting identical bosons orfermions in the canonical ensemble. We derive several general representationsfor the $p$-point correlation function of occupation numbers$\overline{n_1\cdots n_p}$. We demonstrate that it can be expressed as a ratioof two $p\times p$ determinants involving the (canonical) mean occupations$\overline{n_1}$, ..., $\overline{n_p}$, which can themselves be convenientlyexpressed in terms of the $k$-body partition functions (with $k\leq N$). Wedraw some connection with the theory of symmetric functions, and obtain anexpression of the correlation function in terms of Schur functions. Ourfindings are illustrated by revisiting the problem of Bose-Einsteincondensation in a 1D harmonic trap, for which we get analytical results. We getthe moments of the occupation numbers and the correlation between ground stateand excited state occupancies. In the temperature regime dominated by quantumcorrelations, the distribution of the ground state occupancy is shown to be atruncated Gumbel law. The Gumbel law, describing extreme value statistics, isobtained when the temperature is much smaller than the Bose-Einsteintemperature.

Correlations of occupation numbers in the canonical ensemble and application to BEC in a 1D harmonic trap