Equivalence of the grand canonical partition functions of particles with different statistics

It is shown that the grand partition function of an ideal Bose system with single particle spectrum $\epsilon_i = (2n+k+3/2)\hbar\omega$ is identical to that of a system of particles with single particle energy $\epsilon_i =(n+1/2)\hbar\omega$ and obeying a particular kind of statistics based on the permutation group.