Abstract: In this talk I consider the relationship between long cycles and Bose-Einstein condensation (BEC) in the case of several models. Previous results on the Ideal Bose Gas, Mean-Field and Perturbed Mean-Field Models indicate that the condensate density exactly equals the long cycle density. I will consider the infinite-range-hopping Bose-Hubbard model, with both a finite and an infinite on-site repulsion interaction, and conclude that while the existence of a non-zero long cycle density coincides with the occurrence of BEC, their respective densities are not necessarily equal.