  We consider a lattice of bosonic atoms, whose number N may be smaller than
the number of lattice sites M. We study the Hartree-Fock wave function built up
from localized wave functios w(\mathbf{r}) of single atoms, with nearest
neighboring overlap. The zero-momentum particle number is expressed in terms of
permanents of matrices. In one dimension, it is analytically calculated to be
\alpha*N(M-N+1)/M, with \alpha=|\int w(\mathbf{r})d\Omega|^2/[(1+2a)l], where a
is the nearest-neighboring overlap, l is the lattice constant. \alpha is of the
order of 1. The result indicates that the condensate fraction is proportional
to and of the same order of magnitude as that of the vacancy concentration,
hence there is off-diagonal long-range order or Bose-Einstein condensation of
atoms when the number of vacancies M-N is a finite fraction of the number of
the lattice sites M.
