  We examine the counts-in-cells probability distribution functions that
describe dark matter halos in the Dark Energy Universe Simulations (DEUS). We
describe the measurements between redshifts $z=0$ to $z=4$ on both linear and
non-linear scales. The best-fits of the gravitational quasi-equilibrium
distribution (GQED), the negative binomial distribution (NBD), the
Poisson-Lognormal distribution (PLN), and the Poisson-Lognormal distribution
with a bias parameter (PLNB) are compared to simulations. The fits agree
reasonably consistently over a range of redshifts and scales. To distinguish
quintessence (RPCDM) and phantom ($w$CDM) dark energy from $\Lambda$ dark
energy, we present a new method that compares the model parameters of the
counts-in-cells probability distribution functions. We find that the mean and
variance of the halo counts-in-cells on $2-25h^{-1}$Mpc scales between
redshifts $0.65<z<4$ show significant percentage differences for different dark
energy cosmologies. On $15-25h^{-1}$Mpc scales, the $g$ parameter in NBD,
$\omega$ parameter in PLN, $b$ and $C_b$ parameters in PLNB show larger
percentage differences for different dark energy cosmologies than on smaller
scales. On $2-6h^{-1}$Mpc scales, kurtosis and the $b$ parameter in the GQED
show larger percentage differences for different dark energy cosmologies than
on larger scales. For cosmologies explored in the DEUS simulations, the
percentage differences between these statistics for the RPCDM and $w$CDM dark
energy cosmologies relative to $\Lambda$CDM generally increases with redshift
from a few percent to significantly larger percentages at $z=4$. Applying our
method to simulations and galaxy surveys can provide a useful way to
distinguish among dark energy models and cosmologies in general.
