  For this paper, we studied the time evolution of a system of coagulating
particles under a generalized electrorheological (ER) kernel with real power,
$K\left(i,j\right) = \left( \frac{1}{i}+\frac{1}{j} \right)^\alpha$, and
monodisperse initial conditions. We used a combinatorial framework in which
time and cluster sizes were discrete and the binary aggregation governed the
time evolution of the system. We modified a previously-known solution for the
constant kernel to cover the generalized ER kernel and used it in the framework
to obtain the exact expression for the cluster size distribution (the average
number of particles of a given size) and the standard deviation. Our
theoretical solution is validated by a comparison to numerically simulated
results for several values of $\alpha$ and to the experimental data of
coagulating polystyrene particles. Theoretical predictions were accurate for
any time of the aggregation process and for a wide range of $\alpha$.
