Combinatorial solutions to coagulation kernel for linear chains

For this paper, we studied the time evolution of a system coagulating particles under 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 combinatorial framework in which cluster sizes were discrete binary aggregation governed system. modified previously-known solution for constant to cover ER it obtain exact expression size distribution (the average number given size) standard deviation. Our theoretical is validated by comparison numerically simulated results several values $\alpha$ experimental data polystyrene particles. Theoretical predictions accurate any process wide range $\alpha$.