  We consider two simple models for the formation of polymers where at the
initial time, each monomer has a certain number of potential links (called arms
in the text) that are consumed when aggregations occur. Loosely speaking, this
imposes restrictions on the number of aggregations. The dynamics of
concentrations are governed by modifications of Smoluchowski's coagulation
equations. Applying classical techniques based on generating functions,
resolution of quasi-linear PDE's, and Lagrange inversion formula, we obtain
explicit solutions to these non-linear systems of ODE's. We also discuss the
asymptotic behavior of the solutions and point at some connexions with certain
known solutions to Smoluchowski's coagulation equations with additive or
multiplicative kernels.
