  This paper links sizes of model classes to the minimum lengths of their
defining formulas, that is, to their description complexities. Limiting to
models with a fixed domain of size n, we study description complexities with
respect to the extension of propositional logic with the ability to count
assignments. This logic, called GMLU, can alternatively be conceived as graded
modal logic over Kripke models with the universal accessibility relation. While
GMLU is expressively complete for defining multisets of assignments, we also
investigate its fragments GMLU(d) that can count only up to the integer
threshold d. We focus in particular on description complexities of equivalence
classes of GMLU(d). We show that, in restriction to a poset of type
realizations, the order of the equivalence classes based on size is identical
to the order based on description complexities. This also demonstrates a
monotone connection between Boltzmann entropies of model classes and
description complexities. Furthermore, we characterize how the relation between
domain size n and counting threshold d determines whether or not there exists a
dominating class, which essentially means a model class with limit probability
one. To obtain our results, we prove new estimates on r-associated Stirling
numbers. As another crucial tool, we show that model classes split into two
distinct cases in relation to their description complexity.
