  The Darwin-Fowler method in combination with the steepest descent approach is
a common tool in the asymptotic description of many models arising from
statistical physics. In this work, we focus rather on the non-asymptotic
behavior of the Darwin-Fowler procedure. By using a combinatorial approach
based on Bell polynomials, we solve it exactly. Due to that approach, we also
show relationships of typical models with combinatorial Lah and Stirling
numbers.
