  When approximating the joint distribution of the component counts of a
decomposable combinatorial structure that is `almost' in the logarithmic class,
but nonetheless has irregular structure, it is useful to be able first to
establish that the distribution of a certain sum of non-negative integer valued
random variables is smooth. This distribution is not like the normal, and
individual summands can contribute a non-trivial amount to the whole, so its
smoothness is somewhat surprising. In this paper, we consider two coupling
approaches to establishing the smoothness, and contrast the results that are
obtained.
