Smallest components in decomposable structures: Exp-log class

The smallest size of components in random decomposable combinatorial struct ures is studied in a general framework. Our results include limit distribut ion and local theorems for the size of the rth smallest component of an obj ect of size n. Expectation, variance and higher moments of the rth smallest component are also derived. The results apply to several combinatorial str uctures in the exp-log class for both labelled and unlabelled objects. We e xemplify with several combinatorial structures like permutations and polyno mials over finite fields.