Sets are a basic combinatorial construction.
Many properties of general structures are direct consequences of their being sets. In all those cases, the statistics of the largest and smallest component are related to the complexity of algorithms operating over these structures. Very explicit results concerning these statistics can be obtained by extracting coefficients from the proper generating functions, which in this case is a refined way of performing an inclusion-exclusion argument. Taking the difference between largest size at most k and largest size at most k -1 gives the formul for largest size exactly k , and similarly for the smallest size. Various combinations of these techniques produce numerous results.