ABSTRACT
The topics of enumeration, combinatorics, and number theory were called higher arithmetic in previous periods, and were classified as pure mathematics. The need to analyze algorithms transformed them to engineering mathematics since nearly all such analyses are reduced at some point to an enumeration of features in combinatorial structures (such as strings, graphs, or Turing machine traces, etc.). For infinite sets, the size and total weight have no useful meaning. It is important to keep in mind that even though the enumerators are used as formal series, convergence issues do not arise here. The methods of counting compositions the authors developed turn out to be useful in counting the ways of making selections of items from sets, when the selection needs to satisfy certain side conditions. The sum and product operations, and the associated closure, have served us well in our discussion of integer compositions, but we need to prepare for more complicated needs ahead.
