This paper discusses two-terminal series-parallel networks occurring in Applied Probability and other contexts where the mathematical theory of reliability is developed. Eighty years ago Mac Mahon successfully discussed the number of different structures built of n identical components [1]; later Knödel [3] and Carlitz and Riordan [4] in the 1950's investigated the number of different structures built of n different components. However, in many new applications, engineers and statisticians have to study structures having n components with a specification defining various types of components and stating how many identical components of each kind should be used. This general problem is investigated in Section 7 whilst in Sections 1–6 results obtained earlier are presented and it is shown there how modern approaches to combinatorics (particularly Pólya's Enumerative Combinatorial Analysis) can simplify some reasoning of previous authors.
