Stirling and Bell numbers
Like the binomial coefficients, the Stirling numbers often arise in counting problems and depend on two variables, n and r. The Stirling number S(n, r)is the number of ways of partitioning a set of n members into r blocks(with no block empty, and the order of the blocks and within the blocks, is immaterial).(Strictly these are called Stirling numbers of the second kind. Those of the first kind,which are related, count something quite different, namely the number of ways we can permute n objects into r cycles.)For instance, the set with members a, b, c can be partitioned into three blocks injust one way:{a},{b},{c}, into two blocks in three ways{a, b},{c};{a},{b, c}, and{a, c},{b}, and into a single block in one way only:{a, b, c};it follows that S(3,1)=1, S(3,2)=3 and S(3,3)=1. Since a set of n members can be partitioned in only one way into either 1 block or into nblocks, we always have S(n,1)=1=S(n, n). Ifwe draw up the triangle of Stirling numbers after the fashion of Pascal’s Triangle, we arrive at the array of Figure 5, and we now explain how the triangle is generated.