chapter  8
58 Pages

Special Numbers

WithMichael Z. Spivey

This chapter is devoted to several collections of numbers related to the binomial coefficients. Each of these sets of numbers—the Fibonacci, Stirling, Bell, Lah, Catalan, and Bernoulli numbers—has an expression featuring the binomial coefficients. Each satisfies a variety of identities involving the binomial coefficients. Each (with the exception of the Bernoulli numbers) has one or more combinatorial interpretations. In fact, because of these combinatorial interpretations this chapter functions to a large degree like a second chapter on using combinatorial arguments to prove binomial identities. In particular, when discussing Catalan numbers we introduce more sophisticated lattice path arguments than we saw in Chapter 3. We don’t just use combinatorial arguments, though; we’ll also see recurrence relations and generating functions in our proofs in this chapter.