Twenty Combinatorial Examples of Asymptotics Derived from Multivariate Generating Functions

Let $\{ a_{\bf r} : {\bf r} \in {\mathbb N}^d \}$ be a $d$-dimensional array of numbers for which the generating function $F({\bf z}) := \sum_{\bf r} a_{\bf r} {\bf z}^{\bf r}$ is meromorphic in a neighborhood of the origin. For example, $F$ may be a rational multivariate generating function. We discuss recent results that allow the effective computation of asymptotic expansions for the coefficients of $F$. Our purpose is to illustrate the use of these techniques on a variety of problems of combinatorial interest. The survey begins by summarizing previous work on the asymptotics of univariate and multivariate generating functions. Next we describe the Morse-theoretic underpinnings of some new asymptotic techniques. We then quote and summarize these results in such a way that only elementary analyses are needed to check hypotheses and carry out computations. The remainder of the survey focuses on combinatorial applications, such as enumeration of words with forbidden substrings, edges and cycles in graphs, polyominoes, and descents in permutations. After the individual examples, we discuss three broad classes of examples, namely, functions derived via the transfer matrix method, those derived via the kernel method, and those derived via the method of Lagrange inversion. These methods have the property that generating functions derived from them are amenable to our asymptotic analyses, and we describe further machinery that facilitates computations for these classes of examples.