@article{pemantle2008twenty,
  title={Twenty combinatorial examples of asymptotics derived from multivariate generating functions},
  author={Pemantle, Robin and Wilson, Mark C.},
  journal={SIAM Review},
  volume={50},
  number={2},
  pages={199-272},
  year={2008},
  publisher={Society for Industrial and Applied Mathematics},
  keywords={ACSV theory},
  url_Paper={https://epubs.siam.org/doi/epdf/10.1137/050643866},
  abstract={Let $F$ be a power series in at least two variables that defines a
meromorphic function in a neighbourhood 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  $F$, uniform in certain explicitly defined
cones of directions.

The purpose of this article is to illustrate the use of these techniques
on a variety of problems of combinatorial interest. The first part
reviews the Morse-theoretic underpinnings of these techniques, and then
summarizes the necessary results so that only elementary analyses are
needed to check hypotheses and carry out computations. The remainder
focuses on combinatorial applications. Specific examples deal with
enumeration of words with forbidden substrings, edges and cycles in
graphs, polyominoes, descents and solutions to integer equations. 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. Generating functions derived in these three ways are amenable
to our asymptotic analyses, and we state some further general results
that apply to these cases.}
}

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