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Abstract for Edward A. Bender and L. Bruce Richmond,
Admissible Functions and Asymptotics for
Labelled Structures by Number of Components


Let $a(n,k)$ denote the number of combinatorial structures of size
$n$ with $k$ components.
One often has $\sum_{n,k} a(n,k)x^ny^k/n! = \Expg{yC(x)}$, where
$C(x)$ is frequently the exponential generating function for connected
structures.
How does $a(n,k)$ behave as a function of $k$ when $n$ is large and
$C(x)$ is entire or has large singularities on its circle of
convergence?
The Flajolet-Odlyzko singularity analysis does not directly apply in
such cases.
We extend some of Hayman's work on admissible functions of a single
variable to functions of several variables.
As applications, we obtain asymptotics and local limit theorems for
several set partition problems, decomposition of vector spaces,
tagged permutations, and various complete graph covering problems.

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