Local central limit theorems and asymptotic enumerative combinatorics

A main goal in the field of enumerative combinatorics is to derive asymptotic formulas for combinatorial sequences. It has been shown in the past that many such sequences obey a central limit theorem, from which asymptotic estimates can be obtained. In particular, for a large class of combinatorial sequences (i.e., assemblies, multisets, and selections), there is a general principle that allows one to derive an asymptotic formula via a local central limit theorem (LCLT) for integer-valued random variables. We present a novel and robust method to derive quantitative LCLTs under conditions which are easy to verify, which then yields quantitative error estimates for the associated asymptotic formulas. In certain cases we are able to provide new quantitative error estimates unknown before. Joint work with Stephen De Salvo.