Sequences , Patterns and Coincidences

This article provides a contemporary exposition at a moderately quantitative level of the distribution theory associated with sequences and patterns in iid multinomial trials, the birthday problem, and the matching problem. The section on patterns includes the classic distribution theory for waiting time for runs and more general patterns, and their means and moments. It also includes the central limit theorem and a.s. properties of the longest run, and modern applications to DNA sequence alignment. The section on birthdays reviews the Poisson approximation in the classical birthday problem, with new bounds on the total variation distance. It also includes a number of variants of the classic birthday problem, including the case of unequal probabilities, similar triplets, and the Bayesian version with Dirichlet priors on birthday probabilities. A new problem called the strong birthday problem with application to criminology is introduced. The section on matching covers the Poisson asymptotics, errors of Poisson approximations, and some variants such as near matches. It also briefly reviews the recent extraordinary developments in the area of longest increasing subsequences in random permutations. The article provides a large number of examples, many not well known, to help a reader have a feeling for these questions at an intuitive level.