ABSTRACT
Theorem 7.5.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474 7.10 Historical notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476
7.1 Introduction Many important sequences in combinatorics are known to be log-concave or unimodal, but many are only conjectured to be so although several techniques using methods from combinatorics, algebra, geometry and analysis are now available. Stanley [90] and Brenti [25] have written extensive surveys of various techniques that can be used to prove real-rootedness, log-concavity or unimodality. After a brief introduction and a short section on probabilistic consequences of real-rootedness, we will complement [25, 90] with a survey over new techniques that have been developed, and problems and conjectures that have been solved. I stress that this is not a comprehensive account of all work that has been done in the area since op. cit.. The selection is certainly colored by my taste and knowledge.
