Unimodality, Log-concavity, Real-rootedness And Beyond

Abstract

Theorem 7.5.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474 7.10 Historical notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4767.1 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.

The authors' abstract, as published at the source., 2015 · DOI ↗