Unimodality, log-concavity, real-rootedness and beyond

TL;DR

This survey reviews recent advances in understanding unimodality, log-concavity, and real-rootedness in combinatorics, highlighting new techniques, solved problems, and ongoing conjectures.

Contribution

It complements existing surveys by introducing new methods and discussing recent solved problems and conjectures in the field.

Findings

Introduction to recent techniques in combinatorics

Summary of solved problems and conjectures

Overview of new developments in unimodality and log-concavity

Abstract

This is a survey on recent developments on unimodality, log-concavity and real-rootedness in combinatorics. Stanley and Brenti have written extensive surveys of various techniques that can be used to prove real-rootedness, log-concavity or unimodality. After a brief introduction, we will complement these surveys with a survey over some new techniques that have been developed, as well as problems and conjectures that have been solved. This is a draft of a chapter to appear in Handbook of Enumerative Combinatorics, published by CRC Press.