Can anyone suggest a good and recent survey on counting problems and/or problems that are #P.
L. Fortnow. Counting complexity. In L. Hemaspaandra and A. Selman, editors, Complexity Theory Retrospective II, pages 81-107. Springer, 1997
This gives more of the structural complexity point of view (complexity classes, oracles, etc.), and discusses other classes related to #P. Although it's from almost 15 years ago, it's really not that out of date in terms of results.
Pinyan Lu published a survey via ECCC in mid 2011. It compares three popular counting frameworks:
- Counting Graph Homomorphisms,
- Counting Constraint Satisfaction (#CSP), and
- the Holant framework
- (and restrictions of these frameworks).
He also discusses the current dichotomy theorems and the proof techniques used to obtain them.
Xi Chen published a survey as a guest column for SIGACT News in late 2011. It discusses the results and techniques leading up to and including his papers with Jin-Yi Cai and Pinyan Lu on dichotomies for counting graph homomorphisms defined by an undirected target graph with complex weights (arXiv) and nonnegatively-weighted #CSPs (arXiv).
Another framework of counting problems comes from computing the Tutte polynomial of a graph. In this framework, any two complex numbers defines a counting problem.
The book Matroid Applications devotes chapter 6 to The Tutte Polynomial and Its Applications. The previous link is to a scan of that chapter from the website of James Oxley, one of the coauthors. Last semester, he taught a course based on that chapter.
Another good reference on this topic is this survey-like paper by Welsh.