The Written Qual Book: In the summer of 2016, the math graduate students at Dartmouth solved all of the written qualifying exam questions that had been posed since the department changed the exam format in 2012. These solutions were carefully edited, compiled, and annotated by Daryl DeFord and David Freund, and can be found here.
Combinatorics Qual Questions: I attempted to recall and record all of the questions that were asked of me during my oral qualifying exam in combinatorics. For those interested in taking a combinatorics qual who want to see the types of questions that may be asked, this document can be found here. I would be happy to discuss preparation for the oral exam, but I have not included solutions -- you should solve them yourself! Unfortunately, I did not do the same with the questions on my probability qual, and have long since forgotten.
The Online Encyclopedia of Integer Sequences contains thousands of lists of numbers that count almost any combinatorial objects you can think of. Here are some of the sequences related to my work:
Number of 0-convex functions $f: [n] \to [p]$ for $n$ sufficiently large, as a function of $p$
Number of 1-convex permutations of $[n]$
Number of permutations of $[n]$ which give rise to the largest-possible dominating set in the online algorithm
Number of permutations of $[n]$ which give rise to the smallest-possible dominating set in the online algorithm