Counting 3-colorings: http://blog.computationalcomplexity.org/2022/06/a-gadget-for-3-colorings.html and follow-up post https://blog.computationalcomplexity.org/2022/06/counting-number-of-3-colorings-of-graph.html

Like all "natural" NP-complete problems (and many easier problems), the 3-coloring problem should have a #P-complete counting version, but the gadgets needed to prove it are a little subtle and tracking down the history of proof of this result took some effort.