Is counting the words in a finite regular language #P-complete?
See this paper:
Counting the size of regular sequences of given length is #P-complete:
S. Kannan, Z. Sweedyk, and S. R. Mahaney. Counting
and random generation of strings in regular languages.
In ACM-SIAM Symposium on Discrete Algorithms
(SODA), pages 551–557, 1995.