What's the relationship between ASP-complete and #P-complete?
Let an ASP-complete problem Q be given.  There is an ASP-reduction from 3SAT to Q and all ASP-reductions are parsimonious, hence it serves as a parsimonious reduction from #3SAT to #Q.  Since #Q is in #P and #3SAT is #P-hard, #Q is #P-complete.