  In this note we study the computational complexity of feedback arc set
counting problems in directed graphs, highlighting some subtle yet common
properties of counting classes. Counting the number of feedback arc sets of
cardinality $k$ and the total number of feedback arc sets are #P-complete
problems, while counting the number of minimum feedback arc sets is only proven
to be #P-hard. Indeed, this latter problem is #.OptP[log n]-complete, hence if
it belongs to #P then P=NP.
