The classical 1991 result by Brightwell and Winkler states that the number of
linear extensions of a poset is #P-complete. We extend this result to posets
with certain restrictions. First, we prove that the number of linear extension
for posets of height two is #P-complete. Furthermore, we prove that this holds
for incidence posets of graphs. Finally, we prove that the number of linear
extensions for posets of dimension two is #P-complete