  We explore inequalities on linear extensions of posets and make them
effective in different ways. First, we study the Bj\"orner--Wachs inequality
and generalize it to inequalities on order polynomials and their $q$-analogues
via direct injections and FKG inequalities. Second, we give an injective proof
of the Sidorenko inequality with computational complexity significance, namely
that the difference is in $\#P$. Third, we generalize the Sidorenko inequality
to posets with small chain intersections and give complexity theoretic
applications.
