We consider the standard continuous and discrete linear programs P_1 -> max {c'x | Ax <=b; x >=0}and P_2->max {c'x | Ax <=b; x integer}"

as well as their respective "integral "and "counting" versions Q_1-> int_{Ax<=b;x>=0} exp{c'x} dx and Q_2 -> sum_{Ax<=b;x integer}

exp{c'x}

There is a well known duality for P_1.

Perhaps less known are also dualities for Q_1 and Q_2;