Let N(a,b) be the number of strict order preserving maps from a given partial order P to the first r natural numbers which take a particular element x of P to a and take y in P to b. It is shown here that in appropriate ranges of the parameters, this function must obey the inequalities: \[ N(r,u+v+w)N(r+s+t,u)\leq N(r+s,u+w)N(r+t,u+v) \] \[ and\quad N(r,u)N(r+s+t,u+v+w)\leq N(r+s,u+w)N(r+t,u+v), \] and so must the analogous function for not necessarily strict order preserving maps. These results include an inequality of Daykin, Daykin and Paterson, which is the analogue for strict and order preserving maps of a theorem of Stanley that holds for injections. It is shown that these new inequalities do not hold in general for injections. A number of further results of the same kind are also given.