Discriminating integer partitions

Question

Given a fixed positive integer, say $n$, letn $P_n$ be the set all partitions of $n$, where each partition itself is a set i.e. order is discarded and each part is less than 5. Can we establish a bijective mapping $f:P_n \to [1, |Pn|]$ such that for a partition $x$, $f(x)$ is computable without explicitly using $P_n$ (otherwise, we would lexicographically order them and establish the bijection.). Rather, I want $f$ to be computable only from the parts of the partition. My interest is in discriminating differnt partition. If this is not possible, I would look for $f$ which is injective, and co-domain is not necessary to be as defined as above, but preferably have cardinality not very large compared to $|P_n|$. It is fine even if this conditions are met for $n < 30$, though general function would be better.