Storing $K$ numbers with ordering Let $S$ be a sequence of $k$ many numbers. The position of the number in the $S$ matters. In the $S$ there are only $j$ many different numbers are there means $S$ contains many duplicates. Position is a query which needed to be answered, position(i) returns the element at position $i$ in $S$.

Trivial way to store them is using array which will takes $O(S)$ space and position can be solved in $O(1)$ time.

I am looking for a representation that takes $O(j)$ space (or somewhere near to it) and such that position can be solved in $O(1)$(even in more time is also ok)?

Please suggest some papers also.
What you ask is impossible, assuming you don't really mean sets (as sets are orderless), but ordered tuples. Consider $j=2$, then you effectively have $k$-length binary strings. And those require $\Omega(k)$ storage.