The Robinson-Schensted correspondence has already been mentioned in another answer, as providing a bijection between permutations of $\{1,\dots,n\}$ and pairs of standard Young tableaux of the same shape (corresponding to partitions of $n$).
But this correspondence goes even further, since it provides three equivalence relations on the permutations (same left tableau, same right tableau, same shape), and in fact these equivalence relations are the same as those defining the left-, right- and twosided Kazhdan-Lusztig cells, when $S_n$ is seen as the Weyl group in type $A_{n-1}$.
Further, the twosided order on the twosided cells coincides with the dominance order of the tableau (i.e. partitions) determined by the twosided cells.
So this is an example of a combinatorial thing having a representation theoretic interpretation.