RestrictedGrowthString from SetPartition
A set partition's restricted growth string: each position labelled with its block's index, from 0.
An order isomorphism: the k-th set partition of n, in the order SetPartitions lists them, goes to the k-th restricted growth string of length n. So everything defined on one carrier is available on the other through it.
Called as CombinatorialMap(x, "RestrictedGrowthString"). The head's page has its examples.