Sorting permutation preservation ($B_i=A_{\pi (i)}$)

The output of a sorting algorithm must be a permutation of the complete input sequence. For equal lengths $n$, require a permutation $\pi$ of $\{0,\dots ,n-1\}$ such thatEquivalently, the input and output have the same multiset of complete entries: each entry has the same multiplicity. Equality of the underlying sets is insufficient; $(1,1,2)$ and $(1,2,2)$ have the same set of values and the same length but different multisets. When tracking occurrences rather than contents, retain original identity labels to check that each one is moved exactly once. A swap preserves this multiset automatically; an overwrite need not.