I'm looking for a way to enumerate the $k$-permutations of a $n$ element set; that is, if $A=\left\{1, \dots, n\right\}$ and $K=\left\{(p_1, \dots, p_k)\mid p_i\in A\wedge i\neq j\implies p_i\neq p_j\right\}$; then I'd like a bijection
$f:\left\{1, \dots, \frac{n!}{(n-k)!}\right\}\to K$
along with it's inverse. In particular I'd like something computable rather than enumerative since clearly I could enumerate them by hand, but this becomes memory inefficient for large $n$ and $k$.