Suppose you have a deck of $n$ cards; e.g., $n{=}12$: $$ (1,2,3,4,5,6,7,8,9,10,11,12) \;. $$ Cut the deck into $k$ equal-sized pieces, where $k|n$; e.g., for $k{=}4$, the $12$ cards are partitioned into $4$ piles, each of $m=n/k=3$ cards: $$ \left( \begin{array}{ccc} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \\ 10 & 11 & 12 \\ \end{array} \right) \;. $$ Now perfectly shuffle them by selecting the top card from stack $1$, the top card from stack $2$, and so on, walking down the columns of the matrix above, resulting in this shuffled deck of cards: $$ (1,4,7,10,2,5,8,11,3,6,9,12) \;. $$ Continue in this manner until the deck of cards returns to its initial sorting: $$ \left( \begin{array}{cccccccccccc} 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 \\ 1 & 4 & 7 & 10 & 2 & 5 & 8 & 11 & 3 & 6 & 9 & 12 \\ 1 & 10 & 8 & 6 & 4 & 2 & 11 & 9 & 7 & 5 & 3 & 12 \\ 1 & 6 & 11 & 5 & 10 & 4 & 9 & 3 & 8 & 2 & 7 & 12 \\ 1 & 5 & 9 & 2 & 6 & 10 & 3 & 7 & 11 & 4 & 8 & 12 \\ 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 \\ \end{array} \right) \;. $$ Here, for $n{=}12$ cards partitioned into $k{=}4$ parts, it requires $s{=}5$ perfect shuffles to cycle. Let us say that $f(n,k)=s$, i.e., $f(12,4)=5$. Similarly I can calculate that $$ f(8,2)=3,\; f(18,3)=16,\; f(33,3)=8, \; f(52,2)=8, $$ etc. The last represents a perfect "outer-shuffle" of a standard $52$-card deck, which is known to take $8$ shuffles to cycle. It seems likely this function is known to combinatorialists:

. What is $f(n,k)$?Q