I don't know much about group theory and card-shuffling theory, so this may already have a name I don't know about. I often shuffle a deck of cards using a method that is defined by a particular element of $S_{52}$, the symmetric group on 52 letters. In Mathematica notation, it is:
p = Cycles[{{1, 46, 35, 29, 8, 11, 47, 30, 3, 36, 24, 33, 39, 9, 6,
21, 48, 25, 28, 13, 37, 19, 7, 16, 22, 43, 50, 15, 27, 18, 12,
42, 51, 10}, {2, 41, 52, 5, 26, 23, 38, 14, 32, 44, 45, 40, 4,
31, 49, 20}}];
Similar to how a Faro shuffle returns a deck into its original order after 8 shuffles, the method I use returns the deck into it's original order after 272 shuffles:
G = PermutationGroup[{p}];
o = GroupOrder[G]
(*272*)
Unfortunately, that's not the end of the story. By applying a crude "sortedness metric" to the sequence of decks obtained by the shuffles, it's apparent that every 34 shuffles, the deck is in an "almost-ordered" state:
d = NestList[Permute[#, p] &, Range[52], o - 1];
s = Total@(Differences[#]^-2) & /@ d;
ListLinePlot[s, PlotRange -> All]
The sortedness metric is just the sum of the inverses of the squares of the elements of the delta sequence of the card-numbers of the deck, as can be inferred from above. The large spike at zero is just the original sorted deck, and the periodic spikes are correspond to almost-sorted states.
Here are the initial and 7 subsequent almost-sorted states (right-click the image and open in a new tab to get the full-size 1500 x 453 pixel size):
ListPlot[d[[#]], PlotLabel -> #] & /@ Flatten@Position[s, x_ /; x > 20]
These oscillations mean that in practice, the shuffle can only be used 34 times before the deck is suspiciously similar to its initial state.
So my question is:
- Is there a name for these oscillations?
