A quick bound on shuffles

God plays dice

David Eppstein asks: how many riffles does it take until all permutations are possible?

A riffle shuffle permutation, in the mathematics of shuffling cards, is a permutation of the cards that can be obtained by a single riffle shuffle — that is, you cut the deck into two packets and then interleave the packets. There are $latex 2^n – n$ distinct riffle shuffles of n cards. For example, consider a five-card deck, which is initially in the order 1, 2, 3, 4, 5. Then a riffle shuffle consists of:

  • cutting the deck in one of the six possible ways: 12345/ (no cut), 1234/5, 123/45, 12/345, 1/2345, /12345 (also no cut) where the slash represents the cut.
  • riffling. For example look at 123/45. There are $latex {5 \choose 2} = 10$ possible ways to make a permutation from this – we decide which two slots to put the 4 and…

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