How is a "riffle shuffle" mathematically defined? According to http://mathworld.wolfram.com/RiffleShuffle.html : (1) 8 out-shuffles return an ordinary deck of 52 cards to its original order. (2) 7 riffle shuffles are needed to get to close to random. [Hence 8 riffle shuffles would be even better, I think.] So clearly the "riffle shuffles" of (2) are not meant to be identical to "out-shuffles". But then how is a "riffle shuffle" of (2) mathematically defined?

Aigner & Ziegler explore this fully in Chapter 24 of their Proofs From The Book.
Essentially, a riffle shuffle is a permutation $\pi$ such that the sequence $(\pi(1),\dots,\pi(52))$ is precisely two interleaved increasing sequences.
Consider a deck of 10 cards, as an easier case, and suppose they started in sorted order: 1 at the top, 10 at the bottom. If we cut the deck in half perfectly and interleave one-by-one, the result is $(1,6,2,7,3,8,4,9,5,10)$. Notice we can pick out $(1,2,3,4,5)$ and $(6,7,8,9,10)$ as the two increasing sequences, so there's actually a bijection between the result of the shuffle and the set of ways to cut the deck and then interleave the halves.
