How many riffles mix a deck?

Ideal Gilbert–Shannon–Reeds (GSR) riffles of 52 distinct cards, as described by Bayer and Diaconis. Each riffle cuts the deck after k cards with probability C(52, k)/2⁵². Cards then fall one at a time from the bottom of either half, with probability proportional to the cards left in that half. Every compatible cut and interleaving has probability 2⁻⁵². Both halves keep their order, and every card keeps its identity.

Convention: deck[position] = card, position 1 = top; a card’s number is its place in the fresh deck. With pos[card] = position, the number of rising sequences is r = 1 + descents(pos): runs of consecutive card numbers still in their original order (the threads). It is not the number of increasing stretches you read along the spread. One riffle leaves r ≤ 2; m riffles leave r ≤ 2^m.

The chance of one exact order after m riffles is P_m(π) = C(2^m + 52 − r, 52) / 2^52m. That is the mass on a single order, not a distance. A perfectly mixed deck gives every order 1 in 52! ≈ 8.07 × 10⁶⁷. The distance from uniform is the total variation distance of the whole distribution, d = ½ Σ_π |P_m(π) − 1/52!|. It is computed exactly by grouping all 52! orders by r with Eulerian numbers, in double precision (error estimated below 10⁻¹²), with no sampling. Bayer and Diaconis report 0.614 after six riffles and 0.334 after seven; the bars use the same formula.

The sample deck is one seeded realization (mulberry32 generator, seed shown), replayed riffle by riffle from the same stream. It is not a frequency estimate: its rising-sequence count can fall as well as rise, and no single deck measures the distance. The 0.5 yardstick is a convention for ‘close enough’ in this ideal model. Seven is not perfect randomness, and other yardsticks or measures of mixing give other counts. Not modeled: real hands (clumpy drops, uneven cuts), overhand shuffles, casino machines, and repeated or marked cards. Domain: 52 cards, 0–10 riffles. Animation timing is for viewing, not the speed of a real shuffle.

Source: D. Bayer and P. Diaconis, Trailing the Dovetail Shuffle to Its Lair, Annals of Applied Probability 2 (1992), §§1–3. The paper credits the riffle model to Gilbert, Shannon and Reeds.