Random-to-top card shuffle

The random-to-top shuffle chooses one uniformly random card and moves it to the top of the deck. The time at which every card has been chosen is a strong stationary time, yielding cutoff at $n\log n$ with a window of order $n$.

The top-to-random shuffle removes the top card and inserts it in a uniformly random position. Under the uniform measure on permutations it is the time reversal of the random-to-top card shuffle, so the two shuffles have the same total-variation mixing profile.