Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 215 / 1 / b / i / Solution

One shuffle removes the ordered top $k$ cards and inserts them, in a uniformly random person-order, into successively chosen uniform slots. The transition rule depends on a deck only through relabeling of its cards, so its transition matrix on the symmetric group is doubly stochastic. Hence the uniform distribution on all $n!$ permutations is invariant.