In the top to random shuffle, the first a cards are removed from a deck of n cards 12 . . . n and then inserted back into the deck. This action can be studied by treating the top to random shuffle as an element Ba of the algebra Q(Sn). For a = 1, Adriano Garsia in “On the Powers of Top to Random Shuffling” (2002) derived an expansion formula for B 1 for k ≤ n, though his proof for the formula w...