  In the top to random shuffle, the first a cards are removed from a deck of n
cards 12 \cdots n and then inserted back into the deck. This action can be
studied by treating the top to random shuffle as an element B_a, which we
define formally in Section 2, of the algebra Q[S_n]. For a = 1, Adriano Garsia
in "On the Powers of Top to Random Shuffling" (2002) derived an expansion
formula for B_1^k for k \leq n, though his proof for the formula was
non-bijective. We prove, bijectively, an expansion formula for the arbitrary
finite product B_{a_1}B_{a_2} \cdots B_{a_k} where a_1, \ldots, a_k are
positive integers, from which an improved version of Garsia's aforementioned
formula follows. We show some applications of this formula for B_{a_1}B_{a_2}
\cdots B_{a_k}, which include enumeration and calculating probabilities. Then
for an arbitrary group G we define the group of G-permutations S_n^G := G \wr
S_n and further generalize the aforementioned expansion formula to the algebra
Q[S_n^G] for the case of finite G, and we show how other similar expansion
formulae in Q[S_n] can be generalized to Q[S_n^G].
