Generalization of Garsia’s Formula

Notice that Bk a is a sum of terms σ1σ2 . . . σk where σi ∈ Sn is a term of the ith copy of Ba. Each σi will be called an a-shuffle, because it shuffles the first a cards back into the deck. We can thus denote σi = (b(i−1)a+1, b(i−1)a+2, . . . , bia) where b(i−1)a+m = σi(m) (i.e. σi acted on the mth card of the deck). Thus, the sequence (σ1, . . . , σk) gives rise to the ka-tuple (b1, b2, . . . , bka). Determining (σ1, . . . , σk) is thus the same as determining (b1, b2, . . . , bka). σi acts on the mth card by sending it to position σi(m) = b(i−1)a+m; we say that the mth card is hit by b(i−1)a+m in this situation. We say that cards 1, 2, . . . , j are hit by σ1 . . . σk if each of them is hit by some bl. Thus, each such term σ1σ2 . . . σk hits cards 1, 2, . . . , j for a unique j ∈ [a,min(ka, n)]; it must hit at least the cards 1, 2, . . . , a, and