Generalizations of an Expansion Formula for Top to Random Shuffles

Abstract

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 ${{\mathbb{Q}[S_n]}}$. For ${a = 1}$, Garsia in “On the powers of top to random shuffling” (2002) derived an expansion formula for ${{B^k_1}}$ for ${{k \leq n}}$, though his proof for the formula was non-bijective. We prove, bijectively, an expansion formula for the arbitrary finite product ${B_{a1} B_{a2} \cdots B_{ak}}$ where ${a_{1}, \cdots , a_{k}}$ are positive integers, from which an improved version of Garsia’s aforementioned formula follows. We show some applications of this formula for ${B_{a1} B_{a2} \cdots B_{ak}}$, which include enumeration and calculating probabilities. Then for an arbitrary group ${G}$ we define the group of ${G}$-permutations ${{S^G_n} := {G \wr S_n}}$ and further generalize the aforementioned expansion formula to the algebra ${{\mathbb{Q} [ S^G_n ]}}$ for the case of finite ${G}$, and we show how other similar expansion formulae in ${{\mathbb{Q} [S_n]}}$ can be generalized to ${{\mathbb{Q} [S^G_n]}}$.