\begin{definition}[Definition:Modified Perfect Faro Shuffle]
A '''modified perfect Faro shuffle''' is a shuffle performed on a deck of cards $D$ of size $2 n$.
Cut $D$ into $2$ piles $D_1$ (the top half) and $D_2$ (the bottom half) of exactly $n$ cards each.
The shuffled deck $D'$ is assembled by alternating the cards in $D$:
:The cards in positions $2, 4, 6, \ldots, 2 n$ of $D'$ are to be occupied by the cards $1, 2, \ldots, n$ of $D_1$ in the same order
:The cards in positions $1, 3, 5, \ldots, 2 n - 1$ of $D'$ are to be occupied by the cards $1, 2, \ldots, n$ of $D_2$ in the same order.
Thus a card in position $k$ in $D$ ends up in position $2 k \bmod \paren {2 n + 1}$ in $D'$.
\end{definition}