Problems Section: Problems 1371 - 1380

Q1371 Consider shuffles of a standard $52$-card pack.  (See the article in this issue for terminology and basic information.)  Let $\mathop{rev}\nolimits$ be the shuffle which reverses the pack -- that is, the first card is swapped with the last, the second with the second last, and so on.  As in the article, $\mathop{out}\nolimits$ denotes the outshuffle.
 
(a) Write a formula for $\mathop{rev}\nolimits(k)$ in terms of $k$; also, write $\mathop{rev}\nolimits$ as a product of cycles.
(b) Without any calculation, write down the cycle type of the composite shuffle $\mathop{rev}\circ\mathop{out}\circ\mathop{rev}$.
(c) Show that if we shuffle a pack of cards with both an outshuffle and a reverse shuffle, it makes no difference which one we do first.
 
Q1372 Show that a composition of $n-1$ cycles
$$(\,1\ 2\,)\circ(\,1\ 3\,)\circ(\,1\ 4\,)\circ\cdots\circ(\,1\ n\,)$$
can be written as a single cycle.  Is
$$(\,1\ n\,)\circ\cdots\circ(\,1\ 4\,)\circ(\,1\ 3\,)\circ(\,1\ 2\,)$$
the same cycle?  For any numbers $a_1,a_2,\ldots,a_m$, describe the shuffle
$$(\,a_1\ a_2\ \cdots\ a_m\,)\circ(a_m\ \cdots\ a_2\ a_1\,)\ .$$