Unshuffling a deck of cards (Q6631536)

Perfect shuffle is a well-known card shuffling method. Given a deck of \(2n\) cards, one separates the upper half and lower half into two piles and then interleaves the two piles one card at a time. There are two different ways to do the interleaving: the \textit{in shuffle} (denoted by \(I\)) is the one where the new bottom card comes from the upper pile and the \textit{out shuffle} (denoted by \(O\)) is the one where the new bottom card comes from the lower pile.\N\NPerfect shuffle can be used in magic tricks. However, it is difficult to master and it is natural to wonder whether there are easier shuffling techniques that can do the same job.\N\NThe unshuffle was introduced by [\textit{D. Ensley}, Math Horiz. 11, No. 3, 13--16 (2004; \url{doi:10.1080/10724117.2004.12021753})] with this motivation. Given a deck of \(2n\) cards, one deals the cards alternately into two piles with the top card going to the left pile and then stacks one of the piles onto the other. There are two different ways to do the stacking: the \textit{left shuffle} (denoted by \(L\)) is the one where the left pile gets placed at the top and the \textit{right shuffle} (denoted by \(R\)) is the one where the right pile gets placed at the top.\N\NOne can view the aforementioned shuffling methods \(I,O,L,R\) as elements of the symmetric group on \(2n\) letters. The structure of the subgroup \(\langle I,O\rangle\) generated by the perfect shuffles is already known by \textit{P. Diaconis} et al. [Adv. Appl. Math. 4, 175--196 (1983; Zbl 0521.05005)]. In Theorem 1.1 of the paper under review, the authors determined the structure of the subgroup \(\langle L,R\rangle\) generated by the unshuffles. In particular, they proved that \(\langle L,R\rangle = \langle I,O\rangle\) when \(n\not\equiv 3\) (mod \(4\)) and \([\langle L,R\rangle:\langle I,O\rangle] =2\) when \(n\equiv 3\) (mod \(4\)). Therefore, from a group-theoretic point of view, unshuffle can realize all the card arrangements arising from perfect shuffle, even though it is an easier card shuffling method.