Honestly had no idea how to title the question, so apologies for any confusion that may arise when reading the title. Feel free to edit accordingly.
Let $M_i$ denote the $i^{th}$ man and $W_i$ denote the $i^{th}$ woman in an infinite set of people $\Omega$ and denote this as the identity permutation
$$i:= \left(\begin{array}{cccc} ...,W_1, M_1, W_2, M_2, W_3, M_3, W_4, M_4, W_5, M_5, W_6,... \\ ...,W_1, M_1, W_2, M_2, W_3, M_3, W_4, M_4, W_5, M_5, W_6,... \end{array}\right).$$
Now let $s$ be a reflection in the vertical axis down the middle of the middle man, and let $r$ be the translation to the right that takes one man to the next man.
(a) Write out the permutations for $r$ and $s$.
(b) Suppose we had some random person $A_{\sigma}$ where $A$ is of any gender. Where would he/she "map" to after $r$ is performed. What about $s$?
So part (a) is pretty easy:
$$s\circ i:=s= \left(\begin{array}{cccc} ...,W_1, M_1, W_2, M_2, W_3, M_3, W_4, M_4, W_5, M_5, W_6,... \\ ...,W_6, M_5, W_5, M_4, W_4, M_3, W_3, M_2, W_2, M_1, W_1,... \end{array}\right),$$
$$r\circ i:=r= \left(\begin{array}{cccc} ...,W_1, M_1, W_2, M_2, W_3, M_3, W_4, M_4, W_5, M_5, W_6,... \\ ...,W_0, M_0, W_1, M_1, W_2, M_2, W_3, M_3, W_4, M_4, W_5,... \end{array}\right).$$
In part (b), the general mapping of A for $r$ isn't too hard
$$A_{\sigma} -> A_{\sigma-1},$$
but I come a bit unstuck when trying to do the same for $s$, any hints/tips will be much appreciated here. Thanks in advance!