  In this paper, we study permutations $\pi \in S_n$ with exactly $m$
transpositions. In particular, we are interested in the expected value of
$\pi(1)$ when such permutations are chosen uniformly at random. When $n$ is
even, this expected value is approximated closely by $(n+1)/2$, with an error
term that is related to the number isometries of the $(n/2-m)$-dimensional
hypercube that move every face. Furthermore, when $k \mid n$, this construction
generalizes to allow us to compute the expected value of $\pi(1)$ for
permutations with exactly $m$ $k$-cycles. In this case, the expected value has
an error term which is related instead to the number derangements of the
generalized symmetric group $S(k,n/k-m)$.
  When $k$ does not divide $n$, the expected value of $\pi(1)$ is precisely
$(n+1)/2$. Indirectly, this suggests the existence of a reversible algorithm to
insert a letter into a permutation which preserves the number of $k$-cycles,
which we construct.
