  We study the distribution of entries of a random permutation matrix under a
"randomized basis," i.e., we conjugate the random permutation matrix by an
independent random orthogonal matrix drawn from Haar measure. It is shown that
under certain conditions, the linear combination of entries of a random
permutation matrix under a "randomized basis" converges to a sum of independent
variables $sY + Z$ where $Y$ is Poisson distributed, $Z$ is normally
distributed, and $s$ is a constant.
