We present an easily defined countable family of permutations of the natural numbers for which explicit rearrangements (i.e., the sums induced by the permutations) can be computed. The digamma function proves to be the key tool for the computations found here for the alternating harmonic series. The permutations $φ$ under consideration are simple in a sense: they are involutions ($φ\circφ$ is the identity function). We show that the countable set of rearrangements obtained from the simple involutions considered below are dense in the reals.