  Pearson's correlation is among the mostly widely reported measures of
association. The strength of the statistical evidence for linear association is
determined by the p-value of a hypothesis test. If the true distribution of a
dataset is bivariate normal, then under specific data transformations a
t-statistic returns the exact p-value, otherwise it is an approximation.
Alternatively, the p-value can be estimated by analyzing the distribution of
the sample correlation under permutations of the data. Moment approximations of
this distribution are not as widely used since estimation of the moments
themselves are numerically intensive with greater uncertainties. In this paper
we derive an inductive formula allowing for the analytic expression of the
sample moments of the sample correlation under permutations of the data in
terms of the central moments of the data. These formulas placed in a proper
statistical framework could open up the possibility of new estimation methods
for computing the p-value.
