We propose a new nonparametric test for the supposition of independence
between two continuous random variables. The test is based on the size of the
longest increasing subsequence of a random permutation. We identified the
independence assumption between the two continuous variables with the space of
permutation equipped with the uniform distribution and we show the exact
distribution of the statistic. We calculate the distribution for several sample
sizes. Through a simulation study we estimate the power of our test for diverse
alternative hypothesis under the null hypothesis of independence.