Statistical dependence and shape of Young tableau

Given two continuous random variables, X and Y, we study the relationship between their statistical dependence and the Young tableau of the permutation defined from the graph of a bivariate sample coming from (X, Y). From a sample of size n of (X, Y), we identify the Young tableau of the permutation which maps the ranks of the X observations on the ranks of the Y observations. Procedures to detect statistical dependence between pairs of random variables, based on statistics calculated on the permutation defined by the graph of a bivariate sample have been developed, see García and González-López (2020) [Symmetry 12, 9, 1415. https://doi.org/10.3390/sym12091415] and García and González-López (2014) [J Multivar Anal 127, 126–146. https://doi.org/10.1016/j.jmva.2014.02.010]. In those papers, the information used is the length of the longest increasing (decreasing) subsequence, identified as the first line (the first column) of the Young tableau of the permutation. In this paper, we expose the information captured by the shape of the Young tableau of the permutation.