The study of orderings for positive dependence on bivariate empirical distributions can be viewed as the study of partial orderings on the set $S_N$ of all permutations of the integers $1,\ldots,N$. This paper extends earlier bivariate results to multivariate empirical distributions, with focus on the trivariate case. In terms of a newly defined notion of relative rearrangement, characterizations are given of the more positively upper orthant dependent ordering and related orderings. A new partial ordering describing concordance on $(S_N)^m$ is also introduced and connected with the positively upper orthant dependence ordering.
"Orderings for Positive Dependence on Multivariate Empirical Distributions." Ann. Appl. Probab. 3 (4) 1241 - 1251, November, 1993. https://doi.org/10.1214/aoap/1177005281