Positive Dependence, Upper Sets, and Multidimensional Partitions

Let U be an upper set contained in the finite discrete lattice L = {1, …, D(1)} × ⋯ × {1, …, D(p)}. Representations for U are obtained and shown to correspond to certain multidimensional partitions of integers. It is shown that for p = 3, the number of possible upper sets in L is . Various other representation and enumeration results are obtained for related settings. Considered are a variety of applications to multivariate positive dependence and various notions in statistics and reliability theory.