A review: The arrangement increasing partial ordering

Let λ=(λ1,…,λn), λ1 … λn, and x =(x1, x2,…, xn). A function f(λ,x) is said to be arrangement increasing (AI) if (i) ƒ is permutation invariant in both arguments λ and x, and (ii) ƒ(λ, x) ƒ(λ, x′) whenever x and x′ differ in two coordinates only, say i and j, (xi-xj)(i–j) 0, and xi′=xj, xj′=xi. This paper reviews concepts and many of the basic properties of AI functions, their preservation properties under mixtures, compositions and integral transformations. The AI class of functions includes as special cases other well-known classes of functions such as Schur functions, totally positive functions of order two and positive set functions. We present a number of applications of AI functions to problems in probability, statistics, reliability theory and mathematics. A multivariate extension of the arrangement ordering is also reviewed.