A REVIEW - THE ARRANGEMENT INCREASING PARTIAL ORDERING

Let lambda = (lambda(1),...,lambda(n)),lambda(1) less than or equal to ...less than or equal to lambda(n), and x = (x(1),x(2),...,x(n)). A fu nction f(lambda, x) is said to be arrangement increasing (AI) if (i) f is permutation invariant in both arguments lambda and x, and (ii) f(l ambda, x) greater than or equal to f(lambda, x') whenever x and x' dif fer in two coordinates only, say i and j, (x(i)- x(j))(i-j)greater tha n or equal to 0, and x'(i) = x'(j), x'(j) = x(i). This paper reviews c oncepts and many of the basic properties of Al functions, their preser vation properties under mixtures, compositons and integral transformat ions. The AI class of functions includes as special cases other well-k nown classes of functions such as Schur functions, totally positive fu nctions of order two and positive set functions. We present a number o f applications of Al functions to problems in probability, statistics, reliability theory and mathematics. A multivariate extension of the a rrangement ordering is also reviewed.