Extremal Permutations with Respect to Weak Majorizations

LetZ=(z1, z2, . . . , zn)denote a permutation of ann-set. DefineFl(Z)={f(z1, z2), . . . , f(zn-1, zn)}and its cyclic versionFc(Z)={f(z1, z2), . . . , f (zn-1, zn), f(zn, z1)},wheref(x, y)increases in max{x, y}and decreases in min{x, y}.We give conditions onfsuch that extremal permutations with respect to weak majorization can be found. We then use the weak majorization property to obtain extremal permutations forLf(Z)=g(f(z1, z2), . . . , f(zn-1, zn))andCf(Z)=g(f(z1, z2), . . . , f(zn-1, zn), f(zn, z1))whengis either convex increasing or concave decreasing.