Integral majorization Polytopes

Let n be a positive integer. We may write (or split) n as sums of n nonincreasing nonnegative integers p1, p2, . . . , pn in different ways (or partitions; see Section 2). For example, 3 = 3 + 0 + 0 = 2 + 1 + 0 = 1 + 1 + 1. If we denote by P (n) the number of different partitions of n, then P (3) = 3. One may check that P (5) = 7. As n gets large, P (n) increases rapidly. It is astounding that P (200) is about 4 trillion [1, p. 68]. The determination of P (n) is an intriguing and difficult problem in number theory and combinatorics (see, e.g., [12] and [9, Chapter 15]). It has much to do with the theories of majorization and polytopes. In the language of majorization, P (5) = 7 means that there are 7 nonincreasing integral vectors in R that are majorized by the vector (5, 0, 0, 0, 0). Equivalently, there are 7 nonincreasing integral vectors in R that are contained in the majorization polytope generated by (5, 0, 0, 0, 0). In this paper we study majorization polytopes for more general integral vectors. For vectors x and a in R, we say that x is majorized by a, denoted by x a,