More on the generalized macaulay theorem - II

Let k,ec,G-Sk, be givc.:n positive integers an+ let S denote the set of vectors x =qX‘,X2*..*, x,) with integer components satisfying 0 6 ,I, g k,, i = II 2,. . . , n. Let X be a sub.set of S. (I)X denotes the subset of X consisting of vectors with component sum I; F(m, X) denotes the lexicographicalfy first m vectors of X; fX denotes the set of vectors in S obtainable by subtracting 1 from a component of a vector in X; 1 Xi is the number of vectors in X. In this paper it is shown that 1 ZT(e, (1)s rj is an increasing function of 1 for fixed e and is a wbadditive function of d for fixed 1.