Abstract Let M be a finite set consisting of ki elements of type i, i = 1, 2,…, n and let S denote the set of subsets of M or, equivalently, the set of all vectors x = (x1, x2,…,xn) with integral coefficients xi satisfying 0 ⩽ xi ⩽ ki, i = 1, 2,…, n. An antichain is a subset of S in which there is no pair of distinct vectors x and y such that x is contained in y (that is, there is no pair of distinct vectors x and y such that the inequalities xi ⩽ yi, i = 1, 2,…, n all hold). Let ∥Y ∥ denote the number of vectors in S which are contained in at least one vector in and let ∥B ∥=∑ x∈ (X 1 +X 2 +⋯+X n ) , the number of basic elements in . For given m we give procedures for calculating min ∥Y ∥ and min ∥B ∥ , where the minima are taken over all m-element antichains in S.
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