A Generalization of a Combinatorial Theorem of Macaulay
Let E denote the set of all vectors of dimension n (n ≽ 2) with non-negative integral components. E is ordered in the lexicographic order. Let E. denote the subset of all vectors in E with component sum v. If H v denotes any subset of E v let LHv denote the set of the I Hv I last elements in Ev , where I H v I is the number of elements of H v . Let PH v denote the set of all vectors of E v+1 , which are obtained by the addition of 1 to a component of a vector in H v. In [3) Macaulay proved the inclusion P(LH v) C L(PH v). Sperner gave a shorter proof in [4). Let k l ≼ k 2 ≼ … ≼ k n be given positive integers and let F denote the set of all vectors (a l , … , a n) with integer components and 0 ≼ a i ≼ k i i ═ 1, … , n. We shall prove Macaulay–s inclusion for subsets Hv of Fv even if the operators P and L are restricted to operate in F. This will follow from our theorem. As another application we prove a generalization of the main result ill [2). By a different method Katona proved the theorem when k l ═ k 2 ═ … ═ k n ═ 1 (see [1, Theorem 1))
KeywordsPositive Integer Number System Multivalued Function Combinatorial Problem Short Proof
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