A Generalization of a Combinatorial Theorem of Macaulay

  • G. F. Clements
  • B. Lindström
Part of the Modern Birkhäuser Classics book series (MBC)


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 lk 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 ik 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 lk 2 ═ … ═ k n ═ 1 (see [1, Theorem 1))


Positive Integer Number System Multivalued Function Combinatorial Problem Short Proof 
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  1. 1.
    G. Katona, A Theorem of Finite Sets, Theory of Graphs (Proceedings of the colloqui um held at Tihany, Hungary September 1966), ed. by P. Erdös and G. Katona, Academic Press, New York and London, 1968.Google Scholar
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    B. Lindström and H.-O. Zetterström, A Combinatorial Problem in the &-adic Number System, Proc. Amer. Math. Soc. 18 (1967), 166–170.Google Scholar
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    F. S. Macaulay, Some Properties of Enumeration in the Theory of Modular Systems, Proc. London Math. Soc. 26 (1927), 531–555.Google Scholar
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    E. Sperner, Über einen kombinatorischen Satz von Macaulay und seine Anwendung auf die Theorie der Polynomideale, Abh. Math. Sem. Univ. Hamburg 7 (1930); 149–163.CrossRefGoogle Scholar
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    E. Sperner, Ein Satz über Untermengen einer endlichen Menge, Math. Z. 27 (1928), 544–548.MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Birkhäuser Boston, a part of Springer Science+Business Media, LLC 2009

Authors and Affiliations

  • G. F. Clements
    • 1
  • B. Lindström
    • 2
  1. 1.University of ColoradoBoulderUSA
  2. 2.University of StockholmStockholmSweden

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