More on the generalized Macaulay theorem

AbstractLet k1 ⩽ k2 ⩽ … ⩽ kn be given positive integers and let F denote the set of vectors (l1, …, ln) with integer components satisfying 0 ⩽ li ⩽ ki, i = 1, 2, …, n. If H is a subset of F, let (l)H denote the subset of H consisting of those vectors with component sum l, and let C((l)H) denote the smallest [(l)H] elements of (l)F. The generalized Macaulay theorem due to the author and B. Lindström [3] shows that |Gamma;((C)(l)(H)|, ⩾ |Γ(C((l)H))|, where Γ((l)H) is the setof vectors in F obtainable by subtracting l from a single component of a vector in (l)H. A method is given for computing [Γ(C((l)H)] in this paper. It is analogous to the method for computing |Γ(C(l)H))| in the k1 = … = kn = 1 case which has been given independently by Katona [4] and Kruskal [5]

Discrete Macaulay-Steiner Geometry

This thesis is concerned with discrete isoperimetric inequalities and Hilbert functions. Two generalizations of the Ahlswede-Cai local global principle are presented. These results give positive answers to two questions posed by Harper. One of these results is achieved by proving uniqueness of the lexicographic and colexicographic orders in two dimensions. The other result generalizes the technique which is commonly known as compression and includes almost all previously published results in this direction. The Ahlswede-Cai local global principle is a direct corollary of this result. Optimal downsets are studied in rectangles and triangles. All optimal downsets are found. The main result in this direction gives a unified description, optimal downsets are those that are a symmetrization/stabilization of initial segments of the lexicographic and colexicographic orders. Lindsay’s Theorem and the Ahlswede-Katona Edge Isoperimetric Theorem are corollaries. The theory of Macaulay posets is connected to that of Hilbert functions. Several old and new results in both commutative algebra and extremal combinatorics are obtained. Hoefel’s questions on applying Macaulay poset theory to commutative algebra is answered in the affirmative as a by product. A question of Bezrukov and Leck on taking the product of a Macaulay poset with a chain is answered by using a result of Mermin and Peeva. Several answers are given to a problem of Mermin and Peeva