Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: Shifted set families, degree sequences, and plethysm. | Shifted set families degree sequences and plethysm C. Klivans Depts. of Mathematics and Computer Science Univ. of Chicago cjk@ V. Reined School of Mathematics Univ. of Minnesota reiner@ Submitted Jan 1 2007 Accepted Jan 7 2008 Published Jan 14 2008 Mathematics Subject ClassiEcation 05C07 05C65 05E05 Abstract We study in three parts degree sequences of k-families or k-uniform hypergraphs and shifted k-families. The Erst part collects for the Erst time in one place various implications such as Threshold Uniquely Realizable Degree-Maximal Shifted which are equivalent concepts for 2-families simple graphs but strict implications for k-families with k 3. The implication that uniquely realizable implies degree-maximal seems to be new. The second part recalls Merris and Roby s reformulation of the characterization due to Ruch and Gutman for graphical degree sequences and shifted 2-families. It then introduces two generalizations which are characterizations of shifted k-families. The third part recalls the connection between degree sequences of k-families of size m and the plethysm of elementary symmetric functions em ek . It then uses highest weight theory to explain how shifted k-families provide the top part of these plethysm expansions along with oEering a conjecture about a further relation. -Partially supported by NSF VIGRE grant DMS-0502215. yPartially supported by NSF grant DMS-0601010. THE ELECTRONIC JOURNAL OF COMBINATORICS 15 2008 R14 1 Contents 1 Introduction 2 2 Definitions and Preliminaries 3 The basic definitions. 3 Cancellation conditions. 6 Vicinal preorder. 7 The zonotope of degree sequences. 8 Swinging and shifting. 8 3 Some relations between the concepts 9 4 How to characterize degree sequences 13 The problem and an unsatisfactory answer. 13 Some data on degree sequences . 15 Reconstructing families. 15 Some promising geometry . 17 5 Shifted families and plethysm of elementary symmetric .
