Shifted set families, degree sequences, and plethysm

Abstract

We study, in three parts, degree sequences of k-families (or k-uniform hypergraphs) and shifted k-families. The first part collects for the first time in one place, various implications such as: Threshold implies Uniquely Realizable implies Degree-Maximal implies Shifted, which are equivalent concepts for 2-families (=simple graphs), but strict implications for k-families with k > 2. 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 e_m[e_k]. It then uses highest weight theory to explain how shifted k-families provide the ``top part'' of these plethysm expansions, along with offering a conjecture about a further relation.