Shifted set families, degree sequences, and plethysm

TL;DR

This paper explores the relationships among various properties of degree sequences in hypergraphs and shifted families, introduces new characterizations, and connects these concepts to plethysm of symmetric functions, providing new insights and conjectures.

Contribution

It establishes new implications among properties of degree sequences for hypergraphs, generalizes known characterizations to shifted k-families, and links these to plethysm and highest weight theory.

Findings

Threshold implies Uniquely Realizable implies Degree-Maximal implies Shifted for 2-families.

Introduces new characterizations of shifted k-families.

Connects degree sequences to plethysm of elementary symmetric functions.

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.