Shifted set families, degree sequences, and plethysm

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dc.creatorKlivans, Caroline
dc.creatorReiner, Victor
dc.date2006-10-26
dc.date2008-01-10
dc.date.accessioned2026-07-07T08:53:42Z
dc.date.available2026-07-07T08:53:42Z
dc.descriptionWe 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.
dc.descriptionFinal version, 26 pages, 3 figures
dc.identifierhttps://arxiv.org/abs/math/0610787
dc.identifierhttp://arxiv.org/abs/math/0610787
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/145713
dc.subjectCombinatorics
dc.subject05C07, 05C65, 05E05
dc.titleShifted set families, degree sequences, and plethysm
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