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 ⇒ Uniquely Realizable ⇒ Degree-Maximal ⇒ Shifted which are equivalent concepts for 2-families (= simple graphs), but strict implications for kfamilies 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[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.2000 Mathematics Subject Classification. 05C07,05C65,05E05.

Optimization over Degree Sequences

“…and the vertices of D n 2 were characterized in [13] as precisely the degree sequences of threshold graphs. More recently, the polytopes D n k for k ≥ 3 were studied in [12,17,19], but neither a complete inequality description nor a complete characterization of vertices is known.…”

“…, m} n → R which is not necessarily separable as considered above. For this we discuss the degree sequence polytopes studied in [12,13,17,19,21] and references therein, introduce and study degree sequence polytopes of hypergraphs with prescribed number of edges, and show that for k = 2 their vertices correspond to suitable threshold graphs [18].…”

Nonconvexity of the Set of Hypergraph Degree Sequences

“…Suppose it were, so that we could write p = S∈K e S for some K ⊂ K 0 . Let B = (b rs ) be the 6 × 6 matrix such that b rs counts the number of q for which {v 1 q , v 2 r , v 3 s } ∈ K. Then the sequence of row and column sums of B must both be (2,4,6,8,3,7). Since we also know that 0 ≤ b rs ≤ 5, this means that:…”

“…) We consider the analogous question for k-uniform hypergraphs when k > 2. Klivans and Reiner [3] verified computationally that the set of degree sequences for k-uniform hypergraphs is the intersection of a lattice and a convex polytope for k = 3 and n ≤ 8 and asked whether this holds in general. We will show in Section 2 that it does not hold for k ≥ 3 and n ≥ k + 13.…”

Minimal and maximal constituents of twisted Foulkes characters

“…We remark that Theorem 2.6 in the authors' earlier paper [22] is the special case of Theorem 1.3 when m is odd and ν = (n). The authors recently learned of work by Klivans and Reiner [17,Proposition 5.10] which gives a result equivalent to this special case. The proofs in this paper use some similar ideas to [22], but are considerably shorter, and give more general results.…”