Positive Sum Systems

Abstract

Let x ₁, x ₂, …, x _n  be real numbers summing to zero, and let $\mathcal{P}^{+}$ be the family of all subsets $J \subseteq [n]:=\{ 1,2,\ldots,n\}$ such that $\sum _{j\in J}x_{j} > 0$. Subset families arising in this way are the objects of study here.

We prove that the order complex of $\mathcal{P}^{+}$, viewed as a poset under set containment, triangulates a shellable ball whose f-vector does not depend on the choice of x, and whose h-polynomial is the classical Eulerian polynomial. Then we study various components of the flag f-vector of $\mathcal{P}^{+}$ and derive some inequalities satisfied by them.

It has been conjectured by Manickam, Miklós and Singhi in 1986 that $\binom{n - 1}{k - 1}$ is a lower bound for the number of k-element subsets in $\mathcal{P}^{+}$, unless n∕k is too small. We discuss some related results that arise from applying the order complex and flag f-vector point of view.

Some remarks at the end include brief discussions of related extensions and questions. For instance, we mention positive sum set systems arising in matroids whose elements are weighted by real numbers.