On the Number of Nonnegative Sums for Semi-partitions

Abstract

Let $[n]=\{1,2,\dots , n\}$. Let $\left( {\begin{array}{c}[n]\\ k\end{array}}\right)$ be the family of all subsets of [n] of size k. Define a real-valued weight function w on the set $\left( {\begin{array}{c}[n]\\ k\end{array}}\right)$ such that $\sum _{X\in \left( {\begin{array}{c}[n]\\ k\end{array}}\right) } w(X)\ge 0$. Let ${\mathcal {U}}_{n,t,k}$ be the set of all ${\mathbf {P}}=\{P_1,P_2,\dots ,P_t\}$ such that $P_i\in \left( {\begin{array}{c}[n]\\ k\end{array}}\right)$ for all i and $P_i\cap P_j=\varnothing$ for $i\ne j$. For each ${\mathbf {P}}\in \mathcal {U}_{n,t,k}$, let $w({\mathbf {P}})=\sum _{P\in {\mathbf {P}}} w(P)$. Let $\mathcal {U}_{n,t,k}^+(w)$ be set of all ${\mathbf {P}}\in \mathcal {U}_{n,t,k}$ with $w({\mathbf {P}})\ge 0$. In this paper, we show that $\vert \mathcal {U}_{n,t,k}^+(w)\vert \ge \frac{\prod _{1\le i\le (t-1)k} (n-tk+i)}{(k!)^{t-1}((t-1)!)}$ for sufficiently large n.