On the number of nonnegative sums for semi-partitions

Abstract

Let [ n] = { 1 , 2 , ⋯ , n}. Let ([n]k) be the family of all subsets of [n] of size k. Define a real-valued weight function w on the set ([n]k) such that ∑X∈([n]k)w(X)≥0. Let Un,t,k be the set of all P= { P1, P2, ⋯ , Pt} such that Pi∈([n]k) for all i and Pi∩ Pj= ∅ for i≠ j. For each P∈ Un,t,k, let w(P) = ∑ P∈Pw(P). Let Un,t,k+(w) be set of all P∈ Un,t,k with w(P) ≥ 0. In this paper, we show that |Un,t,k+(w)|≥∏1≤i≤(t-1)k(n-tk+i)(k!)t-1((t-1)!) for sufficiently large n.