On the number of nonnegative sums for certain function - Ku, Cheng Yeaw (2022)
Let [n] = {1 , 2 , ⋯ , n}. For each i ∈ [k] and j ∈ [n], let wᵢ(j) be a real number. Suppose that ∑ i∈[k], j∈[n] wᵢ(j) ≥ 0. Let F be the set of all functions with domain [k] and codomain [n]. For each f ∈ F, let w(f) = w₁(f(1)) + w₂(f(2)) + ⋯ + wk (f(k)). A function f ∈ F is said to be nonnegative if w(f) ≥ 0. Let F⁺(w) be set of all nonnegative functions, i.e., F⁺(w) = {f ∈ F: w(f) ≥ 0}. In this paper, we show that |F⁺(w)| ≥ nᵏ⁻¹ for n ≥ 3 (k -1)².