Generalized reverse order sum in rearrangement inequality

We introduce a generalization of the Rearrangement Inequality. For integers m, n, m≥ 3, let ai,1,ai,2, ⋯ , ai,n be a permutation of $1$, 2 , ⋯ , n \ (∀ i ∈ [m]). We consider the minimum of S=∑ ⱼ₌₁ⁿ∏ ₖ₌₁ᵐak,j, denoted as Sₙ. Since the exact value of Sₙ is usually unknown, we focus on its asymptotic behaviour as n tends to infinity, namely computing Tₙ=Sₙnᵐ⁺¹. We provide a lower bound for Tₙ, using partial inequalities, strengthened proposition and perturbation method. And we conjecture that the lower bound is actually lim n → ∞ Tₙ. Then we provide an upper bound for Tₙ by giving examples and that when $m=3$, for sufficiently large n there holds 0.0548<Tₙ<0.05488. Finally we calculate some Sₙ using Python and list the example when $m=3,n=201$.