Optimality of the rearrangement inequality with applications to Lorentz-type sequence spaces

We characterize the sequences $(w_i)_{i=1}^infty$ of non-negative numbers for which [ sum_{i=1}^infty a_i w_i quad text{ is of the same order as } quad sup_n sum_{i=1}^n a_i w_{1+n-i} ] when $(a_i)_{i=1}^infty$ runs over all non-increasing sequences of non-negative numbers. As a by-product of our work we settle a problem raised in [F. Albiac, Jose L. Ansorena and B. Wallis; arXiv:1703.07772[math.FA]] and prove that Garling sequences spaces have no symmetric basis.