We study the problem of assigning objects to a group of agents, when each agent has ordinal preferences over the objects. We focus on probabilistic methods, in particular, the serial rule, introduced by Bogomolnaia and Moulin (2001). Liu and Pycia (2011) show that for each economy with support, the serial rule is the only one satisfying stochastic dominance efficiency and stochastic dominance no-envy (for short, we use the prefix sd'' for stochastic dominance in other expressions below). We first generalize their results by introducing a weaker restriction on preference profile, which we call support on partition.''Given this restriction, we turn to the domain of weak preferences. We provide a characterization of the serial correspondence,'' Proposed by Katta and Sethuraman (2006) as an adaptation of the serial rule. Our main result is on the domain of preferences with strict full Support on partition: for each economy with strict full support on partition, the set of assignment matrices selected by the extended serial correspondence coincides with the set of matrices that are sd-efficient and sd envy-free. The key to these results is to exploit a structural property of ES assignment matrix (Heo, 2011b) and use preference-decreasing consumption schedules.