Is it permissible for a divorced woman to sit alone in her home with her two children (8 and 6 years old)?

There is no impediment to a divorced woman living in her home with her two young children. The default ruling is that it is permissible, and no evidence has been reported to prohibit it. However, if her residence is not safe, she should seek one of her mahrams (unmarriageable kin) or sisters to live with her.