Is one absolved and is the expiation for an oath waived by washing a small portion of the dishes, after having sworn to wash them all, and the mother refused that?

If you intended to wash all the dishes, washing only some of them is not sufficient. If the time has passed, you have broken your oath, and an expiation for breaking an oath (kaffarat yamin) becomes incumbent upon you. The expiation for an oath is not waived by your mother's oath or the obligation to be dutiful to her. Whoever breaks their oath must offer expiation. The expiation is: feeding ten poor persons, or clothing them, or emancipating a slave; or, for one who cannot afford these, fasting for three days.