It depends. in order to file as Head of Household, you would have to have an eligible dependent. If you do not have an eligible dependent, then no, you cannot file as head of household. If only one of you have an eligible dependent, then that person would be able to claim head of household while the other would have to file as married filing separately.
Additionally, you said you did not live together for the majority of last year. The criteria is that you can not live together for any part of the last 6 months of the year.
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