No, you and your partner cannot BOTH file Head of Household (HOH) because per IRS rules, there is only ONE Head of Household per home.
Assuming neither of you is married, one of you can file HOH and one as Single, each claiming their own biological child as a dependent. To determine which one of you claims HOH, you will need to determine which one of you pays over half the cost of the home, even if by just $1. Normally the person with the higher income will file as HOH.
See more details about the HOH Filing Status and Requirements for Claiming a Dependent in the TurboTax FAQ's below:
https://ttlc.intuit.com/replies/3288637 https://ttlc.intuit.com/replies/5187060
