The person who actually pays more than half of the household expenses would be the head of household and the other person would be single with a dependent, or married if that is the case.
With the head of household status, it ends up coming down to the rule that states "Heads of Household must pay more than half of the household expenses." By definition, two people can't pay more than half of the expenses for the same household.
In your case, it seems likely the homeowner would be the head of household and the person paying a reduced rent would not.
Total household expenses include mortgage or rent, property taxes, utilities, insurance, repairs, etc.
As far as trying to establish a different arrangement, if you are all related, I am not sure you could defend any arrangement to the IRS.
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