Only the parent who had actual physical custody more than half the nights of the year can use the children to qualify for EIC, HOH, and the dependent care credit. (183 or more nights, they do not have to be consecutive.) The IRS goes by actual presence in the home, and does not follow state court orders or agreements. There is no such thing as 50/50 custody in a year with an odd number of days.
If that is your ex, then you can only claim the child tax credit, and only if you submit form 8332 with your tax return. You can’t use any of the children to qualify for HOH.
In a future year, if you arrange actual physical presence of the children so that at least one child lived in your home more than half the nights, and at least one child lived more than half the year with your ex, then you could each file HOH using the corresponding child. The child who lived in your home would also automatically by your dependent and you would not need form 8332 to claim that child.
