You can't claim your children as dependents unless they lived in your home more than half the year, unless you get a signed release from the parent who does have custody. The IRS follows actual physical custody -- where the children lived -- not any court orders or financial support. Even if you get a signed release from the other parent, that only allows you to claim the child tax credit, not HOH, because that always stays with the parent who had custody more than half the year.
If it is the case that both parents had physical custody more than half the year (such as, because you separated in August and both parents lived in the same home as the children for 6+ months before that), then the first tiebreaker goes to whichever parent the children lived with the greater number of nights.
See IRS publication 501 for the full rules.
https://www.irs.gov/forms-pubs/about-publication-501
