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You must be separated since July 1 (or earlier) and pay more than half the cost of maintaining your home.  If the children lived with you more than half the nights of the year, AND more nights than they lived with the other parent, the other parent’s permission is not needed for you to claim them.  If they lived with you more than half the nights of the year but with the other parent more nights than you, you can still claim them and written permission is not required, but verbal permission is a good idea.