I'm assuming that you are the custodial parent (the parent that the children lived with more than 1/2 of the year.)
Paying child support is not a criteria for claiming a dependent. So long as you transfer the exemption to him with Form 8332, he can claim either or both children. Even if you do, only you can claim Head of Household tax status, the Earned Income Tax Credit and the Child and Dependent Care Credit. He cannot, as he is not the custodial parent. You can claim those even if you transfer the exemption to him with Form 8332.
Transferring the exemption to him with form 8332 is your choice. Assuming there is nothing in a decree mandating that you transfer the exemption you don't have to.
He can claim the only the children's exemption and child tax credit.
