If you want to allow the ex to claim the child, you send him for 8332. If you don't, don't sent it to him (it would be polite to tell you will be claiming the child, so he cant). The splitting is so common that TurboTax can handle it. It's a matter of putting the right information on the tax forms.
You are the only one that fills out form 8332, you send it to him not the IRS. You do not both, technically, "claim him as a dependent".
He claims the child as a dependent and you have a "Qualifying child" for Earned income credit, head of household filing status and the day care credit.
If you do let him claim the children, you might want to point out, to him, the above info (that his claim is limited).
How do you not know this? Blame your congressman or woman. This is just one small part of very complex tax laws.
