With whom do the children live?

IF you have actually released the claim to the father, you will be unable to claim the children. The IRS looks first at who has custody of the child or where does the child spend most nights. If you have custody of the children, you may need to sign a form 8332 releasing the claim.

Please review this article on Form 8332. The article includes your rights as the custodial parent and what you lose by not claiming your child along with who needs the form signed.

We need more information.  Who do they live with?  Or do all of you live together?  If you do not live together then who had the children for the most nights in 2019?

If all of you live together then only one of you can claim the children and the other parent does not enter anything on the return about the kids.

No, if you all live together. If you and the other parent live together, either one of you (but not both) may claim the child. You may decide between you which one will claim the child. Only if you can’t agree, do the IRS tie breaker rules apply, to see who has first choice. It may be worthwhile to prepare trial returns, both ways,  to see which way the family comes out best. This tool may be useful: https://turbotax.intuit.com/tax-tools/calculators/taxcaster/?s=1.

____________________________________________________________________

Yes, if the children live with you and not their father.  There is a special rule in the case of divorced & separated (including never married) parents. When the non-custodial parent is claiming the child as a dependent/exemption/child tax credit; the custodial parent is still allowed to claim the same child for Earned Income Credit, Head of Household filing status, and day care credit. This "splitting of the child" is not available to parents who lived together at any time during the last 6 months of the year; then only one of you can claim the child for any tax reasons. The tax benefits may not be split in any other manner.

Note in particular that the non-custodial parent can never claim the Earned Income Credit, Head of Household filing status or the day care credit, based on that child, even when the custodial parent has released the dependency to him.

 So, it's good idea to let the other parent know that you will be claiming those items, as many first time divorced parents are not aware of this rule and may try to claim those items, which will cause the IRS to send out letters.

Ref: https://www.irs.gov/publications/p17#en_US_2017_publink1000170897

Scroll down to "Children of divorced or separated parents (or parents who live apart)"

You can if you are the custodial parent.  The custodial  parent is (the parent the child lived with for more than 183 days in 2016.