Which divorced parent gets to claim a 20-year-old full-time college student??

Hi there!  

I was the sole custodial parent for my daughter from ages birth to 18.  In my custody and divorce decree, I was granted the right to claim my daughter as a dependent on taxes.  Fast forward to now, where she is a full-time college student.  Her father is paying her tuition and her rent while she is at college, but when she is not at school (ie:  all school breaks including summer , and some weekends, are spent living with me).  In other words, I maintain and share with her the home that she comes home to from college.  A week ago her father texted me and alerted me that he was going to be claiming her on his taxes as his dependent.  I told him that I didn't think that was possible, given my interpretation the dependency tests, but he is quite clear that he has the legal right to claim her and I do not.  I presume this is either because he believes or has been told by somebody, that his financial support of her at college then gives him this right?  Basically, I don't want to file as head of household and with her as my dependent. If I'm not supposed to, given the above conditions. Can somebody please help me be sure?  Am I missing something?   My thanks.  Krisa