Every horse has 32 pairs of chromosomes. In each pair, one chromosome stems from the mother and one from the father. Homozygous means that a trait is the same on both copies of the chromosomes, heterozygous means that the two copies are different.
E is the black/red status. Black is dominant and no black is recessive.
EE (= homozygous black) means that the horse has the dominant genetic trait to express black on both copies. A chestnut has ee and can only pass on a recessive e (= no black = red). Your horse will not have a chestnut parent and will never produce a chestnut or red dun foal.
A stands for agouti and is responsible for limiting the black to certain areas (mane, tail, legs). A is dominant and makes the horse look bay/brown. Any black horse is aa, otherwise it would look brown.
T stands for tobiano, which also has simple dominant/recessive genetics. TT means that your horse got tobiano from both parents and will pass it on in all his offspring.
That his dad is homozygous for tobiano (=TT) is irrelevant, since you already know that he is homozygous too. His mom and his dad both passed on one tobiano to him.
Grullo means that the horse has at least one copy of dun. This turns the black to grullo.
5 panel negative means what DraftyAires said, he is negative for the 5 most important genetic diseases in Quarter horses.
No matter what color the mare is, this horse will never produce a chestnut or red dun.
No matter what color the mare is, this horse will always produce a tobiano pattern.
When crossed with a black horse, this horse will always produce a black or grullo tobiano.
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