It actually is "rocket science" for the poor nine year old boy.
He regards someone as his best friend, and naturally expects to be invited to his birthday party, however small it is.
The best friend tells him an invitation has been sent to his mum, but the mum has not received it.
The party happens without the boy attending, and he is left heartbroken and sobbing, not having a clue why he was left out.
I don't think it is easy for the OP to clarify the situation with the other mum. She would most probably not get an honest answer anyway.
She should perhaps find out from her DS if his best friend was disappointed by his absence.
As many have suggested, the other boy might not be considering him as his best friend.
But I think it is the mum who for whatever reason didn't want to invite him.
Luckily, it was just before the long holidays and the DS will have other things to do and forget about this.
The lesson to be learnt here is that such things should be clarified straight away by parents for the sake of the little children, in this case before the birthday party.