And the irony of this whole issue/conversation is that if the birthday girl's mom had said "What? Your DD did not get her invitation? I mailed it out last week!" everyone here would have been saying "Good thing you called or your daughter would have missed the party."
But, then, you might have wondered if it was really a case of a mis-directed or late invite or whether the mom was so put-on-the-spot that she issued the invitation just then (and hoping you'd decline it because she wasn't *really* invited).