Need help identifying a type of fencing?

Hi! I currently have a split rail fence on my property that has seen better days. I want to replace it with another type of fence that seems a little better quality. The posts are similar, three big slots. But instead of split rails, you would use a board (plank), maybe 1.5×6? And, the boards would overlap inside the post. I have seen this, but I don’t know what to call it. I want to try and track down some photos on the web so I can sell this idea to my wife. Any thoughts?