I was just scanning through my Marlin book by William Brophy and saw your question. I too was curious to know the answer to your question, so I looked the Model 81 up. Here's what I found out.
The Model 81 was introduced in 1937 and is similar to the Model 80 except it has the tubular magazine. There are several different versions of the Mod. 81 and they are; 81, 81E, 81B and 81BE.
Back in those days serial numbers weren't required on these rifles, so it's not uncommon to find early 22 caliber rifles without serial numbers.
There's no mention as to how many were produced, so I can't help you there.
I hope that helps a little