For $1 bills there are no real mules after the switch in notes in 1963. The whole thing about number size was because of the backs from the silver certs being used on the newer fed notes. Mules listed for 1981 is just found by knowing the plate numbers (no difference in size). They started over with plate 1. So large numbers had to come from 1977a. If your 1981 plate number is a four digit it is from the 1977a series. (ie 3269) Then again on the 1981a series plate numbers above 300 came from the 1981 series. These are collected as mules but there is no physical difference other that plate numbers that did not exist in that series. I am sure there are others that could be mules like this but to my knowledge the bep does not give out the plate numbers used for a series. This was only noticed by the large difference. Later on the bep started using a larger back plate number for the fort worth bills. This resulted in the fw 295 plate number error where the back of some bills had the smaller dc plate number 295. This occurred in 1995.
hope this helps. I know this is an old post, I just read it. This is info i got from other people.