[math-fun] Pairs (f,g) of maps [n] --> [n] such that f g = g f

Hope it is OK to occasionally(!) raise an OEIS related question. This is about A181162 ( https://oeis.org/draft/A181162 ). I have no better idea than the author: exhaustive search. It seems almost certain that this (nice) problem has been investigated somewhere. Any pointers? Best, jj