This is a little bit of a lark, but I would argue it still marks important progress in combinatorics: $T=1+T^2\implies T^7=T$. Specifically, this is the (loose) initial justification for the "Seven Trees In One" (categorically) natural bijection between the set of binary trees and the set of 7-tuples of binary trees. It's an exhibition of the still-being-explored connections between combinatorial species, types, and category theory.