On Hilbert series of Koszul operads and a classification result for set-operads

Motivated by numerous examples in the literature, we state a conjecture on the Hilbert series of Koszul symmetric operads generated by one element of arity $2$. We prove this conjecture for all Koszul symmetric set-operads generated by one element of arity $2$ by explicitly classifying those. There are $11$ such operads; $4$ of them are new.