User blog comment:Alemagno12/Defining stage n cardinals in terms of stationary sets/@comment-1605058-20180612183310/@comment-1605058-20180612213736

If some set is stationary in a cardinal, then so is any superset of it (since if every club meets a set, it sure will meet a larger set).

The first part of the argument shows that 1-hyper-Mahlos are enumerated by S(T,a), while the second part shows that no other cardinals are.

By the way, all weakly compact cardinals are 1-hyper-Mahlo, so they are included somewhere in the enumeration, but I don't see how it's of any help - already S(0,a) includes all weakly compacts in the enumeration.