There are two types of partners:

(1) Star Alliance partners, which allows the *A airline to get reward seats from another.

(2) Bilateral partners, where one airline gets into an agreement with another airline.

To complicate things a little more, some Star Alliance members form agreements with others to release additional seats (in the case of NH, for example, they have agreement with OZ) and some restrict other *A member seats (UA blocking off SQ on SIN-HKG)

Then of course there are search engine and staffing issues which are very subjective.