User blog comment:Wythagoras/Friendship function/@comment-5529393-20140718081359

Interesting idea, Wythagoras. I'm assuming that you also want n <= n'.

It's not clear to me how you are ordering your brackets. You say that the elements in each bracket should be decreasing, but then you write them in increasing order in your examples.

The bracket expressions that you describe are not well-quasi-ordered. An example of an infinite antichain is:

(2,2)

(1,3) (2,2)

(1,3) (1,3) (2,2)

(1,3) (1,3) (1,3) (2,2)