A partition is self-conjugate if in terms of Ferrers diagram.
The bijection can be shown graphically. See wiki.
In your example, (4,2,1,1) corresponds to 7+1 in the link.
The generating function for this is .
An unrestricted partition (order does not count, equality of sizes is ok) partition is called self-conjugate if it is identical with its conjugate. E. g 8 = 4 + 2 + 1 + 1.
Show that the number of self conjugate unrestricted partitions of n is equal to the number of partitions of n into distinct odd parts
(optional) express this result as an identity of generating functions.
Yes. If your definition of an "unrestricted" partition is simply a partition without any further constraint being given.
See the claim in the above link:
Claim: The number of self-conjugate partitions is the same as the number of partitions with distinct odd parts.
For instance, (5, 5, 4, 3, 2) |- n, where n=19, is a self-conjugate partition. It converts into the 9+7+3=19. Read the link I provided. It will give you some explanations graphically.
The link I provided only shows the sketch of the proof. I encourage you to make a full proof on your own by using the sketch of the proof in the link.