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I know Major Macmahon conjectured the formula $$ \prod_{m=1}^\infty \frac{1}{(1-q^m)^m}=1 + \sum_{n=1}^\infty PL(n)q^n$$ but who was the first to prove it?

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I think you should split-off as a separate question (or omit) the 'bonus points' questions. IMO this mix of precise and broad, does in general not work well in one question. – user9072 Jul 24 '12 at 19:23
That has been done. – Daniel Parry Jul 24 '12 at 19:47
up vote 6 down vote accepted

The answer is MacMahon himself, who proved this in his book Combinatory Analysis as a corollary of a more general theorem about plane partitions. See Sections IX and X.

There is some additional historical information in the Notes to Chapter 7 of Richard Stanley's book Enumerative Combinatorics, volume 2.

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