Elementary proof of MacMahon's generating function for plane partitions

Question

Recall Macmahon's elegant and beautiful generating function for plane partitions $$ \sum_{n=0}^{\infty} pp(n) q^n = \frac{1}{(1 - q)^1(1 - q^2)^2(1-q^3)^3\cdots}= \prod_{j=1}^{\infty}\frac{1}{(1-q^j)^j} $$ Does an elementary proof of this formula exist?

The reason I ask is that I believe I have discovered such a proof... I have searched the literature and found nothing but I wanted to confirm that my proof is novel.

Answer 1

There are many proofs, most of them are elementary. For example:

• non-intersecting lattice paths and LGV-lemma — see e.g. Bressoud. Proofs and Confirmations (ch. 3)
• RSK-correspondence — see e.g. Stanley. Enumerative combinatorics (vol. 2, 7.20)
• counting lozenge tilings using condensation