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.


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
  • $\begingroup$ Thanks for the pointers. My proof is actually more elementary than those. I have Bressoud right in front of me as a matter of fact! $\endgroup$ – calculemur Jul 6 '16 at 7:42
  • $\begingroup$ IMO, it's certainly possible to find a new proof of Macmahon's formula. Yet it's even easier to rediscover some old proof (my list is by no means exhaustive). Perhaps you should consult some expert(s)… $\endgroup$ – Grigory M Jul 6 '16 at 8:46
  • $\begingroup$ I agree that it is very easy to rediscover old proofs, and I am acutely aware of how people can fool themselves into believing they've discovered something new when they are just being cranks. However, my proof is considerably more elementary than and different to any existing proof that I have read about. I have gone over it again and again and I can't see an error. So I am going to write my proof up properly and send it to some experts to see whether they think it is novel. Thanks for your help. $\endgroup$ – calculemur Jul 6 '16 at 9:03

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