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?
$\begingroup$
$\endgroup$
2
asked Jul 24, 2012 at 18:10
-
$\begingroup$ 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. $\endgroup$– user9072Commented Jul 24, 2012 at 19:23
-
$\begingroup$ That has been done. $\endgroup$– Daniel ParryCommented Jul 24, 2012 at 19:47
Add a comment
|
1 Answer
$\begingroup$
$\endgroup$
2
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.
answered Jul 25, 2012 at 1:19
-
$\begingroup$ I have not read this section of MacMahon's book carefully, but many of his proofs are quite vague and would not be considered proofs by today's standards. $\endgroup$ Commented Sep 13, 2020 at 21:07
-
$\begingroup$ Stanley calls it a proof, albeit "lengthy and indirect." If MacMahon's proof is unsatisfactory, then the next published proof (about 15 years later) seems to be T. W. Chaundy, "Partition-generating functions," Quarterly J. Math. (Oxford) 2 (1931), 234-240. But Chaundy does not say anything to suggest that MacMahon's proof was vague or incomplete. $\endgroup$ Commented Sep 13, 2020 at 21:47