# Why were plane partitions invented?

I realize that these objects were originally created by Major Percy Macmahon and today have many applications but what was the original motivation for studying them?

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I guess the observation that they are the obvious step after usual partitions is not a satisfactory answer? – Mariano Suárez-Alvarez Feb 25 '11 at 3:43
Was it really that simple? – Daniel Parry Feb 25 '11 at 3:44
Is there any reason to suspect it wasn't? (It seems like the way to answer this question is to go read whatever MacMahon wrote about them, doesn't it?) – JBL Feb 25 '11 at 4:52
We'd never refer to Felix Klein as "Mr Klein", or even as "Dr Klein", so why does Percy MacMahon always get his title? "Sister Celine" is another example, as is "Lord Rayleigh". – Kevin O'Bryant Feb 25 '11 at 19:22
Well, there's a statistic of permutations called the "major index". This is so-called not because it is paired with some "minor index" but because MacMahon came up with it. If you don't call him "Major MacMahon" this stops making sense. At least this is how I've always heard the story -- but in fact MacMahon's book (Google Books: bit.ly/hh7Y1e) uses the terms "major index" and "minor index". No explanation for Sister Celine, though. – Michael Lugo Feb 25 '11 at 23:23

MacMahon invented a technique which he called partition analysis to determine (multivariable) generating functions for many combinatorial objects and as a computational method for solving combinatorial problems in connection with systems of linear diophantine inequalities and equations. This was introduced in his book "Combinatory analysis". At the center was his $\Omega_{\geq}$ operator, for which he proved many properties. He then claimed that plane partitions were a simple toy case to apply these lemmas and was able to compute many interesting generating functions in some limited cases but ran into some problems with the general case of unrestricted plane partitions. He was however lead to some conjectures, some of which he proved later. From there it became clear that there was a lot of interesting mathematics related to plane partitions. I believe you will find some interesting material in the series of papers "MacMahon's partition analysis" I-XII by G.E. Andrews, P.Paule, A Riese and V. Strehl.

Edit: I was a bit rushed to conclude that $\Omega$ had something to do with the motivation to look at plane partitions, see Richard Stanley's answer. I still believe that it was part of the machinery that he built for the same kind of problems that inspired looking at plane partitions. (I mean all of the results about counting tuples of integers satisfying sets of equalities/inequalities.)

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I think you've missed the name of his book, too. Should be "Combinatory Analysis". – Kevin O'Bryant Feb 25 '11 at 19:18
Thanks, I made the correction. – Gjergji Zaimi Feb 25 '11 at 19:34

It does not seem from MacMahon's first mention of plane partitions that the $\Omega_\leq$ operator was relevant. At the end of Article 42 of his paper "Memoirs on the theory of the partitions of numbers---Part I", MacMahon says "This partition may be termed 'graphically regularised' by reason of its origination in a subjacent succession of lines in the bipartite graph. This species of regularisation is the natural extension to three dimensions of Sylvester's graphical method in two dimensions." He then goes on to develop some simple properties of plane partitions (without using that terminology) and to conjecture his famous generating function $\prod_{n\geq 1}(1-x^n)^{-n}$. He also suggests less confidently that three-dimension partitions have the generating function $\prod_{n\geq 1}(1-x^n)^{-{n+1\choose 2}}$ (now known to be false). The $\Omega_\leq$ operator is used implicitly to prove some simple results, but it does not seem to be relevant to MacMahon's original motivation. I believe that MacMahon did not explicitly use his $\Omega_\leq$ operator until "Memoirs on the theory of the partitions of numbers---Part II", about three years after Part I. In Part II he does consider plane partitions as an example.

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Gjergji's answer, while it's a correct answer and a good answer, and the question itself only address the very first mention of plane partitions in the literature, and not the almost-first mention that was clearly in the air at the time. Yes, MacMahon had in mind generating function theory. However, plane partitions are also a special case of semistandard tableaux, which are used to enumerate basis of irreducible representations of the general linear group. They are a variation of standard tableau, which enumerate bases of irreps of the symmetric group and were first described by Alfred Young a little bit before MacMahon's work.

I would say that semistandard Young tableaux are more like a fellow traveler of plane partitions than an "application". MacMahon must have known something about it. (But conceivably some of these connections were only cleaned up decades later.)

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Nitpick: "...plane partitions are a special case of semistandard tableaux..."? I would have thought it's the other way round: semistandard tableaux (or "column-strict tableaux") are plane partitions with the property that column height strictly decrease from top to bottom. – Sheikraisinrollbank Feb 25 '11 at 20:19
Yes, there is an injection from all semistandard tableaux to all plane partitions, since plane partitions are weakly decreasing in both directions. But I had in mind "special case" in a more bijective sense. If you take the plane partitions in an $a \times b \times c$ box, they are bijective with semistandard tableaux with a rectangular shape. However, semistandard tableaux with some other partition shape, and a finite alphabet, are not bijective with any very convenient class of plane partitions. – Greg Kuperberg Feb 26 '11 at 6:47
I researched a lot of this history for the Notes section of Chapter 7 of Enumerative Combinatorics, vol. 2. I write that the first statement that Kostka numbers count semistandard tableaux (and so far as I know the first appearance of such tableaux in the literature) was given by D. E. Littlewood in 1937, well after the work of MacMahon. I would be interested in any corrections to these Chapter 4 Notes. – Richard Stanley Feb 26 '11 at 19:40
@Richard I'm left thinking that my answer just isn't that great. Your books seem incredibly careful in general with citations and history. I said parenthetically that "some of these connections were only cleaned up decades later", and it seems possible now that this last remark is more true than everything else that I said. – Greg Kuperberg Feb 26 '11 at 20:00