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Taken from HAKMEM 18.


A partition of $N$ is a finite string of non-increasing integers that add up to $N$. Thus 7 3 3 2 1 1 1 is a partition of 18. Sometimes an infinite string of zeros is extended to the right, filling a half-line. The number of partitions of $N$, $P(N)$, is a fairly well understood function.
The generating function is:

$$\sum_{n=0}^{\infty} \; P(n) \; x^n = \dfrac1{\prod\limits_{k=1}^{\infty} (1-x^k)}$$

A planar partition is like a partition, but the entries are in a two-dimensional array (the first quadrant) instead of a string. Entries must be non-increasing in both the x and y directions. A planar partition of 34 would be:


Zeros fill out the unused portion of the quadrant. The number of planar partitions of $n$, $PL(n)$, is not a very well understood function.
The generating function is:

$$\sum_{n=0}^{\infty} \; PL(n) \; x^n = \dfrac1{\prod\limits_{k=1}^{\infty} (1-x^k)^k}$$

Similarly, one can define cubic partitions with entries in the first octant, but no one has been able to discover the generating function. Some counts for cubic partitions and a discussion appear in Knuth, Math. Comp. 1970 or so.

Did anyone find a generating function for cubic partitions since then?

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up vote 3 down vote accepted

I'm pretty sure not -- the topic was of some interest to string theorists about 5 years ago, and I think they found some ties to something called a "(2D) dimer model", but I don't think they actually found a generating function.

Did a quick search on the ArXiv, but couldn't find the right papers. I'll try again tomorrow.

Try here:

Also search the hep-th archive for "melting crystal".

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+1 very nice paper – draks ... Mar 30 '12 at 18:43

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