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It is an undergraduate exercise to show that the generating function for the sequence of unrestricted integer partitions $p(n)$ is the celebrated infinite product

$$\prod_{k=1}^{\infty}\frac{1}{1-x^k}$$

More work has to be done if one wants to get asymptotic estimates of the function $p(n)$ (see for instance the pioneering work of Hardy and Ramanujan).

My question is the following: in the course of my research I have found the following generating function (which has a very similar aspect):

$$\prod_{k=1}^{\infty}\frac{1}{(1-x^k)^k}$$

My question is if there are kwnown "natural" combinatorial families enumerated by this generating function.

If yes, I will be happy to know some references.

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This is called the MacMahon function, and counts plane =(3d, confusingly) partitions.

http://en.wikipedia.org/wiki/Plane_partition

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  • $\begingroup$ Plane partitions are plane arrays of numbers, so there is no confusion in the terminology. Only the equivalent of its Young diagram has become 3-dimensional, but that is not really the same thing as the plane partition itself. $\endgroup$ – Marc van Leeuwen Apr 16 '14 at 9:58
  • $\begingroup$ You're right, obviously -- it's just that, being geometrically minded, I think of the Young diagram as being the partition, more centrally than "a list of numbers". $\endgroup$ – Paul Johnson Apr 16 '14 at 21:04

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