Multi dimensional partitions

[Richard Mathar]

There is no indication in the Almkvist paper that his formula
(1-x^k)^(-k^r) counts partitions in r-dimensions, although there is
an explicit statement that r=1 counts plane partition. This looks like a
ad-hoc, nice and short (unfortunately wrong) description of the
sequences through sheer-will extrapolation from r=1 to the cases of higher r>1.
The only available descriptions are "g.f. (1-x^n)^(-n^r)" so far
for the individual r=2,3,.... 

--Richard

> From seqfan-owner at ext.jussieu.fr  Mon Dec  4 19:09:48 2006
> Return-Path: <seqfan-owner at ext.jussieu.fr>
> To: seqfan at ext.jussieu.fr
> Subject: Multi dimensional partitions
> Date: Mon, 04 Dec 2006 13:08:11 -0500
> ...
> http://www.research.att.com/~njas/sequences/A023871 is titled "Number 
> of partitions in 3 dimensions."  This is the Euler transform of the 
> squares; but I know of no sense in which it counts 3-dimensional 
> partitions.  Does anybody?
> 
> (http://www.research.att.com/~njas/sequences/A000293 is the normal 
> sequence for 3D partitions.)
> 
> The same objection applies to A023872-A023878.
>..