[seqfan] [suresh.govindarajan at gmail.com: Enumerating numbers of solid partitions -- new results]

Joerg Arndt arndt at jjj.de
Wed Dec 15 09:27:51 CET 2010

Forwarded with permission from Suresh Govindarajan.

----- Forwarded message from Suresh Govindarajan <suresh.govindarajan at gmail.com> -----

Date:         Mon, 13 Dec 2010 20:35:08 -0600
From: Suresh Govindarajan <suresh.govindarajan at gmail.com>
Reply-To: Number Theory List <NMBRTHRY at LISTSERV.NODAK.EDU>,
	Suresh Govindarajan <suresh.govindarajan at gmail.com>
Subject: Enumerating numbers of solid partitions -- new results
X-Spam-Score: -0.0 (/)

Dear number theorists,

Recently, with some undergraduate students, I embarked on a project to
enumerate the numbers of solid (three-dimensional) partitions of
integers less than or equal to 100. The home page for the project is
here: http://boltzmann.wikidot.com/solid-partitions   For one and
two-dimensional (plane) partitions, there exist generating functions
due to Euler and MacMahon. However, MacMahon's guess for the
generating function of solid partitions  is known to be wrong and
hence it is difficult to enumerate solid partitions. At the start of
this project, the first 50 numbers were known (due to work by Knuth
and subsequently by Mustonen and Rajesh -- see also
http://oeis.org/A000293 ).

Thanks to some nice innovations, so far the project has added 12 new
numbers to this list thus extending the numbers to integers <=62.
Roughly speaking, adding five numbers takes 10 times longer. The last
run took about 30K CPU hours run on the IIT Madras super-clusters.
Here are the twelve numbers (see
http://boltzmann.wikidot.com/solidpartitions-62 for the complete

51 	11622 14153 25837 	57 	2 41380 42828 01444
52 	19379 44766 58112 	58 	3 97140 96826 33930
53	32238 23655 07746 	59	6 52064 95439 12193
54 	53505 67710 14674 	60 	10 68461 42257 15559
55	88603 33844 75166 	61	17 47294 70062 57293
56	1 46400 93392 99229 	62	28 51869 10933 88854

Mustonen and Rajesh also observed that the asymptotics of the wrong
MacMahon generating function seemed to fit numerical data obtained
through Monte Carlo simulations. In particular, they find that

      [n^{-3/4} log p(n )]  --> 1.79 +- 0.01 ,

which is in agreement with the value one obtains from the asymptotic
behaviour of the wrong MacMahon generating function. We observe that
the asymptotic formula given by MacMahon's generating function seems
to work even for small numbers like n=50/60. Using this formula (and
its generalization), we were able to predict the numbers of solid
partitions from 55-62 accurate to 0.02% -- around 3-5 digits. See
http://boltzmann.wikidot.com/solidpartitions-62 for more details.

I look forward to your comments and suggestions on these results and
related issues.

Suresh Govindarajan
Department of Physics
Indian Institute of Technology Madras
Chennai 600036 INDIA

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