This seems like an interesting project. --Rich ----- Forwarded message from suresh.govindarajan@gmail.com ----- Date: Mon, 13 Dec 2010 20:35:08 -0600 From: Suresh Govindarajan <suresh.govindarajan@gmail.com> Reply-To: Number Theory List <NMBRTHRY@listserv.nodak.edu>, Suresh Govindarajan <suresh.govindarajan@gmail.com> Subject: Enumerating numbers of solid partitions -- new results To: NMBRTHRY@listserv.nodak.edu 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 results) 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 Professor Department of Physics Indian Institute of Technology Madras Chennai 600036 INDIA http://www.physics.iitm.ac.in/~suresh ----- End forwarded message -----