Exact enumeration of four-dimensional partitions

This is the homepage for the exact enumeration of four-dimensional partitions. This is sequence A000334 in the OEIS. The goal of this project is to carry out the exact enumeration of the numbers of four-dimensional partitions. We should be able to determine the first 47 numbers and possibly up to 50 with v2.0 of the code.

### Results

Date Jan 2011 July 2011 Aug 2011
N 35 40 45
Code v1.0 v2.0 v2.0

### Current state of the project: Ongoing

• [Jan 2011] Converted the code for solid partitions to compute four-dimensional partitions. This is v1.0 of the code. This generated the first 35 numbers. Nothing more was attempted as all efforts were towards pushing the solid partitions project.
• [June 10, 2011] Creating a parallel version of the code used to obtain the first 35 numbers. It will be run at depth 6 (140 branches) and will generate the first 41 numbers. The next step will be to go to depth 9 (1464 branches) and go up to 46/47. Here are our estimates for these numbers and a one-parameter fit to an asymptotic formula given in arXiv:1005.6231.
(1)
\begin{align} p_4(41)\sim 22674\ 94248\ 55626\ , \qquad p_4(47)\sim 12\ 92594\ 73949\ 60782\ , \qquad p_4(50)\sim 93\ 78932\ 22247\ 17232 \end{align}
• [June 14, 2011] We list below the number of nodes and the number of branches (or equivalence classes) that we need to run in the parallel version of the code once it is ready. The first run will be at depth 6 while the second run will be at depth 9 or 10 depending on the timing of the test runs.
 Depth 6 10 No. of Nodes No. Eq. classes 140 3122 4 5 7 8 9 11 12 76 412 2560 17632 133528 1090696 9518104 88003840 857713528 26 59 307 684 1464 6500 13426
• [June 17, 2011] Timing runs suggest that the second run should be at depth $10$ and for $N=45$. It is looking hard to go beyond this using v2.0 which is not yet ready.
• [July 1, 2011] Srivatsan delivers v2.0 of the code and it has passed preliminary tests. We should have new numbers soon. Branching for depth 6 have been verified and numbers for N=30 have been generated again and agree with results from v1.0 of the code.
• [July 2, 2011] Started the depth 6 run for $N\leq 40$. Expect results in a week to ten days. Carried out preliminary checks for the branching at depth 10 which will generate numbers for $N\leq 45$.
• [July 4, 2011] Started the depth 10 run for $N\leq 45$. We expect results in about forty days. Meanwhile, we have obtained two new numbers using a test run at depth 12. Here are the results
(2)
• [July 13, 2011] The depth 6 run for $N\leq 40$ yields three new numbers.
(3)
• [Aug 8, 2011] The depth 10 run for $N\leq 45$ yields five new numbers.
(4)