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.

\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 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
---|---|---|---|---|---|---|---|---|---|

No. of Nodes | 76 | 412 | 2560 | 17632 | 133528 | 1090696 | 9518104 | 88003840 | 857713528 |

No. Eq. classes | 26 | 59 | 140 | 307 | 684 | 1464 | 3122 | 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

\begin{align} \boxed{ p_4(36)=737\ 51783\ 44373\quad,\quad p_4(37)=1483\ 36792\ 65311 }\ . \end{align}

**[July 13, 2011]**The depth 6 run for $N\leq 40$ yields three new numbers.

\begin{align} \boxed{ p_4(38)= 2972\ 70283\ 83070\quad,\quad p_4(39)=5936\ 34540\ 21280\quad,\quad p_4(40)=11813\ 75569\ 12895 }\ . \end{align}

**[Aug 8, 2011]**The depth 10 run for $N\leq 45$ yields five new numbers.

\begin{align} \boxed{ p_4(41) = 23431\ 04837\ 18511 \quad,\quad p_4(42) = 46319\ 32862\ 45475 \quad,\quad p_4(43) = 91270\ 38659\ 94165 }\ , \end{align}

(5)
\begin{align} \boxed{ p_4(44) = 1\ 79276\ 62211\ 91810 \quad,\quad p_4(45) = 3\ 51052\ 07780\ 98410 } \ . \end{align}

**[Mar 31, 2012]**The next generation of code is being worked on. It will be an implementation of the Knuth algorithm with less memory usage and might run faster. Further, we are simultaneously working on a parallelization that works uniformly for partitions in all dimensions with tweaks for further speedup that is dimension dependent. These should be ready by the end of April 2012.

### Code

**v1.0**of the code is a straight port of the code that generated numbers for solid partitions.**v2.0**is a parallel version of the code should be ready soon

Page maintained by Suresh Govindarajan