Number of solid partitions for N<=55

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Results

Here are the results obtained on November 3, 2010 by running a parallelized and modified version of Knuth's algorithm for about ten days(4500 CPU hours). Below, c[N] denotes the number of solid partitions of N. We add five numbers, c[51] to c[55] to the numbers computed by Knuth and Mustonen-Rajesh. The number of solid partitions for the first fifty numbers are also listed in the On-Line Encyclopedia of Integer Sequences. Our list is given below:

Wed Nov 3 12:34:16 IST 2010

c[1] = 1
c[2] = 4
c[3] = 10
c[4] = 26
c[5] = 59
c[6] = 140
c[7] = 307
c[8] = 684
c[9] = 1464
c[10] = 3122
c[11] = 6500
c[12] = 13426
c[13] = 27248
c[14] = 54804
c[15] = 1 08802
c[16] = 2 14071
c[17] = 4 16849
c[18] = 8 05124
c[19] = 15 41637
c[20] = 29 30329
c[21] = 55 28733
c[22] = 103 62312
c[23] = 192 95226
c[24] = 357 13454
c[25] = 657 15094
c[26] = 1202 56653
c[27] = 2188 93580
c[28] = 3964 18699
c[29] = 7143 99381
c[30] = 12814 03841
c[31] = 22879 86987
c[32] = 40674 28375
c[33] = 72002 10523
c[34] = 1 26938 90803
c[35] = 2 22907 27268
c[36] = 3 89934 10516
c[37] = 6 79590 10130
c[38] = 11 80166 56268
c[39] = 20 42336 54229
c[40] = 35 22457 10866
c[41] = 60 55388 66862
c[42] = 103 76685 22922
c[43] = 177 27009 55975
c[44] = 301 93338 54177
c[45] = 512 76944 84375
c[46] = 868 36766 38832
c[47] = 1466 52339 66068
c[48] = 2470 07526 91832
c[49] = 4149 51768 77972
c[50] = 6953 13056 79518
c[51] = 11622 14153 25837
c[52] = 19379 44766 58112
c[53] = 32238 23655 07746
c[54] = 53505 67710 14674
c[55] = 88603 33844 75166

Credits: B. Srivatsan (a sophomore at IIT Madras) for writing the code which was parallelized by Suresh Govindarajan(SG). All runs were carried out by SG on machines at the High Performance Computing Environment at IIT Madras and the Physics Department at IIT Madras.
Disclaimer: While a lot of effort has gone into ensuring the correctness of our results, this has not been independently verified. Our results do agree with known results up to N=50 and could be considered as an independent verfication of the 2003 Mustonen-Rajesh result. as well as a validation of the numbers for N=51,52 that were generated using different code (v2.0).

Estimating solid partitions

Mustonen and Rajesh provide evidence using Monte-Carlo simulations that the following formula1 captures the asymptotics of solid partitions for large N.

(1)
\begin{eqnarray} q_3(x)\sim \frac{0.2135888793}{x^{61/96}} \exp\left(a x^{3/4}+b x^{1/2}+c x^{1/4}\right)\ , \hspace{1.5in}\\ \textrm{where}\quad a=\frac{ 2^{7/4} \pi }{3 \sqrt[4]{15}}\quad, \quad b=\frac{\sqrt{\frac{15}{2}}\ \zeta (3)}{\pi ^2}\quad, \quad c= -\frac{15 \sqrt[4]{15}\ \zeta (3)^2}{ 2^{7/4} \pi ^5}\ . \end{eqnarray}

where we fixed the multiplicative constant to give the exact answer for 50. The table below lists the values given by rounding off the above formula along with the exact answer — we underline the correct digits. It is easy to see that the above approximation is a good one (worst case is for 55 where it is off by 0.06%) and gives a better number than directly using MacMahon's formula.

$N$ 50 51 52 53 54 55
$p_3(N)$ 69531305679518 116221415325837 193794476658112 322382365507746 535056771014674 886033384475166 Exact
$q_3(N)$ 69531305679518 116241874474137 193858787192598 322532432126512 535364615347267 886618388502748 Approximate

This page has been created by Suresh Govindarajan

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