Number of solid partitions for N<=68

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Predictions

• The first prediction in the table below is being made on Mon Apr 18 15:57:34 IST 2011. and makes use of the following asymptotic formula.

(1)

\begin{eqnarray} q_3(x)= \frac{0.213475761886676013}{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^{5/4}\ \zeta (3)^2}{ 2^{7/4} \pi ^5}\ . \end{eqnarray}

where we have fixed the multiplicative constant to give the exact answer for 62. We anticipate that the numbers will be of the correct order with the first three digits being correct.

• The second and third predictions are being made on Sun May 8 06:13:26 IST 2011 and makes use of a formula similar to $q_3(x)$ with more parameters added. The first parameter that is added in $r_3(x)$ is a shift i.e., $x=n+\textrm{shift}$. Thus $r_3(x)$ has two free parameters. The third prediction is based on adding a third parameter (to the shift and the overall scale) obtained by letting $c$ in the formula for $q_3(x)$ be a free parameter. This is represented by the function $s_3(x)$ — we now let $c$ in the formula for $q_3(x)$. These parameters were determined by fitting to the data in the range $[58,62]$. We expect that the predictions with larger number of parameters will provide better fits.

(2)

\begin{align} r_3(x) =\exp\Big[ a\ (x+sh)^{3/4} +b\ \sqrt{x+sh}+c\ \sqrt[4]{x+sh} -\frac{61}{96} \log (x+sh)-1.530354723\Big] \end{align}

with $sh=-0.02844026154$ and

(3)

\begin{align} s_3(x) =\exp\Big[ a\ (x+sh)^{3/4} +b\ \sqrt{x+sh}+0.2571306945 \sqrt[4]{x+sh}-\frac{61}{96} \log (x+sh)-3.210696955\Big] \end{align}

with $sh=1.689159515$.

$N$	61	62	63	64
$q_3(N)$	17474590403967699	used for fit	46453074905306481	75522726337662733
$r_3(N)$	17473162727829722	28517941261598224	46454377905586507	75528871134371025
$s_3(N)$	17472949259247056	28518689103596432	46458481008470927	75541939909077818
Exact	17472947006257293	28518691093388854	46458506464748807	75542021868032878
$N$	65	66	67	68
$q_3(N)$	122556018966297693	198518226269824763	320988410810838956	518102330350099210
$r_3(N)$	122572402629704284	198554960326767292	321063993755480353	518249988167731396
$s_3(N)$	122606641782139492	198635634117555516	321241553598715473	518622446910206277
Exact	122606799866017598	198635761249922839	321241075686259326	518619444932991189

Results

Here are the results obtained on Wed Jun 1 13:01:58 IST 2011 by running a parallelized and modified version of Knuth's algorithm for about five months (360000 CPU hours). Below, c[N] denotes the number of solid partitions of N. So far we have added eighteen numbers (the last six are new), c[51] to c[68] to the numbers computed by Knuth and Mustonen-Rajesh. The number of solid partitions appear in the On-Line Encyclopedia of Integer Sequences as sequence A000293. Our list is given below:

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] = 108802
c[16] = 214071
c[17] = 416849
c[18] = 805124
c[19] = 1541637
c[20] = 2930329
c[21] = 5528733
c[22] = 10362312
c[23] = 19295226
c[24] = 35713454
c[25] = 65715094
c[26] = 120256653
c[27] = 218893580
c[28] = 396418699
c[29] = 714399381
c[30] = 1281403841
c[31] = 2287986987
c[32] = 4067428375
c[33] = 7200210523
c[34] = 12693890803
c[35] = 22290727268
c[36] = 38993410516
c[37] = 67959010130
c[38] = 118016656268
c[39] = 204233654229
c[40] = 352245710866
c[41] = 605538866862
c[42] = 1037668522922
c[43] = 1772700955975
c[44] = 3019333854177
c[45] = 5127694484375
c[46] = 8683676638832
c[47] = 14665233966068
c[48] = 24700752691832
c[49] = 41495176877972
c[50] = 69531305679518

c[51] = 116221415325837
c[52] = 193794476658112
c[53] = 322382365507746
c[54] = 535056771014674
c[55] = 886033384475166
c[56] = 1464009339299229
c[57] = 2413804282801444
c[58] = 3971409682633930
c[59] = 6520649543912193
c[60] = 10684614225715559
c[61] = 17472947006257293
c[62] = 28518691093388854
c[63] = 46458506464748807
c[64] = 75542021868032878
c[65] = 122606799866017598
c[66] = 198635761249922839
c[67] = 321241075686259326
c[68] = 518619444932991189

Credits: B. Srivatsan (a sophomore at IIT Madras) for writing all 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.
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); N=53-55 that were generated using v3.0 of the code and N=56-62 that were generated using v3.2 of the code.

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© 2011 Suresh Govindarajan