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

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