Numbers of Solid Partitions of N (<= 52)

Here are the results obtained on October 17, 2010 by running a parallelized version of Knuth's algorithm for about a month. Below, c[N] denotes the number of solid partitions of N. We add two numbers, c[51] and c[52] 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:

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
For comparison, MacMahon's wrong formula gives: c[50]= 71742322621601; c[51]=119938400508354 and c[52]=200023400885346

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.

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{align} c[x]\sim \frac{0.213699}{x^{61/96}} \exp\left(1.7898 x^{3/4}+0.3335 x^{1/2}-0.0414 x^{1/4}\right)\ , \end{align}

where we fixed the multiplicative constant to give the exact answer for 50. This gives values for N=51,52 that are within 0.03% of their actual value. For instance, it shows that c[60] is about 55 times c[52].

This page has been created by Suresh Govindarajan

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