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An ordinary (i.e. one-dimensional) partition is a row of nonnegative integers in nonincreasing (weakly decreasing) order whose sum is . A plane partition is a two-dimensional arrangement of nonnegative integers, nonincreasing (weakly decreasing) in the x- and y-directions, which sum to .
A solid partition of is a three-dimensional arrangement of nonnegative integers
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which are nonincreasing (weakly decreasing) in all rows, columns and stacks:
and which sum to :
-
Number of solid partitions of n
The number of solid partitions of a positive integer is an unsolved problem in combinatorial number theory.
The number of solid partitions of gives the sequence (Cf. A000293)
- {1, 1, 4, 10, 26, 59, 140, 307, 684, 1464, 3122, 6500, 13426, 27248, 54804, 108802, 214071, 416849, 805124, 1541637, 2930329, 5528733, 10362312, 19295226, 35713454, 65715094, 120256653, 218893580, ...}
Recurrence
...
Generating function
Conjectured (but false at n=6) by MacMahon
The generating function of point (degenerate) partitions is
The generating function of ordinary or line (one-dimensional) partitions is the reciprocal of Euler's function
In 1912, Major Percy A. MacMahon proved that the generating function for plane partitions is
MacMahon conjectured, but doubted it later[1], (but it was shown to disagree at = 6) that the generating function for solid partitions is
where is the th triangular number.
Very tempting conjecture indeed, but false for = 6. MacMahon had actually guessed the following formula for the generating function of -dimensional partitions
where is the th ()-dimensional simplex number.
However, Rajesh[2] and Mustonen seem to claim that the formula nevertheless captures the asymptotics correctly.
Asymptotic behaviour
Asymptotic behaviour of p_3(n)
...
Asymptotic behaviour log(p_3(n))
-
where is the number of solid partitions of the integer .
Noting that
could this be the actual limit? (Cf. A016629 Decimal expansion of ln(6).)
This would produce the result (if true...)
This gives the asymptotic behaviour on the number of digits of in any fixed radix representation, e.g. base 10, of
and in base 6
Table
Observe that for = 1, 2 and 3 we obtain the -dimensional simplex numbers (i.e. triangular gnomon numbers, triangular numbers, tetrahedral numbers.) Would this be true for ?
Solid, plane, line and point partitions
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0
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1
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1
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1
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1
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1
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1
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1
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1
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1
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2
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4
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3
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2
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1
|
3
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10
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6
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3
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1
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4
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26
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13
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5
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1
|
5
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59
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24
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7
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1
|
6
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140
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48
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11
|
1
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7
|
307
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86
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15
|
1
|
8
|
684
|
160
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22
|
1
|
9
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1464
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282
|
30
|
1
|
10
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3122
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500
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42
|
1
|
11
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6500
|
859
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56
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1
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12
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13426
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1479
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77
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1
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13
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27248
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2485
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101
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1
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14
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54804
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4167
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135
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1
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15
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108802
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6879
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176
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1
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16
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214071
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11297
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231
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1
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17
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416849
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1
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18
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805124
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1
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19
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1541637
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1
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20
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2930329
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1
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21
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5528733
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1
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22
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10362312
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1
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23
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19295226
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1
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24
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35713454
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1
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25
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65715094
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1
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26
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120256653
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1
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27
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218893580
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1
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28
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396418699
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1
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29
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714399381
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1
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30
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1281403841
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1
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31
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2287986987
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1
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32
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4067428375
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1
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33
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7200210523
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|
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1
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34
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12693890803
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1
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35
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22290727268
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1
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36
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38993410516
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1
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37
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67959010130
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|
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1
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38
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118016656268
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1
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39
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204233654229
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|
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1
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40
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352245710866
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|
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1
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41
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605538866862
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1
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42
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1037668522922
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|
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1
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43
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1772700955975
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|
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1
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44
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3019333854177
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|
|
1
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45
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5127694484375
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|
|
1
|
46
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8683676638832
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|
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1
|
47
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14665233966068
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|
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1
|
48
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24700752691832
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|
|
1
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49
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41495176877972
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|
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1
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50
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69531305679518
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|
|
1
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51
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116221415325837
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|
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1
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52
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193794476658112
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|
|
1
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53
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322382365507746
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|
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1
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54
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535056771014674
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|
|
1
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55
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886033384475166
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|
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1
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56
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1464009339299229
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|
|
1
|
57
|
2413804282801444
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|
|
1
|
58
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3971409682633930
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|
|
1
|
59
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6520649543912193
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|
|
1
|
60
|
10684614225715559
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|
|
1
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61
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17472947006257293
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1
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62
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28518691093388854
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1
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See also
Notes
References
- P. A. MacMahon, Combinatory Analysis, Vol. 2. New York: Chelsea, pp. 75-176, 1960.
External links
- Donald E. Knuth, A Note on Solid Partitions, 1970.
- Boltzmann | Physics enthusiasts at IIT Madras, Counting Solid Partitions.
- Boltzmann | Physics enthusiasts at IIT Madras, Number of solid partitions for N <= 62.
- Boltzmann | Physics enthusiasts at IIT Madras, Solid Partitions using Volunteer Computing.
- Ville Mustonen and R. Rajesh, Numerical estimation of the asymptotic behaviour of solid partitions of an integer, 2003 J. Phys. A: Math. Gen. 36 6651.
- Weisstein, Eric W., Solid Partition, from MathWorld—A Wolfram Web Resource..