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Solid partitions

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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





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
0 1 1 1 1
1 1 1 1 1
2 4 3 2 1
3 10 6 3 1
4 26 13 5 1
5 59 24 7 1
6 140 48 11 1
7 307 86 15 1
8 684 160 22 1
9 1464 282 30 1
10 3122 500 42 1
11 6500 859 56 1
12 13426 1479 77 1
13 27248 2485 101 1
14 54804 4167 135 1
15 108802 6879 176 1
16 214071 11297 231 1
17 416849     1
18 805124     1
19 1541637     1
20 2930329     1
21 5528733     1
22 10362312     1
23 19295226     1
24 35713454     1
25 65715094     1
26 120256653     1
27 218893580     1
28 396418699     1
29 714399381     1
30 1281403841     1
31 2287986987     1
32 4067428375     1
33 7200210523     1
34 12693890803     1
35 22290727268     1
36 38993410516     1
37 67959010130     1
38 118016656268     1
39 204233654229     1
40 352245710866     1
41 605538866862     1
42 1037668522922     1
43 1772700955975     1
44 3019333854177     1
45 5127694484375     1
46 8683676638832     1
47 14665233966068     1
48 24700752691832     1
49 41495176877972     1
50 69531305679518     1
51 116221415325837     1
52 193794476658112     1
53 322382365507746     1
54 535056771014674     1
55 886033384475166     1
56 1464009339299229     1
57 2413804282801444     1
58 3971409682633930     1
59 6520649543912193     1
60 10684614225715559     1
61 17472947006257293     1
62 28518691093388854     1


See also




Notes

  1. P. A. MacMahon, Combinatory Analysis, vol. 2, p. 175, first edition, 1916.
  2. R. Rajesh, Partitions of an Integer.

References

  • P. A. MacMahon, Combinatory Analysis, Vol. 2. New York: Chelsea, pp. 75-176, 1960.

External links