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A000293 a(n) = number of solid (i.e., three-dimensional) partitions of n.
(Formerly M3392 N1371)
33
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, 396418699, 714399381, 1281403841, 2287986987, 4067428375, 7200210523, 12693890803, 22290727268, 38993410516, 67959010130, 118016656268, 204233654229, 352245710866, 605538866862, 1037668522922, 1772700955975, 3019333854177, 5127694484375, 8683676638832, 14665233966068, 24700752691832, 41495176877972, 69531305679518 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

An ordinary partition is a row of numbers in nondecreasing order whose sum is n. Here the numbers are in a three-dimensional pile, nondecreasing in the x-, y- and z-directions.

Finding a g.f. for this sequence is an unsolved problem. At first it was thought that it was given by A000294.

Equals A000041 convolved with A002836: [1, 0, 2, 5, 12, 24, 56, 113, ...] and row sums of the convolution triangle A161564. - Gary W. Adamson, Jun 13 2009

REFERENCES

P. Bratley and J. K. S. McKay, Algorithm 313: Multi-dimensional partition generator, Comm. ACM, 10 (Issue 10, 1967), p. 666.

P. A. MacMahon, Memoir on the theory of partitions of numbers - Part VI, Phil. Trans. Roal Soc., 211 (1912), 345-373.

P. A. MacMahon, Combinatory Analysis. Cambridge Univ. Press, London and New York, Vol. 1, 1915 and Vol. 2, 1916; see vol. 2, p 332.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

Suresh Govindarajan, Table of n, a(n) for n = 0..72

A. O. L. Atkin, P. Bratley, I. G. McDonald and J. K. S. McKay, Some computations for m-dimensional partitions, Proc. Camb. Phil. Soc., 63 (1967), 1097-1100. [Annotated scanned copy], DOI

Srivatsan Balakrishnan, Suresh Govindarajan and Naveen S. Prabhakar, On the asymptotics of higher-dimensional partitions, arXiv:1105.6231.

Nicolas Destainville and Suresh Govindarajan, Estimating the asymptotics of solid partitions, J. Stat. Phys. 158 (2015) 950-967; arXiv:1406.5605.

Suresh Govindarajan, Solid Partitions Project Dec 14, 2010

D. E. Knuth, A Note on Solid Partitions, Math. Comp. 24, 955-961, 1970.

P. A. MacMahon, Combinatory analysis.

Ville Mustonen and R. Rajesh, Numerical Estimation of the Asymptotic Behaviour of Solid Partitions of an Integer, J. Phys. A 36 (2003), no. 24, 6651-6659.

S. P. Naveen, On The Asymptotics of Some Counting Problems in Physics, Thesis, Bachelor of Technology, DEPARTMENT OF PHYSICS, INDIAN INSTITUTE OF TECHNOLOGY, MADRAS, May 2011.

Eric Weisstein's World of Mathematics, Solid Partition

Wikipedia, Solid partition

EXAMPLE

Examples for n=2 and n=3.

a(2) = 4: 2; 11 where the first 1 is at the origin and the second 1 is in the x, y or z direction.

a(3) = 10: 3; 21 where the 2 is at the origin and the 1 is on the x, y or z axis; 111 (a row of 3 ones on the x, y or z axes); and three 1's with one 1 at the origin and the other two 1's on two of the three axes.

CROSSREFS

Cf. A000041, A000219 (2-dim), A000294, A000334 (4-dim), A000390 (5-dim), A002835, A002836, A005980, A037452 (inverse Euler trans.), A080207, A082535, A000416 (6-dim), A000427 (7-dim), A179855 (8-dim).

Cf. A161564. - Gary W. Adamson, Jun 13 2009

Sequence in context: A022812 A192306 A276432 * A000294 A133086 A285186

Adjacent sequences:  A000290 A000291 A000292 * A000294 A000295 A000296

KEYWORD

nonn,nice,changed

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from the Mustonen and Rajesh article, May 02 2003

a(51)-a(62) found by Suresh Govindarajan and students, Dec 14 2010

a(63)-a(68) found by Suresh Govindarajan and students, Jun 01 2011

a(69)-a(72) found by Suresh Govindarajan and Srivatsan Balakrishnan, Jan 03 2013

STATUS

approved

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Last modified August 21 17:39 EDT 2017. Contains 290891 sequences.