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 A000293 a(n) = number of solid (i.e., three-dimensional) partitions of n. (Formerly M3392 N1371) 37
 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. 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 Alimzhan Amanov and Damir Yeliussizov, MacMahon's statistics on higher-dimensional partitions, arXiv:2009.00592 [math.CO], 2020. Mentions this sequence. 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 [cond-mat.stat-mech], 2011. P. Bratley and J. K. S. McKay, Algorithm 313: Multi-dimensional partition generator, Comm. ACM, 10 (Issue 10, 1967), p. 666. Nicolas Destainville and Suresh Govindarajan, Estimating the asymptotics of solid partitions, arXiv:1406.5605 [cond-mat.stat-mech], 2014; J. Stat. Phys. 158 (2015) 950-967. 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, arXiv:cond-mat/0303607 [cond-mat.stat-mech], 2003; 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. From Gus Wiseman, Jan 22 2019: (Start) The a(1) = 1 through a(4) = 26 solid partitions, represented as chains of chains of integer partitions:   ((1))  ((2))       ((3))            ((4))          ((11))      ((21))           ((22))          ((1)(1))    ((111))          ((31))          ((1))((1))  ((2)(1))         ((211))                      ((11)(1))        ((1111))                      ((2))((1))       ((2)(2))                      ((1)(1)(1))      ((3)(1))                      ((11))((1))      ((21)(1))                      ((1)(1))((1))    ((11)(11))                      ((1))((1))((1))  ((111)(1))                                       ((2))((2))                                       ((3))((1))                                       ((2)(1)(1))                                       ((21))((1))                                       ((11))((11))                                       ((11)(1)(1))                                       ((111))((1))                                       ((2)(1))((1))                                       ((1)(1)(1)(1))                                       ((11)(1))((1))                                       ((2))((1))((1))                                       ((1)(1))((1)(1))                                       ((1)(1)(1))((1))                                       ((11))((1))((1))                                       ((1)(1))((1))((1))                                       ((1))((1))((1))((1)) (End) MATHEMATICA planePtns[n_]:=Join@@Table[Select[Tuples[IntegerPartitions/@ptn], And@@(GreaterEqual@@@Transpose[PadRight[#]])&], {ptn, IntegerPartitions[n]}]; solidPtns[n_]:=Join@@Table[Select[Tuples[planePtns/@y], And@@(GreaterEqual@@@Transpose[Join@@@(PadRight[#, {n, n}]&/@#)])&], {y, IntegerPartitions[n]}]; Table[Length[solidPtns[n]], {n, 10}] (* Gus Wiseman, Jan 23 2019 *) CROSSREFS Cf. A000041, A000219 (2-dim), A000294, A000334 (4-dim), A000390 (5-dim), A002835, A002836, A005980, A037452 (inverse Euler trans.), A080207, A007326, A000416 (6-dim), A000427 (7-dim), A179855 (8-dim). Cf. A161564. - Gary W. Adamson, Jun 13 2009 Cf. A001970, A002974, A003293, A007713, A114736, A117433, A323657. Sequence in context: A192306 A276432 A308817 * A000294 A308723 A133086 Adjacent sequences:  A000290 A000291 A000292 * A000294 A000295 A000296 KEYWORD nonn,nice AUTHOR 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 15 00:08 EDT 2022. Contains 356122 sequences. (Running on oeis4.)