A179855 -id:A179855 - cp's OEIS frontend

A000293 a(n) = number of solid (i.e., three-dimensional) partitions of n.

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

Offset: 0

N. J. A. Sloane

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

			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)

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).

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
Damir Yeliussizov, Bounds on the number of higher-dimensional partitions, arXiv:2302.04799 [math.CO], 2023.

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.

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 *)

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

A096751 Square table, read by antidiagonals, where T(n,k) equals the number of n-dimensional partitions of k.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 3, 1, 1, 1, 4, 6, 5, 1, 1, 1, 5, 10, 13, 7, 1, 1, 1, 6, 15, 26, 24, 11, 1, 1, 1, 7, 21, 45, 59, 48, 15, 1, 1, 1, 8, 28, 71, 120, 140, 86, 22, 1, 1, 1, 9, 36, 105, 216, 326, 307, 160, 30, 1, 1, 1, 10, 45, 148, 357, 657, 835, 684, 282, 42, 1

Offset: 0

Paul D. Hanna, Jul 07 2004

Main diagonal forms A096752. Antidiagonal sums form A096753. Row with index n lists the row sums of the n-th matrix power of triangle A096651, for n>=0.

			n-th row lists n-dimensional partitions; table begins with n=0:
  [1,1,1,1,1,1,1,1,1,1,1,1,...],
  [1,1,2,3,5,7,11,15,22,30,42,56,...],
  [1,1,3,6,13,24,48,86,160,282,500,859,...],
  [1,1,4,10,26,59,140,307,684,1464,3122,...],
  [1,1,5,15,45,120,326,835,2145,5345,...],
  [1,1,6,21,71,216,657,1907,5507,15522,...],
  [1,1,7,28,105,357,1197,3857,12300,38430,...],
  [1,1,8,36,148,554,2024,7134,24796,84625,...],
  [1,1,9,45,201,819,3231,12321,46209,170370,...],
  [1,1,10,55,265,1165,4927,20155,80920,...],...
Array begins:
      k=0:  k=1:  k=2:  k=3:  k=4:  k=5:  k=6:  k=7:  k=8:
  n=0:  1     1     1     1     1     1     1     1     1
  n=1:  1     1     2     3     5     7    11    15    22
  n=2:  1     1     3     6    13    24    48    86   160
  n=3:  1     1     4    10    26    59   140   307   684
  n=4:  1     1     5    15    45   120   326   835  2145
  n=5:  1     1     6    21    71   216   657  1907  5507
  n=6:  1     1     7    28   105   357  1197  3857 12300
  n=7:  1     1     8    36   148   554  2024  7134 24796
  n=8:  1     1     9    45   201   819  3231 12321 46209
  n=9:  1     1    10    55   265  1165  4927 20155 80920

G. E. Andrews, The Theory of Partitions, Add.-Wes. 1976, pp. 189-197.

Pontus von Brömssen, Table of n, a(n) for n = 0..275 (first 23 antidiagonals)
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]

Rows: A000012 (n=0), A000041 (n=1), A000219 (n=2), A000293 (n=3), A000334 (n=4), A000390 (n=5), A000416 (n=6), A000427 (n=7), A179855 (n=8).
Columns: A008778 (k=4), A008779 (k=5), A042984 (k=6).
Cf. A096651, A096752, A096753.
Cf. A096806.
Cf. A042984.
Cf. A144150, A213427, A290353, A323718, A323719.

trans[x_]:=If[x=={},{},Transpose[x]];
levptns[n_,k_]:=If[k==1,IntegerPartitions[n],Join@@Table[Select[Tuples[levptns[#,k-1]&/@y],And@@(GreaterEqual@@@trans[Flatten/@(PadRight[#,ConstantArray[n,k-1]]&/@#)])&],{y,IntegerPartitions[n]}]];
Table[If[sum==k,1,Length[levptns[k,sum-k]]],{sum,0,10},{k,0,sum}] (* Gus Wiseman, Jan 27 2019 *)

T(0, n)=T(n, 0)=T(n, 1)=1 for n>=0.

Inverse binomial transforms of the columns is given by triangle A096806.

cp's OEIS Frontend

A000293 a(n) = number of solid (i.e., three-dimensional) partitions of n.

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A096751 Square table, read by antidiagonals, where T(n,k) equals the number of n-dimensional partitions of k.

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Mathematica

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A000427 Number of 7-dimensional partitions of n.

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