A000334 -id:A000334 - 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.

A096651 Lower triangular matrix T, read by rows, such that the row sums of T^n form the n-dimensional partitions.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 1, 0, 1, 1, 3, 1, 1, 0, 1, 3, 1, 4, 1, 1, 0, 1, -1, 7, 1, 5, 1, 1, 0, 1, 15, -17, 14, 1, 6, 1, 1, 0, 1, -78, 133, -61, 25, 1, 7, 1, 1, 0, 1, 632, -1020, 529, -152, 41, 1, 8, 1, 1, 0, 1, -6049, 9826, -4989, 1506, -314, 63, 1, 9, 1, 1, 0, 1, 68036, -110514, 56161, -16668, 3532, -576, 92, 1, 10, 1, 1, 0, 1, -878337, 1427046, -724881, 214528, -44703, 7276, -972, 129, 1, 11, 1, 1, 0, 1, 12817659, -20827070, 10576885, -3123249, 647092, -103476, 13644, -1541, 175, 1, 12, 1, 1

Offset: 0

Paul D. Hanna and Wouter Meeussen, Jul 02 2004

Hanna's Triangle: There exists a unique lower triangular matrix T, with ones on its diagonal, such that the row sums of T^n yields the n-dimensional partitions for all n>0. Specifically, row sums of T form A000041 (linear partitions); row sums of T^2 form A000219 (planar partitions); row sums of T^3 form A000293 (solid partitions); row sums of T^4 form A000334(4-D); row sums of T^5 form A000390(5-D); row sums of T^6 form A000416(6-D); row sums of T^7 form A000427(7-D). Rows indexed 9-13 were calculated by Wouter Meeussen.

Existence and integrality of Hanna's triangle has been proved in arXiv:1203.4419. (Suresh Govindarajan)

			Triangle T begins:
  {1},
  {0,1},
  {0,1,1},
  {0,1,1,1},
  {0,1,2,1,1},
  {0,1,1,3,1,1},
  {0,1,3,1,4,1,1},
  {0,1,-1,7,1,5,1,1},
  {0,1,15,-17,14,1,6,1,1},
  {0,1,-78,133,-61,25,1,7,1,1},
  {0,1,632,-1020,529,-152,41,1,8,1,1},
  {0,1,-6049,9826,-4989,1506,-314,63,1,9,1,1},
  {0,1,68036,-110514,56161,-16668,3532,-576,92,1,10,1,1},
  {0,1,-878337,1427046,-724881,214528,-44703,7276,-972,129,1,11,1,1},
  ...
  with row sums: {1,1,2,3,5,7,11,15,22,...} (A000041).
T^2 begins:
  {1},
  {0,1},
  {0,2,1},
  {0,3,2,1},
  {0,5,5,2,1},
  {0,7,7,7,2,1},
  {0,11,16,9,9,2,1},
  {0,15,15,31,11,11,2,1},
  {0,22,59,-4,54,13,13,2,1},
  ...
  with row sums: {1,1,3,6,13,24,48,86,...} (A000219).

S. Govindarajan Notes on higher-dimensional partitions, arXiv:1203.4419 [math.CO], 2012.
Wouter Meeussen, Rows 14-17 added

Cf. A000041, A000219, A000293, A000334, A000390, A000416, A000427, A096652(T^2), A096653(T^3), A096642-A096645(columns).

Cf. A096800, A096751.

For n>=0: T(0, 0)=1, T(n+1,0)=0, T(n+1,1)=1. For n>=1: T(n, n)=1, T(n+1, n)=1, T(n+2, n)=n, T(n+3, n)=1, T(n+4, n)=n*(5+n^2)/6, T(n+5, n)=(-48+90*n-7*n^2-6*n^3-5*n^4)/24, T(n+6, n)=(400-382*n-55*n^2+30*n^3+35*n^4+12*n^5)/40 (Wouter Meeussen). Corrected entry for the zeroth and first columns of the matrix T -- entry had columns and rows interchanged (Corrected by Suresh Govindarajan)

G.f.: A(x, y) = Product_{n>=1} 1/(1-x^n)^[P_n(y)/n], where P_n(y) is the n-th row polynomial of triangle A096800.

Rows 14-17 calculated (using extra terms in A096642-A096645 provided by Sean A. Irvine) by Wouter Meeussen, Jan 08 2011

A000390 Number of 5-dimensional partitions of n.

Original entry on oeis.org

1, 6, 21, 71, 216, 657, 1907, 5507, 15522, 43352, 119140, 323946, 869476, 2308071, 6056581, 15724170, 40393693, 102736274, 258790004, 645968054, 1598460229, 3923114261, 9554122089, 23098084695, 55458417125, 132293945737, 313657570114

Offset: 1

N. J. A. Sloane

nonn

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 = 1..30
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
S. Balakrishnan, S. Govindarajan and N. S. Prabhakar, On the asymptotics of higher-dimensional partitions, arXiv:1105.6231 [cond-mat.stat-mech], 2011.
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.

Cf. A000012 (0-dim), A000041 (1-dim), A000219 (2-dim), A000293 (3-dim), A000334 (4-dim), A000416 (6-dim).

Cf. A096751 (See row 5).

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[levptns[n, 5] // Length, {n, 1, 7}] (* Robert P. P. McKone, Dec 18 2020 *)

More terms from Sean A. Irvine, Nov 14 2010

More terms found by Suresh Govindarajan, May 30 2011

A007327 Difference between two partition g.f.s.

Original entry on oeis.org

0, 0, 0, 0, 0, 5, 20, 69, 200, 521, 1294, 3126, 7364, 17309, 40577, 95460, 224971, 531368, 1252664, 2943095, 6870029, 15911618, 36507381, 82930347, 186414619, 414654766, 912766795, 1989007381, 4292038414, 9175624264, 19442250125, 40851448761, 85157787033, 176200110937

Offset: 1

N. J. A. Sloane, Mira Bernstein

nonn

George E. Andrews, The Theory of Partitions, Addison-Wesley, 1976, p. 190.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes 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; alternative link.
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]

Cf. A000334, A000335.

Cf. A007326, A007328, A007329, A007330, A008780, A042984.

a(n) = A000335(n) - A000334(n). - Sean A. Irvine, Dec 18 2017

a(11)-a(23) from Sean A. Irvine, Dec 18 2017

More terms from Amiram Eldar, May 11 2024

A119267 Number of 4-dimensional partitions of n up to conjugacy.

Original entry on oeis.org

1, 1, 1, 2, 4, 7, 14, 27, 55, 111, 232, 486, 1039, 2226, 4820, 10449, 22727, 49354, 107117, 231774, 500040, 1074476, 2299589, 4899650

Offset: 0

Franklin T. Adams-Watters, May 11 2006

Partitions are considered as generalized Ferrers diagrams; any permutation of the axes produces a conjugate.

Cf. A119269, A000334.

Cf. A119266, A119338, A119340, A119341, A119342.

a(9)-a(20) from Max Alekseyev, May 16 2006

a(21)-a(23) from Max Alekseyev, Mar 20 2025

A323657 Number of strict solid partitions of n.

Original entry on oeis.org

1, 1, 1, 4, 4, 7, 16, 19, 28, 40, 82, 94, 145, 190, 274, 463, 580, 802, 1096, 1486, 1948, 3148, 3811, 5314, 6922, 9394, 11971, 16156, 23044, 28966, 38368, 50002, 65116, 83872, 108706, 137917, 192070, 236242, 308698, 390772, 506935, 633982, 817324, 1018090

Offset: 0

Gus Wiseman, Jan 22 2019

nonn

A strict solid partition is an infinite three-dimensional array of distinct positive integers (and any number of zeros) summing to n such that all one-dimensional sections are strictly decreasing until they become all zeros.

			The a(1) = 1 through a(6) = 16 strict solid partitions, represented as chains of chains of integer partitions:
  ((1))  ((2))  ((3))       ((4))       ((5))       ((6))
                ((21))      ((31))      ((32))      ((42))
                ((2)(1))    ((3)(1))    ((41))      ((51))
                ((2))((1))  ((3))((1))  ((3)(2))    ((321))
                                        ((4)(1))    ((4)(2))
                                        ((3))((2))  ((5)(1))
                                        ((4))((1))  ((31)(2))
                                                    ((32)(1))
                                                    ((4))((2))
                                                    ((5))((1))
                                                    ((31))((2))
                                                    ((3)(2)(1))
                                                    ((32))((1))
                                                    ((3)(1))((2))
                                                    ((3)(2))((1))
                                                    ((3))((2))((1))

Alois P. Heinz, Table of n, a(n) for n = 0..740 (first 401 terms from John Tyler Rascoe)

Cf. A000219, A000293 (solid partitions), A000334, A001970, A002974, A008289, A114736, A117433 (strict plane partitions), A207542, A321662, A323657.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
ptnplane[n_]:=Union[Map[Reverse@*primeMS,Join@@Permutations/@facs[n],{2}]];
strplptns[n_]:=Join@@Table[Select[ptnplane[Times@@Prime/@y],And[And@@GreaterEqual@@@#,And@@(GreaterEqual@@@Transpose[PadRight[#]])]&],{y,Select[IntegerPartitions[n],UnsameQ@@#&]}]
Table[Length[Join@@Table[Select[Tuples[strplptns/@y],And[UnsameQ@@Flatten[#],And@@(GreaterEqual@@@Transpose[Join@@@(PadRight[#,{n,n}]&/@#)])]&],{y,IntegerPartitions[n]}]],{n,10}]

a(n) = Sum_{k=1..n} A008289(n,k)*A207542(k) for n > 0. - John Tyler Rascoe, Dec 19 2024

a(21) onwards from John Tyler Rascoe, Dec 19 2024

A002974 Number of restricted solid partitions of n.

Original entry on oeis.org

1, 1, 4, 7, 11, 20, 35, 59, 99, 165, 270, 443, 723, 1161, 1861, 2961, 4654, 7279, 11317, 17476, 26879, 41132, 62601, 94878, 143172, 215115, 321995, 480216, 713655, 1057192

Offset: 1

N. J. A. Sloane

Definition, based on Math. Review MR0297583: By a solid partition of n is meant a 3-dimensional arrangement of positive integers N(x,y,z) satisfying the conditions (i) the integer N(x,y,z) is located at the point with Cartesian coordinates (x,y,z); N(x,y,z) is defined only for certain integers x,y,z >= 0, and (ii) if N(x,y,z) is defined and 0 <= x' <= x, 0 <= y' <= y, 0 <= z' <= z then N(x,y,z) is defined and N(x',y',z') <= N(x,y,z). A solid partition is said to correspond to an (ordinary) partition of n=n_1+n_2+...+n_t, n_k>0, if there is a one-to-one correspondence between the summands n_k and the points (x_k,y_k,z_k) for which N is defined so that n_k=N(x_k,y_k,z_k). Finally, a restricted solid partition is a solid partition such that x'<=x, y'<=y, z'<=z and N(x',y',z')=N(x,y,z) implies x'=x, y'=y, z'=z.

Alternatively, a restricted solid partition is an infinite three-dimensional array of nonnegative integers summing to n such that all one-dimensional sections are strictly decreasing until they become all zeros. - Gus Wiseman, Jan 22 2019

			From _Gus Wiseman_, Jan 22 2019: (Start)
The a(1) = 1 through a(6) = 20 restricted solid partitions, represented as chains of chains of integer partitions:
  ((1))  ((2))  ((3))       ((4))          ((5))           ((6))
                ((21))      ((31))         ((32))          ((42))
                ((2)(1))    ((3)(1))       ((41))          ((51))
                ((2))((1))  ((21)(1))      ((3)(2))        ((321))
                            ((3))((1))     ((4)(1))        ((4)(2))
                            ((21))((1))    ((31)(1))       ((5)(1))
                            ((2)(1))((1))  ((3))((2))      ((31)(2))
                                           ((4))((1))      ((32)(1))
                                           ((31))((1))     ((41)(1))
                                           ((3)(1))((1))   ((4))((2))
                                           ((21)(1))((1))  ((5))((1))
                                                           ((31))((2))
                                                           ((3)(2)(1))
                                                           ((32))((1))
                                                           ((41))((1))
                                                           ((3)(1))((2))
                                                           ((3)(2))((1))
                                                           ((4)(1))((1))
                                                           ((31)(1))((1))
                                                           ((3))((2))((1))
(End)

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

H. Gupta, Restricted solid partitions, J. Combin. Theory, A 13 (1972), 140-144.

Cf. A000219, A000293 (solid partitions), A000334, A001970, A114736 (restricted plane partitions), A117433 (strict plane partitions), A321662, A323657 (strict solid partitions).

srcplptns[n_]:=Join@@Table[Select[Tuples[IntegerPartitions/@ptn],And[And@@(GreaterEqual@@@Transpose[PadRight[#]]),And@@Greater@@@#,And@@(Greater@@@DeleteCases[Transpose[PadRight[#]],0,{2}])]&],{ptn,IntegerPartitions[n]}];
srcsolids[n_]:=Join@@Table[Select[Tuples[srcplptns/@y],And[And@@(GreaterEqual@@@Transpose[Join@@@(PadRight[#,{n,n}]&/@#)]),And@@(Greater@@@DeleteCases[Transpose[Join@@@(PadRight[#,{n,n}]&/@#)],0,{2}])]&],{y,IntegerPartitions[n]}]
Table[Length[srcsolids[n]],{n,10}] (* Gus Wiseman, Jan 23 2019 *)

More terms from Sean A. Irvine, Dec 15 2014

A179855 Number of 8-dimensional partitions of n.

Original entry on oeis.org

1, 9, 45, 201, 819, 3231, 12321, 46209, 170370, 621316, 2240838, 8011584, 28395213, 99845553, 348333411, 1205925033, 4142850423

Offset: 1

Suresh Govindarajan, Jan 11 2011

nonn

Andrews, George E., The Theory of Partitions, Cambridge University Press, 1984. Cambridge Books Online. Cambridge University Press.

Cf. A000041, A000219, A000293, A000334, A000390, A000416, A000427, A096651.

A096652 Lower triangular matrix T, read by rows, such that the row sums of T^n form the (2n)-dimensional partition numbers.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 3, 2, 1, 0, 5, 5, 2, 1, 0, 7, 7, 7, 2, 1, 0, 11, 16, 9, 9, 2, 1, 0, 15, 15, 31, 11, 11, 2, 1, 0, 22, 59, -4, 54, 13, 13, 2, 1, 0, 30, -109, 313, -72, 87, 15, 15, 2, 1, 0, 42, 1314, -1922, 1122, -225, 132, 17, 17, 2, 1, 0, 56, -11804, 19468, -9671, 3087, -509, 191, 19, 19, 2, 1, 0, 77, 133957, -217176, 110734, -32581

Offset: 0

Paul D. Hanna, Jul 06 2004

Row sums of T form A000219 (planar partitions); row sums of T^2 form A000334(4-D); row sums of T^3 form A000416(6-D).

			Triangle T begins:
{1},
{0,1},
{0,2,1},
{0,3,2,1},
{0,5,5,2,1},
{0,7,7,7,2,1},
{0,11,16,9,9,2,1},
{0,15,15,31,11,11,2,1},
{0,22,59,-4,54,13,13,2,1},
{0,30,-109,313,-72,87,15,15,2,1},
{0,42,1314,-1922,1122,-225,132,17,17,2,1},
{0,56,-11804,19468,-9671,3087,-509,191,19,19,2,1},
{0,77,133957,-217176,110734,-32581,7137,-980,266,21,21,2,1},
{0,101,-1728760,2809257,-1426436,422732,-87714,14601,-1704,359,23,23,2,1},...
Row sums are: {1,1,3,6,13,24,48,86,160,282,500,859,...} (A000219).
T^2 begins:
{1},
{0,1},
{0,4,1},
{0,10,4,1},
{0,26,14,4,1},
{0,59,38,18,4,1},
{0,140,109,50,22,4,1},
{0,307,256,179,62,26,4,1},
{0,684,709,370,273,74,30,4,1},
{0,1464,1240,1683,438,395,86,34,4,1},...
with row sums: {1,1,5,15,45,120,326,835,2145,5345,...} (A000334).

Cf. A096651, A000219, A000334, A000416.

Matrix square of triangle A096651.

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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A096651 Lower triangular matrix T, read by rows, such that the row sums of T^n form the n-dimensional partitions.

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A000390 Number of 5-dimensional partitions of n.

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A007327 Difference between two partition g.f.s.

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A119267 Number of 4-dimensional partitions of n up to conjugacy.

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A323657 Number of strict solid partitions of n.

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A002974 Number of restricted solid partitions of n.

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