A000293 -id:A000293 - cp's OEIS frontend

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

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

A094504 T(n,m) equals number of solid partitions of n containing m plane partitions.

Original entry on oeis.org

1, 3, 1, 6, 3, 1, 13, 9, 3, 1, 24, 22, 9, 3, 1, 48, 54, 25, 9, 3, 1, 86, 120, 63, 25, 9, 3, 1, 160, 267, 153, 66, 25, 9, 3, 1, 282, 559, 357, 162, 66, 25, 9, 3, 1, 500, 1158, 805, 390, 165, 66, 25, 9, 3, 1, 859, 2314, 1761, 898, 399, 165, 66, 25, 9, 3, 1, 1479, 4559, 3761, 2025, 931, 402, 165, 66, 25, 9, 3, 1

Offset: 1

Wouter Meeussen, Jun 05 2004

First column equals the number of plane partitions of n, corresponding to the 'single layer' solid partitions.

Rows read backward tend to limiting sequence 1, 3, 9, 25, 66, 165, 402, ... A096322.

			T(5,3) = 9 since these 9 solid partitions are [{{3}},{{1}},{{1}}], [{{2,1}},{{1}},{{1}}], [{{1,1,1}},{{1}},{{1}}], [{{2},{1}},{{1}},{{1}}], [{{1,1},{1}},{{1}},{{1}}], [{{1},{1},{1}},{{1}},{{1}}], [{{2}},{{2}},{{1}}], [{{1,1}},{{1,1}},{{1}}], [{{1},{1}},{{1},{1}},{{1}}].
Triangle begins:
   1;
   3,  1;
   6,  3,  1;
  13,  9,  3, 1;
  24, 22,  9, 3, 1;
  48, 54, 25, 9, 3, 1;
  ...

Wouter Meeussen, Table of n, a(n) for n = 1..136 (16 rows)
Wouter Meeussen, Mma functions for plane and solid partitions
Wouter Meeussen, Mma functions for plane and solid partitions

Cf. A000293, A090984, A089924, A096322.

(* uses "Mma functions for plane and solid partitions" also used in A090984, A089924 *)
 Table[Length/@Split[Sort[Length/@Flatten[solidformBTK/@Partitions[n]]]], {n, 16}]

Finding a G.f. for the solid partitions is an open problem.

Renewed linked Mma program file.Wouter Meeussen, Feb 20 2025

A096272 Triangle read by rows: T(n,k) counts solid partitions of n such that the maximum of planes, rows, columns and values is k.

Original entry on oeis.org

1, 0, 4, 0, 6, 4, 0, 10, 12, 4, 0, 13, 30, 12, 4, 0, 18, 70, 36, 12, 4, 0, 19, 142, 94, 36, 12, 4, 0, 24, 274, 234, 100, 36, 12, 4, 0, 19, 501, 534, 258, 100, 36, 12, 4, 0, 18, 872, 1186, 630, 264, 100, 36, 12, 4, 0, 13, 1449, 2486, 1482, 654, 264, 100, 36, 12, 4, 0, 10, 2336, 5080, 3346, 1578, 660, 264, 100, 36, 12, 4

Offset: 1

Wouter Meeussen, Jun 22 2004, Sep 21 2008

Solid partitions of n that fit inside a 4-dimensional k X k X k X k box. Regard solid partitions as safe pilings of boxes in a corner, stacking height does not increase away from the corner and each box contains an integer and this integer too does not increase away from the corner.

If k > 1+(n/2) then T(n,k) = T(n-1,k-1). For large n and k, each row ends as the reverse of 4, 12, 36, 100, 264, 660, 1608, 3772, 8652, 19340, 42392, 91140, 192860, 401880, 836480, ... = 4*A096322(i), i>=1.

			Triangle T(n,k) begins:
  1;
  0,  4;
  0,  6,  4;
  0, 10, 12,  4;
  0, 13, 30, 12,  4;
  0, 18, 70, 36, 12, 4;
  ...
T(16,2) = 1 because only { {{2,2},{2,2}}, {{2,2},{2,2}} } has only two planes, each plane has no more than 2 columns, each column no more than 2 rows and each element is no larger than 2.

Wouter Meeussen, Table of n, a(n) for n = 1..171 (18 rows)
Wouter Meeussen, Mma functions for plane and solid partitions

Cf. A000293, A007760, A094504, A094508, A096322.

Table[Count[Max[Max@(Flatten@(List@@#)),Max@@Map[Length,#,{-2}],Length/@List@@#,Length[#]]&/@Flatten[solidformBTK/@IntegerPartitions[n]] ,#]&/@Range[n],{n,1,12}]; (* see link for function definition *)

A000334 Number of 4-dimensional partitions of n.

Original entry on oeis.org

1, 5, 15, 45, 120, 326, 835, 2145, 5345, 13220, 32068, 76965, 181975, 425490, 982615, 2245444, 5077090, 11371250, 25235790, 55536870, 121250185, 262769080, 565502405, 1209096875, 2569270050, 5427963902, 11404408525, 23836421895, 49573316740, 102610460240

Offset: 1

N. J. A. Sloane

			From _Gus Wiseman_, Jan 23 2019: (Start)
The a(1) = 1 through a(3) = 15 four-dimensional partitions, represented as chains of chains of chains of integer partitions:
  (((1)))  (((2)))         (((3)))
           (((11)))        (((21)))
           (((1)(1)))      (((111)))
           (((1))((1)))    (((2)(1)))
           (((1)))(((1)))  (((11)(1)))
                           (((2))((1)))
                           (((1)(1)(1)))
                           (((11))((1)))
                           (((2)))(((1)))
                           (((1)(1))((1)))
                           (((11)))(((1)))
                           (((1))((1))((1)))
                           (((1)(1)))(((1)))
                           (((1))((1)))(((1)))
                           (((1)))(((1)))(((1)))
(End)

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..40
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. A000219 (2-dim), A000293 (3-dim), A000390 (5-dim), A096751 (k-dim).

Cf. A002974, A007714, A050340.

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[Length[levptns[n,4]],{n,8}] (* Gus Wiseman, Jan 24 2019 *)

More terms from Sean A. Irvine, Nov 14 2010

A096573 Number of fixed points of mirroring operation on solid partitions.

Original entry on oeis.org

1, 2, 4, 8, 13, 24, 39, 68, 110, 182, 288, 468, 728, 1150, 1770, 2751, 4175, 6388, 9597, 14495, 21571, 32200, 47498

Offset: 1

Wouter Meeussen, Jun 27 2004

Uses function "solidformBTK" from link below.

			Solid partition [{{3, 1, 1, 1}, {3}}, {{2, 1}}, {{1}}, {{1}}, {{1}}] mirrors into [{{3, 3}, {1}, {1}, {1}}, {{2}, {1}}, {{1}}, {{1}}, {{1}}] by mirroring each layer as a plane partition.

Wouter Meeussen, Mma functions for plane and solid partitions
Luigi Togliani, Forms of Crossed and Simple Polygons, Science & Philosophy (2019) Vol. 7, No. 2, 71-82.

Cf. A000293, A094504, A094508, A096272, A096574, A096575, A096576, A096577, A096578, A096579, A096580, A096581, A119266.

Tr/@Table[Count[solidformBTK[par],arg_z/;flip[arg]==arg],{n,20},{par,IntegerPartitions[n]}] (* Wouter Meeussen, Feb 05 2025  *)

a(16)-a(23) from Wouter Meeussen, Feb 05 2025

A096574 Number of asymmetric solid partitions under mirroring operation.

Original entry on oeis.org

0, 1, 3, 9, 23, 58, 134, 308, 677, 1470, 3106, 6479, 13260, 26827, 53516

Offset: 1

Wouter Meeussen, Jun 27 2004

Uses function "solidformBTK" from link above.

			Solid partition [{{3, 1, 1, 1}, {3}}, {{2, 1}}, {{1}}, {{1}}, {{1}}] mirrors into [{{3, 3}, {1}, {1}, {1}}, {{2}, {1}}, {{1}}, {{1}}, {{1}}] by mirroring each layer as a plane partition.

Wouter Meeussen, Mma functions for plane and solid partitions

Cf. A000293, A094504, A094508, A096272, A096573, A096575, A096576, A096577, A096578, A096579, A096580, A096581.

a(n) = (A000293(n) - A096573(n))/2.

A096575 Number of fixed points of solid partitions under rotation operation.

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 4, 6, 6, 8, 11, 13, 17, 24, 28, 36, 47, 56, 69, 94, 114, 138, 177, 218, 262

Offset: 1

Wouter Meeussen, Jun 27 2004

Rotation has permutation cycle length 1 or 3. Uses function "solidformBTK" from link below.
Is this the same sequence as A002722? - R. J. Mathar, Sep 04 2008 [This still seems to be true even after 20 terms. - N. J. A. Sloane, Feb 05 2025]
Rotation of each of the plane partitions in a solid partition appears to lead to the same count of fixed points as rotating the 3D-partition as a whole. - Wouter Meeussen, Feb 05 2025

			Solid partition [{{3, 1, 1, 1}, {3}}, {{2, 1}}, {{1}}, {{1}}, {{1}}] rotates into [{{4, 1}, {1, 1}, {1, 1}}, {{2}, {1}}, {{1}}, {{1}}, {{1}}] by rotating each layer as a plane partition.

Wouter Meeussen, Mma functions for plane and solid partitions

Cf. A000293, A094504, A094508, A096272, A096573, A096574, A096576, A096577, A096578, A096579, A096580, A096581, A119266.

Tr/@Table[Count[solidformBTK[par], arg_z /;turn[arg]==arg],{n,20}, {par, IntegerPartitions[n]}]

a(16)-a(23) from Wouter Meeussen, Feb 05 2025

a(24)-a(25) from Wouter Meeussen, Jul 27 2025

A094508 Triangle read by rows: T[n,m] = number of solid partitions of n with trace m, where the trace of a solid partitions is defined as the sum of the traces of the constituent plane partitions.

Original entry on oeis.org

1, 2, 2, 3, 4, 3, 4, 11, 6, 5, 5, 18, 19, 10, 7, 6, 33, 42, 34, 14, 11, 7, 48, 85, 80, 50, 22, 15, 8, 74, 156, 186, 128, 80, 30, 22, 9, 100, 275, 368, 318, 208, 112, 44, 30, 10, 140, 446, 725, 696, 534, 304, 165, 60, 42, 11, 180, 705, 1300, 1464, 1214, 808, 450, 228, 84, 56

Offset: 1

Wouter Meeussen, Jun 05 2004

Last column equals the partition numbers, corresponding to the 'single column' solid partitions.

			Table starts {1}, {2,2},{3,4,3},{4,11,6,5},..
T[4,3]=6 since these 6 solid partitions with trace 3 are:
[{{3,1}}], [{{3},{1}}], [{{2,1}},{{1}}], [{{2},{1}},{{1}}], [{{1,1}},{{1}},{{1}}], [{{1},{1}},{{1}},{{1}}]

Cf. A000293, A090984, A089924.

uses functions defined in A090984, A089924. solidform[q_?PartitionQ]:=Module[{}, Select[Flatten[Outer[z, Sequence@@(planepartitions/@q), 1]], And@@Apply[coversplaneQ, Partition[ #/.z->List, 2, 1], {1}]&]];tomatrix[par_]:=Block[{l=Max[Length/@ par]}, Map[PadRight[ #, l]&, par]]; Table[Length/@Split[Sort[Plus@@@Map[Tr[tomatrix[ # ]]&, Flatten[solidform/ @Partitions[n]], {2}]]], {n, 12}]

Finding a GF for the solid partitions is an open problem.

A096577 Number of fixed points of solid partitions under 'time-lapse' operation.

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 1, 0, 2, 0, 2, 0, 2, 0, 4, 1, 5, 0, 7, 1, 7, 0, 14

Offset: 1

Wouter Meeussen, Jun 27 2004

Operation 'time lapse', or 'lapse', L, operates on a solid partition by creating a new one, layer by layer. Layer k is defined by its 3-dimensional-Ferrers plot, equal to the (existence of) elements of the solid partition with value >= k. As if taking a time-lapse picture of the solid partition, filtering out elements less than k and projecting the resulting structure (filled with ones) to the base plane. Given there are three planes to project into, together with the starting solid partition, that makes four 'isomers'.

			Solid partition [{{3,1,1,1},{3}},{{2,1}},{{1}},{{1}},{{1}}] lapses (L) into
[{{4,1},{2},{1},{1},{1}},{{1,1},{1}},{{1,1}}], then into
[{{2,1,1,1,1},{2,1},{2}},{{1,1}},{{1}},{{1}}], further into
[{{5,2,1},{2},{1},{1}},{{1,1,1}}] and returns after L^4 to
[{{3,1,1,1},{3}},{{2,1}},{{1}},{{1}},{{1}}].

Wouter Meeussen, Mma functions for plane and solid partitions

Cf. A000293, A094504, A094508, A096272, A096573, A096574, A096575, A096576, A096578, A096579, A096580, A096581, A119266.

(* See link above. *)
Tr/@Table[Count[solidformBTK[par],arg_z/;lapse[arg]==arg],{n,20},{par,IntegerPartitions[n]}] (* Wouter Meeussen, Feb 05 2025 *)

a(16)-a(23) from Wouter Meeussen, Mar 19 2025

A119266 Number of 3-dimensional partitions of n up to conjugacy.

Original entry on oeis.org

1, 1, 1, 2, 4, 7, 13, 25, 49, 93, 181, 351, 687, 1332, 2591, 5003, 9644, 18462, 35208, 66721, 125840, 235914, 440020, 816122, 1505986, 2764303, 5048960, 9176069

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, A000293.
Cf. A119338.
Cf. A096573, A096575, A382247, A096577

a(n) = (A000293(n) + 6*A096573(n) + 8*A096575(n) + 3*A382247(n) + 6*A096577(n))/24 by Burnside's lemma. - Wouter Meeussen, Mar 19 2025

a(9)-a(23) from Max Alekseyev, May 15 2006

a(24)-a(27) from Max Alekseyev, Mar 20 2025

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

Extensions

A094504 T(n,m) equals number of solid partitions of n containing m plane partitions.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A096272 Triangle read by rows: T(n,k) counts solid partitions of n such that the maximum of planes, rows, columns and values is k.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A000334 Number of 4-dimensional partitions of n.

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Mathematica

Extensions

A096573 Number of fixed points of mirroring operation on solid partitions.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

A096574 Number of asymmetric solid partitions under mirroring operation.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A096575 Number of fixed points of solid partitions under rotation operation.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

A094508 Triangle read by rows: T[n,m] = number of solid partitions of n with trace m, where the trace of a solid partitions is defined as the sum of the traces of the constituent plane partitions.

Views

Author

Keywords

Comments