A094504 -id:A094504 - cp's OEIS frontend

A378854 a(n) is the number of solid (3D) partitions of n with 2 layers and second layer a plane partition of 2.

3, 9, 24, 51, 111, 213, 414, 756, 1374, 2409, 4206, 7152, 12096, 20109, 33198, 54111, 87648, 140520, 223953, 354006, 556419, 868584, 1348857, 2082225, 3198927, 4888944, 7438548, 11265141, 16990077, 25516401, 38175240, 56894490, 84490935, 125028489, 184400952

Offset: 4

Wouter Meeussen, Feb 10 2025

nonn

Appears to equal 3*A097391(n-3).

			a(4)=3 since the 3 solid partitions of {2,2} are:
  [{{2}},{{2}}], [{{1,1}},{{1,1}}], [{{1},{1}},{{1},{1}}]
a(5)=9 since the 9 solid partitions of {3,2} are:
  [{{3}},{{2}}], [{{2,1}},{{2}}], [{{2,1}},{{1,1}}], [{{1,1,1}},{{1,1}}], [{{2},{1}},{{2}}], [{{2},{1}},{{1},{1}}],[{{1,1},{1}},{{1,1}}], [{{1,1},{1}},{{1},{1}}], [{{1},{1},{1}},{{1},{1}}]

Wouter Meeussen Mma functions for plane and solid partitions

Cf. A097391, A094504, A000219, A000041.

Table[Length@solidformBTK[{n-2,2}],{n,4,22}] (* uses functions defined in link above *)

A_x(N) = {my(x='x+O('x^N)); Vec(3*x^2*(prod(i=1,N, 1/(1-x^i)^i)-prod(i=1,N, 1/(1-x^i))))}
A_x(40) \\ John Tyler Rascoe, Feb 20 2025

From John Tyler Rascoe, Feb 20 2025: (Start)
a(n) = 3*(A000219(n-2) - A000041(n-2)).
G.f.: 3*x^2 * (Product_{i>0} (1/(1-x^i)^i) - Product_{i>0} (1/(1-x^i)^i)). (End)

a(23) onwards from John Tyler Rascoe, Feb 20 2025

A098052 T(n,k) counts the solid partitions of n that can be extended to a solid partition of n+1 in exactly (k+3) ways. Equivalently, the number of solid partitions of n that have exactly k+3 partitions of n+1 majoring them.

Original entry on oeis.org

1, 4, 4, 6, 10, 12, 0, 4, 4, 30, 12, 12, 0, 0, 1, 16, 48, 18, 48, 0, 6, 4, 4, 70, 72, 100, 27, 12, 22, 20, 102, 114, 232, 76, 66, 68, 6, 10, 114, 231, 448, 232, 180, 201, 48, 16, 204, 330, 728, 628, 462, 546, 184, 24

Offset: 4

Wouter Meeussen, Sep 11 2004

Row sums are A000293 (solid partitions) by definition.
First column is conjectured to be A007426 = tau_4(n).
All solid partitions can be extended in at least 4 ways (hence the offset 4).

			T(5,7)=1 because there is only 1 solid partition of 5 [{{2, 1}, {1}}, {{1}}] that can be extended to a solid partition of 6 in exactly (7+3 =10) ways:
  [{{2,1},{2}},{{1}}], [{{2,1},{1,1}},{{1}}], [{{2,2},{1}},{{1}}],
  [{{3,1},{1}},{{1}}], [{{2,1,1},{1}},{{1}}], [{{2,1},{1},{1}},{{1}}],
  [{{2,1},{1}},{{2}}], [{{2,1},{1}},{{1,1}}], [{{2,1},{1}},{{1},{1}}],
  [{{2,1},{1}},{{1}},{{1}}].
Table starts
  1;
  4;
  4,6;
  10,12,0,4;
  4,30,12,12,0,0,1;
  16,48,18,48,0,6,4;
  4,70,72,100,27,12,22;
  20,102,114,232,76,66,68,6;
  ...

Wouter Meeussen, Mma functions for plane and solid partitions

Cf. A000029, A007426, A097994.

(* functions 'solidform' and 'coversplaneQ', see A096574 *)
Table[ Rest@BinCounts[Count[Flatten[solidformBTK/@IntegerPartitions[n+1]],q_/;coverssolidQ[q,#]]&/@Flatten[solidformBTK/@IntegerPartitions[n]]] ,{n,1,8}] (* Wouter Meeussen, Feb 03 2025 *)

A382247 Number of fixed points of solid partitions under twice the 'time-lapse' operation.

Original entry on oeis.org

1, 0, 2, 2, 3, 4, 7, 12, 16, 22, 32, 50, 68, 96, 134, 195, 261, 364, 497, 701, 941, 1288, 1738

Offset: 1

Wouter Meeussen, Mar 19 2025

Permutes the 4 axes of the 4D-Ferrers plot of the solid partitions as 2143.

			z[{{2},{2}}] -> z[{{1,1}},{{1,1}}] -> z[{{2},{2}}] under the 'lapse' operation.

Wouter Meeussen, Mma functions for plane and solid partitions

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

Tr/@Table[Count[solidformBTK[par], arg_z/; Nest[lapse,arg,2]===arg], {n, 20}, {par, IntegerPartitions[n]}]

A380893 Triangle read by rows: T(n,m) = number of solid partitions of n with shape of a plane partition of m.

Original entry on oeis.org

1, 1, 3, 1, 3, 6, 1, 6, 6, 13, 1, 6, 15, 13, 24, 1, 9, 21, 37, 24, 48, 1, 9, 30, 58, 75, 48, 86, 1, 12, 39, 95, 132, 159, 86, 160, 1, 12, 54, 128, 231, 297, 299, 160, 282, 1, 15, 63, 197, 345, 552, 593, 574, 282, 500, 1, 15, 81, 251, 546, 873, 1156, 1180, 1038, 500, 859, 1, 18, 96, 345, 771, 1452, 1933, 2390, 2208, 1874, 859, 1479, 1, 18, 114, 432, 1110, 2151, 3340, 4154, 4614, 4082, 3268, 1479, 2485, 1, 21, 132, 558, 1491, 3276, 5214, 7430, 8310, 8758, 7276, 5685, 2485, 4167

Offset: 1

Wouter Meeussen, Feb 07 2025

A solid (or 3D) partition of n describes a piling of boxes in a corner with heights nonincreasing away from the corner, and containing integers, similarly nonincreasing, that sum to n.
The shape of a solid partitions is defined as the plane partition containing the heights of the piling, irrespective of the numerical content of the boxes.
Row sums equal A000293, T(n,n) = T(n+1,n) equals A000219;
Equals number of solid partitions with total by layer equal to partitions of n with largest part m.

			Table starts as:
  1,
  1,3
  1,3,6
  1,6,6,13
  1,6,15,13,24
  1,9,21,37,24,48
T(4,2) = 6 since the solid partitions of 4 with shapes a plane partition of 2 are:
 z[{{2,2}}], z[{{3,1}}], z[{{2},{2}}], z[{{3},{1}}], z[{{3}},{{1}}], z[{{2}},{{2}}]
with shapes equal to these plane partitions:
  {{2}}, {{2}}, {{1,1}}, {{1,1}}, {{1},{1}}, {{1},{1}}

Wouter Meeussen, Mma functions for plane and solid partitions

Cf. A000219, A000293, A094504.

Table[Tr@(Count[First[lapse[#]]&/@Flatten[sols=Table[solidformBTK[par],{par,IntegerPartitions[n]}] ],#]&/@planepartitions[k]),{n,10},{k,n}] (* using functions from link above, or with the faster second program: *)
Table[ Sum[Length[solidformBTK[TransposePartition@par]],{par,IntegerPartitions[n,{k}]} ],{n,16},{k,n}] (* with transposePartition[par:{Integer..}]:=Count[par,i/;i>=#]&/@Range[Max[par]] *)

A381265 a(n) is the number of solid (3D) partitions of n with 2 layers and second layer a plane partition of 3.

Original entry on oeis.org

6, 21, 57, 138, 294, 606, 1170, 2208, 4008, 7176, 12492, 21510, 36348, 60801, 100281, 164019, 265263, 425853

Offset: 3

Wouter Meeussen, Feb 18 2025

Conjecture: equal to 3*(2*A000219 -A000990 -2*A000041 +1) tested up to n=20.

			a(3)=6 since the 6 solid partitions of {3,3} are:
  z[{{3}},{{3}}],
  z[{{2,1}},{{2,1}}],
  z[{{1,1,1}},{{1,1,1}}],z[{{2},{1}},{{2},{1}}],
  z[{{1,1},{1}},{{1,1},{1}}],
  z[{{1},{1},{1}},{{1},{1},{1}}].

Wouter Meeussen, Mma functions for plane and solid partitions

Cf. A000041, A000219, A000990, A378854.

Table[Length@solidformBTK[{n,3}],{n,3,20}] (* or *)
g=20;3 CoefficientList[Series[2/Product[(1-x^m)^m,{m,g}]+ 1/(1-x)-(1-x)/Product[(1-x^m)^2,{m,g}]-2/Product[(1-x^m),{m,g}],{x,0,g}],x]

G.f.: 3*(2*Product_{k>0} 1/(1-x^k)^k -(1-x)*Product_{k>0} 1/(1-x^k)^2 - 2*Product_{k>0} 1/(1-x^k) + 1/(1 - x)) (conjectured).

A381332 a(n) is the number of different hooklength lists of the plane partitions of n.

Original entry on oeis.org

1, 1, 2, 4, 6, 11, 19, 31, 52, 86, 146, 231, 392, 615, 1006, 1594, 2612, 4062, 6518, 10116, 15958, 24557, 38565, 58548

Offset: 1

Wouter Meeussen, Feb 20 2025

The hooklength list of a plane partition is the sorted list of 3D hooklengths of its 3D Ferrers plot, analogous to the classic 2D case.

			The plane partition {{2,1},{2}} has hooklengths {{{4,2},{1}},{{2,1}}} and so hooklength list is {4,2,2,1,1}. So a(2) = 1.
The 24 plane partitions of n=5 generate only these 6 hooklength lists: {4,2,2,1,1}, {4,3,2,1,1}, {5,2,1,1,1}, {5,2,2,1,1}, {5,3,2,1,1}, {5,4,3,2,1}. So a(5) = 6.

Wouter Meeussen, Mma functions for plane and solid partitions

Cf. A097391, A094504, A000219, A000041.

Table[Length[Union[planehooks/@planepartitions[n]]],{n,20}]

cp's OEIS Frontend

A378854 a(n) is the number of solid (3D) partitions of n with 2 layers and second layer a plane partition of 2.

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A098052 T(n,k) counts the solid partitions of n that can be extended to a solid partition of n+1 in exactly (k+3) ways. Equivalently, the number of solid partitions of n that have exactly k+3 partitions of n+1 majoring them.

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A382247 Number of fixed points of solid partitions under twice the 'time-lapse' operation.

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A380893 Triangle read by rows: T(n,m) = number of solid partitions of n with shape of a plane partition of m.

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A381265 a(n) is the number of solid (3D) partitions of n with 2 layers and second layer a plane partition of 3.

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Formula

A381332 a(n) is the number of different hooklength lists of the plane partitions of n.

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