A097391 -id:A097391 - 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

A097392 The number of hierarchies of n labeled elements with at least one subhierarchy composed of exactly 3 levels and no subhierarchy with more than 3 levels.

Original entry on oeis.org

0, 0, 1, 4, 12, 32, 78, 183, 408, 886

Offset: 1

Thomas Wieder, Aug 13 2004

			Let : denote the separation between two subhierarchies, e.g. 2:3 are two subhierarchies where subhierarchy s=1 contains two elements and subhierarchy s=2 contains three elements. Let | denote the separation between two levels, e.g. 2|2|1 is a hierarchy composed of three levels with two elements on levels l=1 and l=2 and one element on level l=3. For n=5 one has a(5) = 12 hierarchies where at least one subhierarchy has exactly 3 levels (and no level l > 3 is allowed):
3|1|1; 1|3|1; 1|1|3; 2|2|1; 2|1|2; 1|2|2; 1|1|1:2; 1|1|1:1:1; 1|1|1:1|1; 2|1|1:1; 1|2|1:1; 1|1|2:1.

N. J. A. Sloane and Thomas Wieder, The Number of Hierarchical Orderings, arXiv:math/0307064 [math.CO], 2003; Order 21 (2004), 83-89.

Cf. A034691, A097237, A097391, A000041.

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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A097392 The number of hierarchies of n labeled elements with at least one subhierarchy composed of exactly 3 levels and no subhierarchy with more than 3 levels.

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

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Mathematica