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

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

Author

Wouter Meeussen, Jun 05 2004

Keywords

• nonn
• tabl

Comments

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.

Examples

			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;
  ...
		

Links

• 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

Crossrefs

Cf. A000293, A090984, A089924, A096322.

Programs

Mathematica

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

Formula

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

Extensions

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