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

1,2

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.

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

Formula

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

Example

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;
  ...

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}]

Crossrefs

Cf. A000293, A090984, A089924, A096322.

Sequence in context: A130452 A133085 A039805 * A107884 A185628 A321194

Adjacent sequences: A094501 A094502 A094503 * A094505 A094506 A094507

Keyword

nonn,tabl

Author

Wouter Meeussen, Jun 05 2004

Extensions

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

Status

approved