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.

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

Author

Wouter Meeussen, Sep 11 2004

Keywords

• nonn
• tabf
• hard
• more

Comments

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).

Examples

			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;
  ...
		

Links

• Wouter Meeussen, Mma functions for plane and solid partitions

Crossrefs

Cf. A000029, A007426, A097994.

Programs

Mathematica

(* 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 *)