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

4,2

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).

Links

Table of n, a(n) for n=4..57.

Wouter Meeussen, Mma functions for plane and solid partitions

Example

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;
  ...

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 *)

Crossrefs

Cf. A000029, A007426, A097994.

Sequence in context: A127799 A213375 A226834 * A098530 A213379 A163976

Adjacent sequences: A098049 A098050 A098051 * A098053 A098054 A098055

Keyword

nonn,tabf,hard,more

Author

Wouter Meeussen, Sep 11 2004

Status

approved