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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
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

4,2

COMMENTS

Table starts 1; 4; 4,6; 10,12,0,4; 4,30,12,12,0,0,1; .. Row sums are A000293 (solid partitions) by definition. First column 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, SolidPartitions.xls

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}}]

MATHEMATICA

(* functions 'solidform' and 'coversplaneQ', see A096574 *) coverssolidQ[par_z, chi_z]:=Module[{p, c}, p=Length[par]; c=Length[chi]; And[p>=c, And@@MapThread[coversplaneQ, {List@@Take[par, c], List@@chi}]]]; Table[Frequencies[Count[Flatten[solidform/@Partitions[n+1]], q_/; coverssolidQ[q, # ]]&/ @ Flatten[solidform/@Partitions[n]]], {n, 1, 5}]

CROSSREFS

Cf. A000029, A007426, A097994.

Sequence in context: A127799 A213375 A226834 * A098530 A213379 A163976

Adjacent sequences:  A098049 A098050 A098051 * A098053 A098054 A098055

KEYWORD

hard,more,nonn,tabf

AUTHOR

Wouter Meeussen, Sep 11 2004

STATUS

approved

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Last modified May 15 17:11 EDT 2021. Contains 343920 sequences. (Running on oeis4.)