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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
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).
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;
...
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
KEYWORD
nonn,tabf,hard,more
AUTHOR
Wouter Meeussen, Sep 11 2004
STATUS
approved