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%I #11 Oct 28 2021 06:30:17
%S 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,
%T 12,22,20,102,114,232,76,66,68,6,10,114,231,448,232,180,201,48,16,204,
%U 330,728,628,462,546,184,24
%N 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.
%C Row sums are A000293 (solid partitions) by definition.
%C First column is conjectured to be A007426 = tau_4(n).
%C All solid partitions can be extended in at least 4 ways (hence the offset 4).
%H Wouter Meeussen, <a href="http://users.pandora.be/Wouter.Meeussen/solidPartitions.xls">SolidPartitions.xls</a>
%e 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:
%e [{{2,1},{2}},{{1}}], [{{2,1},{1,1}},{{1}}], [{{2,2},{1}},{{1}}],
%e [{{3,1},{1}},{{1}}], [{{2,1,1},{1}},{{1}}], [{{2,1},{1},{1}},{{1}}],
%e [{{2,1},{1}},{{2}}], [{{2,1},{1}},{{1,1}}], [{{2,1},{1}},{{1},{1}}],
%e [{{2,1},{1}},{{1}},{{1}}].
%e Table starts
%e 1;
%e 4;
%e 4,6;
%e 10,12,0,4;
%e 4,30,12,12,0,0,1;
%e ...
%t (* 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}]
%Y Cf. A000029, A007426, A097994.
%K nonn,tabf,hard,more
%O 4,2
%A _Wouter Meeussen_, Sep 11 2004
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