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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.

%I A098052 #19 Feb 03 2025 17:03:27
%S A098052 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 A098052 12,22,20,102,114,232,76,66,68,6,10,114,231,448,232,180,201,48,16,204,
%U A098052 330,728,628,462,546,184,24
%N 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.
%C A098052 Row sums are A000293 (solid partitions) by definition.
%C A098052 First column is conjectured to be A007426 = tau_4(n).
%C A098052 All solid partitions can be extended in at least 4 ways (hence the offset 4).
%H A098052 Wouter Meeussen, <a href="/A094504/a094504_1.txt">Mma functions for plane and solid partitions</a>
%e A098052 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 A098052   [{{2,1},{2}},{{1}}], [{{2,1},{1,1}},{{1}}], [{{2,2},{1}},{{1}}],
%e A098052   [{{3,1},{1}},{{1}}], [{{2,1,1},{1}},{{1}}], [{{2,1},{1},{1}},{{1}}],
%e A098052   [{{2,1},{1}},{{2}}], [{{2,1},{1}},{{1,1}}], [{{2,1},{1}},{{1},{1}}],
%e A098052   [{{2,1},{1}},{{1}},{{1}}].
%e A098052 Table starts
%e A098052   1;
%e A098052   4;
%e A098052   4,6;
%e A098052   10,12,0,4;
%e A098052   4,30,12,12,0,0,1;
%e A098052   16,48,18,48,0,6,4;
%e A098052   4,70,72,100,27,12,22;
%e A098052   20,102,114,232,76,66,68,6;
%e A098052   ...
%t A098052 (* functions 'solidform' and 'coversplaneQ', see A096574 *)
%t A098052 Table[ Rest@BinCounts[Count[Flatten[solidformBTK/@IntegerPartitions[n+1]],q_/;coverssolidQ[q,#]]&/@Flatten[solidformBTK/@IntegerPartitions[n]]] ,{n,1,8}] (* _Wouter Meeussen_, Feb 03 2025 *)
%Y A098052 Cf. A000029, A007426, A097994.
%K A098052 nonn,tabf,hard,more
%O A098052 4,2
%A A098052 _Wouter Meeussen_, Sep 11 2004