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%I #13 Aug 20 2017 23:16:56
%S 1,0,4,0,6,4,0,10,12,4,0,13,30,12,4,0,18,70,36,12,4,0,19,142,94,36,12,
%T 4,0,24,274,234,100,36,12,4,0,19,501,534,258,100,36,12,4,0,18,872,
%U 1186,630,264,100,36,12,4,0,13,1449,2486,1482,654,264,100,36,12,4,0,10,2336
%N Triangle read by rows: T(n,k) counts solid partitions of n such that the maximum of planes, rows, columns and values is k.
%C Solid partitions of n that fit inside a 4-dimensional k X k X k X k box. Regard solid partitions as safe pilings of boxes in a corner, stacking height does not increase away from the corner and each box contains an integer and this integer too does not increase away from the corner.
%C If k > 1+(n/2) then T(n,k) = T(n-1,k-1). For large n and k, each row ends as the reverse of 4, 12, 36, 100, 264, 660, 1608, 3772, 8652, 19340, 42392, 91140, 192860, 401880, 836480, ... = 4*A096322(i), i>=1.
%H Wouter Meeussen, <a href="http://users.pandora.be/Wouter.Meeussen/SolidPartitions.txt">SolidPartitions.txt</a>
%e Triangle T(n,k) begins:
%e 1;
%e 0, 4;
%e 0, 6, 4;
%e 0, 10, 12, 4;
%e 0, 13, 30, 12, 4;
%e 0, 18, 70, 36, 12, 4;
%e ...
%e T(16,2) = 1 because only { {{2,2},{2,2}}, {{2,2},{2,2}} } has only two planes, each plane has no more than 2 columns, each column no more than 2 rows and each element is no larger than 2.
%t Max[ Max @(Flatten@(List @@ #)), Max @@ Map[Length, #, {-2}], Length /@ List @@ #, Length[ # ]] & /@ Flatten[solidformBTK /@ Partitions[n]]]], {n, 12}]; (* see link for function definition *)
%Y Cf. A000293, A007760, A094504, A094508, A096322.
%K nonn,tabl
%O 1,3
%A _Wouter Meeussen_, Jun 22 2004, Sep 21 2008
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