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A096272
Triangle read by rows: T(n,k) counts solid partitions of n such that the maximum of planes, rows, columns and values is k.
10
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, 4, 0, 24, 274, 234, 100, 36, 12, 4, 0, 19, 501, 534, 258, 100, 36, 12, 4, 0, 18, 872, 1186, 630, 264, 100, 36, 12, 4, 0, 13, 1449, 2486, 1482, 654, 264, 100, 36, 12, 4, 0, 10, 2336
OFFSET
1,3
COMMENTS
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.
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.
EXAMPLE
Triangle T(n,k) begins:
1;
0, 4;
0, 6, 4;
0, 10, 12, 4;
0, 13, 30, 12, 4;
0, 18, 70, 36, 12, 4;
...
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.
MATHEMATICA
Max[ Max @(Flatten@(List @@ #)), Max @@ Map[Length, #, {-2}], Length /@ List @@ #, Length[ # ]] & /@ Flatten[solidformBTK /@ Partitions[n]]]], {n, 12}]; (* see link for function definition *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Wouter Meeussen, Jun 22 2004, Sep 21 2008
STATUS
approved