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.

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, 5080, 3346, 1578, 660, 264, 100, 36, 12, 4

Offset: 1

Author

Wouter Meeussen, Jun 22 2004, Sep 21 2008

Keywords

• nonn
• tabl

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.

Examples

			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.
		

Links

• Wouter Meeussen, Table of n, a(n) for n = 1..171 (18 rows)
• Wouter Meeussen, Mma functions for plane and solid partitions

Crossrefs

Cf. A000293, A007760, A094504, A094508, A096322.

Programs

Mathematica

Table[Count[Max[Max@(Flatten@(List@@#)),Max@@Map[Length,#,{-2}],Length/@List@@#,Length[#]]&/@Flatten[solidformBTK/@IntegerPartitions[n]] ,#]&/@Range[n],{n,1,12}]; (* see link for function definition *)