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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.

%I A096272 #24 Feb 03 2025 19:56:21
%S A096272 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 A096272 4,0,24,274,234,100,36,12,4,0,19,501,534,258,100,36,12,4,0,18,872,
%U A096272 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
%N 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.
%C A096272 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 A096272 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 A096272 Wouter Meeussen, <a href="/A096272/b096272.txt">Table of n, a(n) for n = 1..171</a> (18 rows)
%H A096272 Wouter Meeussen, <a href="/A094504/a094504_1.txt">Mma functions for plane and solid partitions</a>
%e A096272 Triangle T(n,k) begins:
%e A096272   1;
%e A096272   0,  4;
%e A096272   0,  6,  4;
%e A096272   0, 10, 12,  4;
%e A096272   0, 13, 30, 12,  4;
%e A096272   0, 18, 70, 36, 12, 4;
%e A096272   ...
%e A096272 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 A096272 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 *)
%Y A096272 Cf. A000293, A007760, A094504, A094508, A096322.
%K A096272 nonn,tabl
%O A096272 1,3
%A A096272 _Wouter Meeussen_, Jun 22 2004, Sep 21 2008