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A380893 Triangle read by rows: T(n,m) = number of solid partitions of n with shape of a plane partition of m.

%I A380893 #11 Feb 10 2025 01:05:29
%S A380893 1,1,3,1,3,6,1,6,6,13,1,6,15,13,24,1,9,21,37,24,48,1,9,30,58,75,48,86,
%T A380893 1,12,39,95,132,159,86,160,1,12,54,128,231,297,299,160,282,1,15,63,
%U A380893 197,345,552,593,574,282,500,1,15,81,251,546,873,1156,1180,1038,500,859,1,18,96,345,771,1452,1933,2390,2208,1874,859,1479,1,18,114,432,1110,2151,3340,4154,4614,4082,3268,1479,2485,1,21,132,558,1491,3276,5214,7430,8310,8758,7276,5685,2485,4167
%N A380893 Triangle read by rows: T(n,m) = number of solid partitions of n with shape of a plane partition of m.
%C A380893 A solid (or 3D) partition of n describes a piling of boxes in a corner with heights nonincreasing away from the corner, and containing integers, similarly nonincreasing, that sum to n.
%C A380893 The shape of a solid partitions is defined as the plane partition containing the heights of the piling, irrespective of the numerical content of the boxes.
%C A380893 Row sums equal A000293, T(n,n) = T(n+1,n) equals A000219;
%C A380893 Equals number of solid partitions with total by layer equal to partitions of n with largest part m.
%H A380893 Wouter Meeussen, <a href="https://oeis.org/A094504/a094504_1.txt">Mma functions for plane and solid partitions</a>
%e A380893 Table starts as:
%e A380893   1,
%e A380893   1,3
%e A380893   1,3,6
%e A380893   1,6,6,13
%e A380893   1,6,15,13,24
%e A380893   1,9,21,37,24,48
%e A380893 T(4,2) = 6 since the solid partitions of 4 with shapes a plane partition of 2 are:
%e A380893  z[{{2,2}}], z[{{3,1}}], z[{{2},{2}}], z[{{3},{1}}], z[{{3}},{{1}}], z[{{2}},{{2}}]
%e A380893 with shapes equal to these plane partitions:
%e A380893   {{2}}, {{2}}, {{1,1}}, {{1,1}}, {{1},{1}}, {{1},{1}}
%t A380893 Table[Tr@(Count[First[lapse[#]]&/@Flatten[sols=Table[solidformBTK[par],{par,IntegerPartitions[n]}] ],#]&/@planepartitions[k]),{n,10},{k,n}] (* using functions from link above, or with the faster second program: *)
%t A380893 Table[ Sum[Length[solidformBTK[TransposePartition@par]],{par,IntegerPartitions[n,{k}]} ],{n,16},{k,n}] (* with transposePartition[par:{_Integer..}]:=Count[par,i_/;i>=#]&/@Range[Max[par]] *)
%Y A380893 Cf. A000219, A000293, A094504.
%K A380893 nonn,tabl
%O A380893 1,3
%A A380893 _Wouter Meeussen_, Feb 07 2025