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Triangle read by rows: T[n,m] = number of solid partitions of n with trace m, where the trace of a solid partitions is defined as the sum of the traces of the constituent plane partitions.
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%I #6 Mar 30 2012 18:37:44

%S 1,2,2,3,4,3,4,11,6,5,5,18,19,10,7,6,33,42,34,14,11,7,48,85,80,50,22,

%T 15,8,74,156,186,128,80,30,22,9,100,275,368,318,208,112,44,30,10,140,

%U 446,725,696,534,304,165,60,42,11,180,705,1300,1464,1214,808,450,228,84,56

%N Triangle read by rows: T[n,m] = number of solid partitions of n with trace m, where the trace of a solid partitions is defined as the sum of the traces of the constituent plane partitions.

%C Last column equals the partition numbers, corresponding to the 'single column' solid partitions.

%F Finding a GF for the solid partitions is an open problem.

%e Table starts {1}, {2,2},{3,4,3},{4,11,6,5},..

%e T[4,3]=6 since these 6 solid partitions with trace 3 are:

%e [{{3,1}}], [{{3},{1}}], [{{2,1}},{{1}}], [{{2},{1}},{{1}}], [{{1,1}},{{1}},{{1}}], [{{1},{1}},{{1}},{{1}}]

%t uses functions defined in A090984, A089924. solidform[q_?PartitionQ]:=Module[{}, Select[Flatten[Outer[z, Sequence@@(planepartitions/@q), 1]], And@@Apply[coversplaneQ, Partition[ #/.z->List, 2, 1], {1}]&]];tomatrix[par_]:=Block[{l=Max[Length/@ par]}, Map[PadRight[ #, l]&, par]]; Table[Length/@Split[Sort[Plus@@@Map[Tr[tomatrix[ # ]]&, Flatten[solidform/ @Partitions[n]], {2}]]], {n, 12}]

%Y Cf. A000293, A090984, A089924.

%K nonn,tabl

%O 1,2

%A _Wouter Meeussen_, Jun 05 2004