A094508 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.

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, 15, 8, 74, 156, 186, 128, 80, 30, 22, 9, 100, 275, 368, 318, 208, 112, 44, 30, 10, 140, 446, 725, 696, 534, 304, 165, 60, 42, 11, 180, 705, 1300, 1464, 1214, 808, 450, 228, 84, 56

Offset

1,2

Comments

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

Links

Table of n, a(n) for n=1..66.

Formula

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

Example

Table starts {1}, {2,2},{3,4,3},{4,11,6,5},..
T[4,3]=6 since these 6 solid partitions with trace 3 are:
[{{3,1}}], [{{3},{1}}], [{{2,1}},{{1}}], [{{2},{1}},{{1}}], [{{1,1}},{{1}},{{1}}], [{{1},{1}},{{1}},{{1}}]

Mathematica

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}]

Crossrefs

Cf. A000293, A090984, A089924.

Sequence in context: A049787 A084192 A129595 * A183517 A080046 A047675

Adjacent sequences: A094505 A094506 A094507 * A094509 A094510 A094511

Keyword

nonn,tabl

Author

Wouter Meeussen, Jun 05 2004

Status

approved