A098052 T(n,k) counts the solid partitions of n that can be extended to a solid partition of n+1 in exactly (k+3) ways. Equivalently, the number of solid partitions of n that have exactly k+3 partitions of n+1 majoring them.
1, 4, 4, 6, 10, 12, 0, 4, 4, 30, 12, 12, 0, 0, 1, 16, 48, 18, 48, 0, 6, 4, 4, 70, 72, 100, 27, 12, 22, 20, 102, 114, 232, 76, 66, 68, 6, 10, 114, 231, 448, 232, 180, 201, 48, 16, 204, 330, 728, 628, 462, 546, 184, 24
Offset: 4
Examples
T(5,7)=1 because there is only 1 solid partition of 5 [{{2, 1}, {1}}, {{1}}] that can be extended to a solid partition of 6 in exactly (7+3 =10) ways:
[{{2,1},{2}},{{1}}], [{{2,1},{1,1}},{{1}}], [{{2,2},{1}},{{1}}],
[{{3,1},{1}},{{1}}], [{{2,1,1},{1}},{{1}}], [{{2,1},{1},{1}},{{1}}],
[{{2,1},{1}},{{2}}], [{{2,1},{1}},{{1,1}}], [{{2,1},{1}},{{1},{1}}],
[{{2,1},{1}},{{1}},{{1}}].
Table starts
1;
4;
4,6;
10,12,0,4;
4,30,12,12,0,0,1;
16,48,18,48,0,6,4;
4,70,72,100,27,12,22;
20,102,114,232,76,66,68,6;
...
Links
- Wouter Meeussen, Mma functions for plane and solid partitions
Programs
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Mathematica
(* functions 'solidform' and 'coversplaneQ', see A096574 *) Table[ Rest@BinCounts[Count[Flatten[solidformBTK/@IntegerPartitions[n+1]],q_/;coverssolidQ[q,#]]&/@Flatten[solidformBTK/@IntegerPartitions[n]]] ,{n,1,8}] (* Wouter Meeussen, Feb 03 2025 *)
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