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A367094
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Irregular triangle read by rows with trailing zeros removed where T(n,k) is the number of integer partitions of 2n whose number of submultisets summing to n is k.
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16
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0, 1, 1, 1, 2, 2, 1, 5, 3, 3, 8, 4, 9, 1, 17, 6, 16, 1, 2, 24, 7, 33, 4, 9, 46, 11, 52, 3, 18, 1, 4, 64, 12, 91, 6, 38, 3, 15, 1, 1, 107, 17, 138, 9, 68, 2, 28, 2, 12, 0, 2, 147, 19, 219, 12, 117, 6, 56, 3, 34, 2, 9, 0, 3
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OFFSET
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0,5
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LINKS
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FORMULA
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EXAMPLE
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The partition (3,2,2,1) has two submultisets summing to 4, namely {2,2} and {1,3}, so it is counted under T(4,2).
The partition (2,2,1,1,1,1) has three submultisets summing to 4, namely {1,1,1,1}, {1,1,2}, and {2,2}, so it is counted under T(4,3).
Triangle begins:
0 1
1 1
2 2 1
5 3 3
8 4 9 1
17 6 16 1 2
24 7 33 4 9
46 11 52 3 18 1 4
64 12 91 6 38 3 15 1 1
107 17 138 9 68 2 28 2 12 0 2
147 19 219 12 117 6 56 3 34 2 9 0 3
Row n = 4 counts the following partitions:
(8) (44) (431) (221111)
(71) (3311) (422)
(62) (2222) (4211)
(611) (11111111) (41111)
(53) (3221)
(521) (32111)
(5111) (311111)
(332) (22211)
(2111111)
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MATHEMATICA
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t=Table[Length[Select[IntegerPartitions[2n], Count[Total/@Union[Subsets[#]], n]==k&]], {n, 0, 5}, {k, 0, 1+PartitionsP[n]}];
Table[NestWhile[Most, t[[i]], Last[#]==0&], {i, Length[t]}]
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CROSSREFS
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The corresponding rank statistic is A357879 (without empty rows).
A182616 counts partitions of 2n with at least one odd part, ranks A366530.
A365543 counts partitions of n with a submultiset summing to k.
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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