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A365923
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Triangle read by rows where T(n,k) is the number of integer partitions of n with exactly k distinct non-subset-sums.
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13
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1, 1, 0, 1, 1, 0, 2, 0, 1, 0, 2, 1, 1, 1, 0, 4, 0, 2, 0, 1, 0, 5, 1, 0, 3, 1, 1, 0, 8, 0, 3, 0, 3, 0, 1, 0, 10, 2, 1, 2, 2, 3, 1, 1, 0, 16, 0, 5, 0, 3, 0, 5, 0, 1, 0, 20, 2, 2, 4, 2, 6, 0, 4, 1, 1, 0, 31, 0, 6, 0, 8, 0, 5, 0, 5, 0, 1, 0, 39, 4, 4, 4, 1, 6, 6, 3, 2, 6, 1, 1, 0
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OFFSET
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0,7
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COMMENTS
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For an integer partition y of n, we call a positive integer k <= n a non-subset-sum iff there is no submultiset of y summing to k.
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LINKS
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EXAMPLE
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The partition (4,2) has subset-sums {2,4,6} and non-subset-sums {1,3,5} so is counted under T(6,3).
Triangle begins:
1
1 0
1 1 0
2 0 1 0
2 1 1 1 0
4 0 2 0 1 0
5 1 0 3 1 1 0
8 0 3 0 3 0 1 0
10 2 1 2 2 3 1 1 0
16 0 5 0 3 0 5 0 1 0
20 2 2 4 2 6 0 4 1 1 0
31 0 6 0 8 0 5 0 5 0 1 0
39 4 4 4 1 6 6 3 2 6 1 1 0
55 0 13 0 8 0 12 0 6 0 6 0 1 0
71 5 8 7 3 5 3 16 3 6 0 6 1 1 0
Row n = 6 counts the following partitions:
(321) (411) . (51) (33) (6) .
(3111) (42)
(2211) (222)
(21111)
(111111)
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MATHEMATICA
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Table[Length[Select[IntegerPartitions[n], Length[Complement[Range[n], Total/@Subsets[#]]]==k&]], {n, 0, 10}, {k, 0, n}]
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CROSSREFS
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The rank statistic counted by this triangle is A325799.
The complement (positive subset-sum) is A365658.
A046663 counts partitions w/o a submultiset summing to k, strict A365663.
A364350 counts combination-free strict partitions, complement A364839.
A365543 counts partitions with a submultiset summing to k, strict A365661.
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KEYWORD
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AUTHOR
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STATUS
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approved
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