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A365663
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Triangle read by rows where T(n,k) is the number of strict integer partitions of n without a subset summing to k.
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45
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1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 2, 2, 3, 2, 2, 3, 3, 3, 3, 3, 3, 3, 4, 3, 5, 3, 4, 3, 5, 5, 4, 5, 5, 4, 5, 5, 5, 6, 5, 6, 7, 6, 5, 6, 5, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 9, 8, 8, 8, 11, 8, 8, 8, 9, 8, 10, 11, 10, 10, 10, 10, 10, 10, 10, 10, 11, 10, 12, 13, 11, 13, 11, 12, 15, 12, 11, 13, 11, 13, 12
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OFFSET
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2,5
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COMMENTS
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Warning: Do not confuse with the non-strict version A046663.
Rows are palindromes.
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LINKS
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EXAMPLE
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Triangle begins:
1
1 1
1 2 1
2 2 2 2
2 2 3 2 2
3 3 3 3 3 3
3 4 3 5 3 4 3
5 5 4 5 5 4 5 5
5 6 5 6 7 6 5 6 5
7 7 7 7 7 7 7 7 7 7
8 9 8 8 8 11 8 8 8 9 8
Row n = 8 counts the following strict partitions:
(8) (8) (8) (8) (8) (8) (8)
(6,2) (7,1) (7,1) (7,1) (7,1) (7,1) (6,2)
(5,3) (5,3) (6,2) (6,2) (6,2) (5,3) (5,3)
(4,3,1) (5,3) (4,3,1)
(5,2,1)
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MATHEMATICA
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Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&FreeQ[Total/@Subsets[#], k]&]], {n, 2, 15}, {k, 1, n-1}]
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CROSSREFS
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Columns k = 0 and k = n are A025147.
The complement for subsets instead of strict partitions is A365381.
A000009 counts subsets summing to n.
A000124 counts distinct possible sums of subsets of {1..n}.
A124506 appears to count combination-free subsets, differences of A326083.
A364350 counts combination-free strict partitions, complement A364839.
Cf. A002219, A108796, A108917, A122768, A275972, A299701, A304792, A364916, A365311, A365376, A365541.
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KEYWORD
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AUTHOR
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STATUS
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approved
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