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A367412
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Triangle read by rows with all zeros removed where T(n,k) is the number of integer partitions of n with k different semi-sums.
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0
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1, 1, 1, 1, 2, 1, 3, 1, 1, 3, 3, 1, 5, 3, 2, 1, 4, 7, 2, 1, 1, 6, 7, 6, 2, 1, 6, 10, 6, 7, 1, 7, 12, 11, 8, 3, 1, 6, 16, 11, 17, 3, 2, 1, 10, 14, 20, 19, 10, 2, 1, 1, 7, 22, 17, 31, 14, 7, 2, 1, 9, 22, 27, 37, 22, 11, 6, 1, 10, 24, 27, 51, 32, 16, 15
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OFFSET
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0,5
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COMMENTS
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We define a semi-sum of a multiset to be any sum of a 2-element submultiset. This is different from sums of pairs of elements. For example, 2 is the sum of a pair of elements of {1}, but there are no semi-sums.
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LINKS
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EXAMPLE
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Triangle begins:
1
1 1
1 2
1 3 1
1 3 3
1 5 3 2
1 4 7 2 1
1 6 7 6 2
1 6 10 6 7
1 7 12 11 8 3
1 6 16 11 17 3 2
1 10 14 20 19 10 2 1
1 7 22 17 31 14 7 2
1 9 22 27 37 22 11 6
1 10 24 27 51 32 16 15
1 11 27 39 57 43 27 22 4
1 9 33 34 79 57 36 39 7 2
1 13 31 51 86 77 45 62 14 4 1
Row n = 9 counts the following partitions:
(9) (81) (711) (621) (5211)
(72) (6111) (531) (4311)
(63) (522) (432) (4221)
(54) (51111) (33111) (42111)
(333) (441) (222111) (3321)
(111111111) (411111) (2211111) (32211)
(3222) (321111)
(3111111)
(22221)
(21111111)
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MATHEMATICA
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DeleteCases[Table[Length[Select[IntegerPartitions[n], Length[Union[Total/@Subsets[#, {2}]]]==k&]], {n, 10}, {k, 0, n}], 0, 2]
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CROSSREFS
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The non-binary version (with zeros) is A365658.
The strict non-binary version (with zeros) is A365832.
The corresponding rank statistic is A366739.
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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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