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A325987
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Irregular triangle read by rows where T(n,k) is the number of integer partitions of n with k submultisets, k > 0.
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1
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1, 0, 1, 0, 1, 1, 0, 1, 0, 2, 0, 1, 1, 1, 1, 1, 0, 1, 0, 2, 0, 3, 0, 1, 0, 1, 1, 3, 0, 1, 1, 2, 1, 1, 0, 1, 0, 3, 0, 3, 0, 4, 0, 1, 0, 3, 0, 1, 1, 3, 1, 3, 0, 3, 2, 1, 0, 4, 0, 1, 1, 1, 0, 1, 0, 5, 0, 3, 0, 5, 0, 3, 0, 6, 0, 1, 0, 3, 0, 2, 0, 1, 0, 1, 1, 4, 0
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OFFSET
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0,10
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COMMENTS
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The number of submultisets of a partition is the product of its multiplicities, each plus one.
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LINKS
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FORMULA
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EXAMPLE
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Triangle begins:
1
0 1
0 1 1
0 1 0 2
0 1 1 1 1 1
0 1 0 2 0 3 0 1
0 1 1 3 0 1 1 2 1 1
0 1 0 3 0 3 0 4 0 1 0 3
0 1 1 3 1 3 0 3 2 1 0 4 0 1 1 1
0 1 0 5 0 3 0 5 0 3 0 6 0 1 0 3 0 2 0 1
0 1 1 4 0 5 0 7 2 1 1 4 0 1 2 5 0 3 0 2 1 0 0 2
Row n = 7 counts the following partitions (empty columns not shown):
(7) (43) (322) (421) (31111) (3211)
(52) (331) (2221) (22111)
(61) (511) (4111) (211111)
(1111111)
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MATHEMATICA
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Table[Length[Select[IntegerPartitions[n], Times@@(1+Length/@Split[#])==k&]], {n, 0, 10}, {k, 1, Max@@(Times@@(1+Length/@Split[#])&)/@IntegerPartitions[n]}]
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CROSSREFS
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Diagonal n = k is A325830 interspersed with zeros.
Column k = 3 appears to be A137719.
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KEYWORD
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AUTHOR
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STATUS
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approved
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