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A367582
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Triangle read by rows where T(n,k) is the number of integer partitions of n whose multiset multiplicity kernel (in which each multiplicity becomes the least element of that multiplicity), sums to k.
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11
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1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 1, 0, 1, 1, 2, 2, 1, 0, 1, 3, 3, 2, 1, 1, 0, 1, 1, 4, 3, 3, 2, 1, 0, 1, 3, 5, 4, 4, 3, 1, 1, 0, 1, 2, 6, 4, 8, 3, 3, 2, 1, 0, 1, 3, 7, 9, 6, 7, 4, 3, 1, 1, 0, 1, 1, 8, 7, 11, 9, 9, 4, 3, 2, 1
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OFFSET
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0,13
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COMMENTS
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We define the multiset multiplicity kernel MMK(m) of a multiset m by the following property, holding for all distinct multiplicities k >= 1. If S is the set of elements of multiplicity k in m, then min(S) has multiplicity |S| in MMK(m). For example, MMK({1,1,2,2,3,4,5}) = {1,1,3,3,3}, and MMK({1,2,3,4,5,5,5,5}) = {1,1,1,1,5}. As an operation on multisets, MMK is represented by A367579, and as an operation on their Heinz numbers, it is represented by A367580.
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LINKS
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EXAMPLE
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Triangle begins:
1
0 1
0 1 1
0 1 1 1
0 1 2 1 1
0 1 1 2 2 1
0 1 3 3 2 1 1
0 1 1 4 3 3 2 1
0 1 3 5 4 4 3 1 1
0 1 2 6 4 8 3 3 2 1
0 1 3 7 9 6 7 4 3 1 1
0 1 1 8 7 11 9 9 4 3 2 1
0 1 5 10 11 13 10 11 6 5 3 1 1
0 1 1 10 11 17 14 18 10 9 4 3 2 1
0 1 3 12 17 19 18 22 14 12 8 4 3 1 1
0 1 3 12 15 27 19 31 19 19 10 9 5 3 2 1
0 1 4 15 23 27 31 33 24 26 18 12 8 4 3 1 1
0 1 1 14 20 35 33 48 32 37 25 20 11 10 4 3 2 1
Row n = 7 counts the following partitions:
(1111111) (61) (421) (52) (4111) (511) (7)
(2221) (331) (322) (43)
(22111) (31111) (3211)
(211111)
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MATHEMATICA
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mmk[q_]:=With[{mts=Length/@Split[q]}, Sort[Table[Min@@Select[q, Count[q, #]==i&], {i, mts}]]];
Table[Length[Select[IntegerPartitions[n], Total[mmk[#]]==k&]], {n, 0, 10}, {k, 0, n}]
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CROSSREFS
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A072233 counts partitions by number of parts.
A116608 counts partitions by number of distinct parts.
A116861 counts partitions by sum of distinct parts.
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KEYWORD
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AUTHOR
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STATUS
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approved
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