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A371954
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Triangle read by rows where T(n,k) is the number of integer partitions of n that can be partitioned into k multisets with equal sums (k-quanimous).
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2
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1, 0, 1, 0, 2, 1, 0, 3, 0, 1, 0, 5, 3, 0, 1, 0, 7, 0, 0, 0, 1, 0, 11, 6, 4, 0, 0, 1, 0, 15, 0, 0, 0, 0, 0, 1, 0, 22, 14, 0, 5, 0, 0, 0, 1, 0, 30, 0, 10, 0, 0, 0, 0, 0, 1, 0, 42, 25, 0, 0, 6, 0, 0, 0, 0, 1, 0, 56, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 77, 53, 30, 15, 0, 7, 0, 0, 0, 0, 0, 1
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OFFSET
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0,5
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COMMENTS
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A finite multiset of numbers is defined to be k-quanimous iff it can be partitioned into k multisets with equal sums.
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LINKS
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EXAMPLE
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Triangle begins:
1
0 1
0 2 1
0 3 0 1
0 5 3 0 1
0 7 0 0 0 1
0 11 6 4 0 0 1
0 15 0 0 0 0 0 1
0 22 14 0 5 0 0 0 1
0 30 0 10 0 0 0 0 0 1
0 42 25 0 0 6 0 0 0 0 1
0 56 0 0 0 0 0 0 0 0 0 1
0 77 53 30 15 0 7 0 0 0 0 0 1
Row n = 6 counts the following partitions:
. (6) (33) (222) . . (111111)
(51) (321) (2211)
(42) (3111) (21111)
(411) (2211) (111111)
(33) (21111)
(321) (111111)
(3111)
(222)
(2211)
(21111)
(111111)
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MATHEMATICA
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hwt[n_]:=Total[Cases[FactorInteger[n], {p_, k_}:>PrimePi[p]*k]];
facs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[facs[n/d], Min@@#>=d&]], {d, Rest[Divisors[n]]}]];
Table[Length[Select[IntegerPartitions[n], Select[facs[Times@@Prime/@#], Length[#]==k&&SameQ@@hwt/@#&]!={}&]], {n, 0, 10}, {k, 0, n}]
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CROSSREFS
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Row n has A000005(n) positive entries.
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KEYWORD
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AUTHOR
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STATUS
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approved
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