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A371789
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Number of non-quanimous subsets of {1..n}, meaning there is only one set partition with all equal block-sums.
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21
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1, 2, 4, 7, 13, 24, 45, 85, 162, 306, 585
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OFFSET
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0,2
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COMMENTS
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A finite multiset of numbers is defined to be quanimous iff it can be partitioned into two or more multisets with equal sums. Quanimous partitions are counted by A321452 and ranked by A321454.
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LINKS
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EXAMPLE
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The set s = {3,4,6,8,9} has set partitions {{3,4,6,8,9}} and {{3,4,8},{6,9}} with equal block-sums, so s is not counted under a(9).
The a(0) = 1 through a(4) = 13 subsets:
{} {} {} {} {}
{1} {1} {1} {1}
{2} {2} {2}
{1,2} {3} {3}
{1,2} {4}
{1,3} {1,2}
{2,3} {1,3}
{1,4}
{2,3}
{2,4}
{3,4}
{1,2,4}
{2,3,4}
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MATHEMATICA
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sps[{}]:={{}}; sps[set:{i_, ___}]:=Join@@Function[s, Prepend[#, s]& /@ sps[Complement[set, s]]]/@Cases[Subsets[set], {i, ___}];
Table[Length[Select[Subsets[Range[n]], Length[Select[sps[#], SameQ@@Total/@#&]]==1&]], {n, 0, 8}]
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CROSSREFS
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The "bi-" complement for integer partitions is A002219, ranks A357976.
The "bi-" version for strict partitions is A371794 (bisection A321142).
A371783 counts k-quanimous partitions.
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KEYWORD
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nonn,more
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AUTHOR
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STATUS
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approved
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