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A336137
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Number of set partitions of the binary indices of n with equal block-sums.
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1
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1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 1, 1, 1, 3, 2, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1
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OFFSET
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0,8
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COMMENTS
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A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.
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LINKS
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EXAMPLE
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The a(n) set partitions for n = 7, 59, 119, 367, 127:
{123} {12456} {123567} {1234679} {1234567}
{12}{3} {126}{45} {1236}{57} {12346}{79} {1247}{356}
{15}{24}{6} {156}{237} {1249}{367} {1256}{347}
{17}{26}{35} {1267}{349} {1346}{257}
{169}{2347} {167}{2345}
{16}{25}{34}{7}
The binary indices of 382 are {2,3,4,5,6,7,9}, with equal block-sum set partitions:
{{2,7},{3,6},{4,5},{9}}
{{2,4,6},{3,9},{5,7}}
{{2,7,9},{3,4,5,6}}
{{2,3,4,9},{5,6,7}}
{{2,3,6,7},{4,5,9}}
{{2,4,5,7},{3,6,9}}
{{2,3,4,5,6,7,9}}
so a(382) = 7.
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MATHEMATICA
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bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n, 2]], 1];
sps[{}]:={{}}; sps[set:{i_, ___}]:=Join@@Function[s, Prepend[#, s]&/@sps[Complement[set, s]]]/@Cases[Subsets[set], {i, ___}];
Table[Length[Select[sps[bpe[n]], SameQ@@Total/@#&]], {n, 0, 100}]
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CROSSREFS
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These set partitions are counted by A035470.
The version for twice-partitions is A279787.
The version for partitions of partitions is A305551.
The version for factorizations is A321455.
The version for normal multiset partitions is A326518.
The version for distinct block-sums is A336138.
Set partitions of binary indices are A050315.
Normal multiset partitions with equal lengths are A317583.
Normal multiset partitions with equal averages are A326520.
Multiset partitions with equal block-sums are ranked by A326534.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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