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Number of set partitions of the binary indices of n with equal block-sums.
10

%I #8 Jul 13 2020 22:21:51

%S 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,

%T 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,

%U 1,1,1,1,1,1,1,2,2,1,1,1,1,1,2,1,1,2,1

%N Number of set partitions of the binary indices of n with equal block-sums.

%C 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.

%H Gus Wiseman, <a href="/A038041/a038041.txt">Sequences counting and ranking multiset partitions whose part lengths, sums, or averages are constant or strict.</a>

%e The a(n) set partitions for n = 7, 59, 119, 367, 127:

%e {123} {12456} {123567} {1234679} {1234567}

%e {12}{3} {126}{45} {1236}{57} {12346}{79} {1247}{356}

%e {15}{24}{6} {156}{237} {1249}{367} {1256}{347}

%e {17}{26}{35} {1267}{349} {1346}{257}

%e {169}{2347} {167}{2345}

%e {16}{25}{34}{7}

%e The binary indices of 382 are {2,3,4,5,6,7,9}, with equal block-sum set partitions:

%e {{2,7},{3,6},{4,5},{9}}

%e {{2,4,6},{3,9},{5,7}}

%e {{2,7,9},{3,4,5,6}}

%e {{2,3,4,9},{5,6,7}}

%e {{2,3,6,7},{4,5,9}}

%e {{2,4,5,7},{3,6,9}}

%e {{2,3,4,5,6,7,9}}

%e so a(382) = 7.

%t bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];

%t sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];

%t Table[Length[Select[sps[bpe[n]],SameQ@@Total/@#&]],{n,0,100}]

%Y These set partitions are counted by A035470.

%Y The version for twice-partitions is A279787.

%Y The version for partitions of partitions is A305551.

%Y The version for factorizations is A321455.

%Y The version for normal multiset partitions is A326518.

%Y The version for distinct block-sums is A336138.

%Y Set partitions of binary indices are A050315.

%Y Normal multiset partitions with equal lengths are A317583.

%Y Normal multiset partitions with equal averages are A326520.

%Y Multiset partitions with equal block-sums are ranked by A326534.

%Y Cf. A000110, A007837, A032011, A038041, A271619, A275780, A322794.

%K nonn

%O 0,8

%A _Gus Wiseman_, Jul 12 2020