cp's OEIS Frontend

A336137 Number of set partitions of the binary indices of n with equal block-sums.

%I A336137 #8 Jul 13 2020 22:21:51
%S A336137 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 A336137 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 A336137 1,1,1,1,1,1,1,2,2,1,1,1,1,1,2,1,1,2,1
%N A336137 Number of set partitions of the binary indices of n with equal block-sums.
%C A336137 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 A336137 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 A336137 The a(n) set partitions for n = 7, 59, 119, 367, 127:
%e A336137   {123}    {12456}      {123567}      {1234679}    {1234567}
%e A336137   {12}{3}  {126}{45}    {1236}{57}    {12346}{79}  {1247}{356}
%e A336137            {15}{24}{6}  {156}{237}    {1249}{367}  {1256}{347}
%e A336137                         {17}{26}{35}  {1267}{349}  {1346}{257}
%e A336137                                       {169}{2347}  {167}{2345}
%e A336137                                                    {16}{25}{34}{7}
%e A336137 The binary indices of 382 are {2,3,4,5,6,7,9}, with equal block-sum set partitions:
%e A336137   {{2,7},{3,6},{4,5},{9}}
%e A336137   {{2,4,6},{3,9},{5,7}}
%e A336137   {{2,7,9},{3,4,5,6}}
%e A336137   {{2,3,4,9},{5,6,7}}
%e A336137   {{2,3,6,7},{4,5,9}}
%e A336137   {{2,4,5,7},{3,6,9}}
%e A336137   {{2,3,4,5,6,7,9}}
%e A336137 so a(382) = 7.
%t A336137 bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
%t A336137 sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t A336137 Table[Length[Select[sps[bpe[n]],SameQ@@Total/@#&]],{n,0,100}]
%Y A336137 These set partitions are counted by A035470.
%Y A336137 The version for twice-partitions is A279787.
%Y A336137 The version for partitions of partitions is A305551.
%Y A336137 The version for factorizations is A321455.
%Y A336137 The version for normal multiset partitions is A326518.
%Y A336137 The version for distinct block-sums is A336138.
%Y A336137 Set partitions of binary indices are A050315.
%Y A336137 Normal multiset partitions with equal lengths are A317583.
%Y A336137 Normal multiset partitions with equal averages are A326520.
%Y A336137 Multiset partitions with equal block-sums are ranked by A326534.
%Y A336137 Cf. A000110, A007837, A032011, A038041, A271619, A275780, A322794.
%K A336137 nonn
%O A336137 0,8
%A A336137 _Gus Wiseman_, Jul 12 2020