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A336138 Number of set partitions of the binary indices of n with distinct block-sums.

%I A336138 #9 Jul 14 2020 05:38:31
%S A336138 1,1,1,2,1,2,2,4,1,2,2,5,2,4,5,12,1,2,2,5,2,5,4,13,2,4,5,13,5,13,13,
%T A336138 43,1,2,2,5,2,5,5,13,2,5,4,14,5,13,14,42,2,4,5,13,5,14,13,43,5,13,14,
%U A336138 45,14,44,44,160,1,2,2,5,2,5,5,14,2,5,5,14,4,13
%N A336138 Number of set partitions of the binary indices of n with distinct block-sums.
%C A336138 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 A336138 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 A336138 The a(n) set partitions for n = 3, 7, 11, 15, 23:
%e A336138   {12}    {123}      {124}      {1234}        {1235}
%e A336138   {1}{2}  {1}{23}    {1}{24}    {1}{234}      {1}{235}
%e A336138           {13}{2}    {12}{4}    {12}{34}      {12}{35}
%e A336138           {1}{2}{3}  {14}{2}    {123}{4}      {123}{5}
%e A336138                      {1}{2}{4}  {124}{3}      {125}{3}
%e A336138                                 {13}{24}      {13}{25}
%e A336138                                 {134}{2}      {135}{2}
%e A336138                                 {1}{2}{34}    {15}{23}
%e A336138                                 {1}{23}{4}    {1}{2}{35}
%e A336138                                 {1}{24}{3}    {1}{25}{3}
%e A336138                                 {14}{2}{3}    {13}{2}{5}
%e A336138                                 {1}{2}{3}{4}  {15}{2}{3}
%e A336138                                               {1}{2}{3}{5}
%t A336138 bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
%t A336138 sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t A336138 Table[Length[Select[sps[bpe[n]],UnsameQ@@Total/@#&]],{n,0,100}]
%Y A336138 The version for twice-partitions is A271619.
%Y A336138 The version for partitions of partitions is (also) A271619.
%Y A336138 These set partitions are counted by A275780.
%Y A336138 The version for factorizations is A321469.
%Y A336138 The version for normal multiset partitions is A326519.
%Y A336138 The version for equal block-sums is A336137.
%Y A336138 Set partitions with distinct block-lengths are A007837.
%Y A336138 Set partitions of binary indices are A050315.
%Y A336138 Twice-partitions with equal sums are A279787.
%Y A336138 Partitions of partitions with equal sums are A305551.
%Y A336138 Normal multiset partitions with equal block-lengths are A317583.
%Y A336138 Multiset partitions with distinct block-sums are ranked by A326535.
%Y A336138 Cf. A000110, A032011, A035470, A131632, A321455, A326026, A326514, A326517, A326518, A326534, A326565.
%K A336138 nonn
%O A336138 0,4
%A A336138 _Gus Wiseman_, Jul 12 2020