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

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, 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, 45, 14, 44, 44, 160, 1, 2, 2, 5, 2, 5, 5, 14, 2, 5, 5, 14, 4, 13

Offset

0,4

Comments

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.

Links

Table of n, a(n) for n=0..77.

Gus Wiseman, Sequences counting and ranking multiset partitions whose part lengths, sums, or averages are constant or strict.

Example

The a(n) set partitions for n = 3, 7, 11, 15, 23:
  {12}    {123}      {124}      {1234}        {1235}
  {1}{2}  {1}{23}    {1}{24}    {1}{234}      {1}{235}
          {13}{2}    {12}{4}    {12}{34}      {12}{35}
          {1}{2}{3}  {14}{2}    {123}{4}      {123}{5}
                     {1}{2}{4}  {124}{3}      {125}{3}
                                {13}{24}      {13}{25}
                                {134}{2}      {135}{2}
                                {1}{2}{34}    {15}{23}
                                {1}{23}{4}    {1}{2}{35}
                                {1}{24}{3}    {1}{25}{3}
                                {14}{2}{3}    {13}{2}{5}
                                {1}{2}{3}{4}  {15}{2}{3}
                                              {1}{2}{3}{5}

Mathematica

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]], UnsameQ@@Total/@#&]], {n, 0, 100}]

Crossrefs

The version for twice-partitions is A271619.

The version for partitions of partitions is (also) A271619.

These set partitions are counted by A275780.

The version for factorizations is A321469.

The version for normal multiset partitions is A326519.

The version for equal block-sums is A336137.

Set partitions with distinct block-lengths are A007837.

Set partitions of binary indices are A050315.

Twice-partitions with equal sums are A279787.

Partitions of partitions with equal sums are A305551.

Normal multiset partitions with equal block-lengths are A317583.

Multiset partitions with distinct block-sums are ranked by A326535.

Cf. A000110, A032011, A035470, A131632, A321455, A326026, A326514, A326517, A326518, A326534, A326565.

Sequence in context: A151678 A273126 A151681 * A366600 A365461 A131097

Adjacent sequences: A336135 A336136 A336137 * A336139 A336140 A336141

Keyword

nonn

Author

Gus Wiseman, Jul 12 2020

Status

approved