A326512 Number of set partitions of {1..n} where every block has the same average.

1, 1, 1, 2, 2, 5, 5, 18, 16, 75, 64, 405, 302, 2581, 1693, 19872, 11295, 175807, 87524, 1851135, 787515, 21909766, 8185713, 298698113, 96514608, 4538610230, 1285072142

Offset

0,4

Comments

The common average is necessarily (n+1)/2. The number of blocks with this average is given by A070925. - Christian Sievers, Aug 22 2024

Links

Table of n, a(n) for n=0..26.

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

Example

The a(1) = 1 through a(7) = 18 set partitions:
  {1}  {12}  {123}    {1234}    {12345}      {123456}      {1234567}
             {13}{2}  {14}{23}  {1245}{3}    {1256}{34}    {123567}{4}
                                {135}{24}    {1346}{25}    {12467}{35}
                                {15}{234}    {16}{2345}    {1267}{345}
                                {15}{24}{3}  {16}{25}{34}  {13457}{26}
                                                           {1357}{246}
                                                           {1456}{237}
                                                           {147}{2356}
                                                           {156}{2347}
                                                           {17}{23456}
                                                           {1267}{35}{4}
                                                           {1357}{26}{4}
                                                           {147}{26}{35}
                                                           {156}{237}{4}
                                                           {17}{2356}{4}
                                                           {17}{246}{35}
                                                           {17}{26}{345}
                                                           {17}{26}{35}{4}

Mathematica

sps[{}]:={{}}; sps[set:{i_, ___}]:=Join@@Function[s, Prepend[#, s]&/@sps[Complement[set, s]]]/@Cases[Subsets[set], {i, ___}];
Table[Length[Select[sps[Range[n]], SameQ@@Mean/@#&]], {n, 0, 8}]

Crossrefs

Cf. A000110, A007837, A035470, A038041, A070925, A275780, A306017, A306021, A319169, A326513, A326515, A326520, A326536.

Sequence in context: A245844 A373849 A083849 * A325983 A063501 A103892

Adjacent sequences: A326509 A326510 A326511 * A326513 A326514 A326515

Keyword

nonn,more

Author

Gus Wiseman, Jul 11 2019

Extensions

a(12)-a(15) from Alois P. Heinz, Jul 12 2019

a(16)-a(26) from Christian Sievers, Aug 22 2024

Status

approved