A322785 Number of uniform multiset partitions of uniform multisets of size n whose union is an initial interval of positive integers.

1, 1, 4, 4, 12, 4, 48, 4, 183, 297, 1186, 4, 33950, 4, 139527, 1529608, 4726356, 4, 229255536, 4, 3705777010, 36279746314, 13764663019, 4, 14096735197959, 5194673049514, 7907992957755, 2977586461058927, 13426396910491001, 4, 1350012288268171854, 4, 59487352224070807287

Offset

0,3

Comments

A multiset is uniform if all multiplicities are equal. A multiset partition is uniform if all parts have the same size.

Links

Andrew Howroyd, Table of n, a(n) for n = 0..100

Formula

a(n) = 4 <=> n in { A000040 }. - Alois P. Heinz, Feb 03 2022

Example

The a(1) = 1 though a(6) = 48 multiset partitions:
  {1}  {11}    {111}      {1111}        {11111}          {111111}
       {12}    {123}      {1122}        {12345}          {111222}
       {1}{1}  {1}{1}{1}  {1234}        {1}{1}{1}{1}{1}  {112233}
       {1}{2}  {1}{2}{3}  {11}{11}      {1}{2}{3}{4}{5}  {123456}
                          {11}{22}                       {111}{111}
                          {12}{12}                       {111}{222}
                          {12}{34}                       {112}{122}
                          {13}{24}                       {112}{233}
                          {14}{23}                       {113}{223}
                          {1}{1}{1}{1}                   {122}{133}
                          {1}{1}{2}{2}                   {123}{123}
                          {1}{2}{3}{4}                   {123}{456}
                                                         {124}{356}
                                                         {125}{346}
                                                         {126}{345}
                                                         {134}{256}
                                                         {135}{246}
                                                         {136}{245}
                                                         {145}{236}
                                                         {146}{235}
                                                         {156}{234}
                                                         {11}{11}{11}
                                                         {11}{12}{22}
                                                         {11}{22}{33}
                                                         {11}{23}{23}
                                                         {12}{12}{12}
                                                         {12}{12}{33}
                                                         {12}{13}{23}
                                                         {12}{34}{56}
                                                         {12}{35}{46}
                                                         {12}{36}{45}
                                                         {13}{13}{22}
                                                         {13}{24}{56}
                                                         {13}{25}{46}
                                                         {13}{26}{45}
                                                         {14}{23}{56}
                                                         {14}{25}{36}
                                                         {14}{26}{35}
                                                         {15}{23}{46}
                                                         {15}{24}{36}
                                                         {15}{26}{34}
                                                         {16}{23}{45}
                                                         {16}{24}{35}
                                                         {16}{25}{34}
                                                         {1}{1}{1}{1}{1}{1}
                                                         {1}{1}{1}{2}{2}{2}
                                                         {1}{1}{2}{2}{3}{3}
                                                         {1}{2}{3}{4}{5}{6}

Mathematica

sps[{}]:={{}}; sps[set:{i_, ___}]:=Join@@Function[s, Prepend[#, s]&/@sps[Complement[set, s]]]/@Cases[Subsets[set], {i, ___}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
Table[Sum[Length[Select[mps[m], SameQ@@Length/@#&]], {m, Table[Join@@Table[Range[n/d], {d}], {d, Divisors[n]}]}], {n, 8}]

Crossrefs

Row sums of A322788.

Cf. A000040, A038041, A072774, A100778, A299353, A306017, A306018, A306021, A317583, A317584, A319056, A319189, A321721, A322705, A322784, A322788.

Sequence in context: A229253 A272789 A273152 * A189545 A326993 A272990

Adjacent sequences: A322782 A322783 A322784 * A322786 A322787 A322788

Keyword

nonn

Author

Gus Wiseman, Dec 26 2018

Extensions

More terms from Alois P. Heinz, Jan 30 2019

Terms a(14) and beyond from Andrew Howroyd, Feb 03 2022

Status

approved