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

Author

Gus Wiseman, Dec 26 2018

Keywords

• nonn

Comments

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

Examples

			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}
		

Links

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

Crossrefs

Row sums of A322788.

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

Programs

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}]

Formula

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

Extensions

More terms from Alois P. Heinz, Jan 30 2019

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