A322785 - OEIS

%I #17 Feb 03 2022 16:45:59

%S 1,1,4,4,12,4,48,4,183,297,1186,4,33950,4,139527,1529608,4726356,4,

%T 229255536,4,3705777010,36279746314,13764663019,4,14096735197959,

%U 5194673049514,7907992957755,2977586461058927,13426396910491001,4,1350012288268171854,4,59487352224070807287

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

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

%H Andrew Howroyd, <a href="/A322785/b322785.txt">Table of n, a(n) for n = 0..100</a>

%F a(n) = 4 <=> n in { A000040 }. - _Alois P. Heinz_, Feb 03 2022

%e The a(1) = 1 though a(6) = 48 multiset partitions:

%e   {1}  {11}    {111}      {1111}        {11111}          {111111}

%e        {12}    {123}      {1122}        {12345}          {111222}

%e        {1}{1}  {1}{1}{1}  {1234}        {1}{1}{1}{1}{1}  {112233}

%e        {1}{2}  {1}{2}{3}  {11}{11}      {1}{2}{3}{4}{5}  {123456}

%e                           {11}{22}                       {111}{111}

%e                           {12}{12}                       {111}{222}

%e                           {12}{34}                       {112}{122}

%e                           {13}{24}                       {112}{233}

%e                           {14}{23}                       {113}{223}

%e                           {1}{1}{1}{1}                   {122}{133}

%e                           {1}{1}{2}{2}                   {123}{123}

%e                           {1}{2}{3}{4}                   {123}{456}

%e                                                          {124}{356}

%e                                                          {125}{346}

%e                                                          {126}{345}

%e                                                          {134}{256}

%e                                                          {135}{246}

%e                                                          {136}{245}

%e                                                          {145}{236}

%e                                                          {146}{235}

%e                                                          {156}{234}

%e                                                          {11}{11}{11}

%e                                                          {11}{12}{22}

%e                                                          {11}{22}{33}

%e                                                          {11}{23}{23}

%e                                                          {12}{12}{12}

%e                                                          {12}{12}{33}

%e                                                          {12}{13}{23}

%e                                                          {12}{34}{56}

%e                                                          {12}{35}{46}

%e                                                          {12}{36}{45}

%e                                                          {13}{13}{22}

%e                                                          {13}{24}{56}

%e                                                          {13}{25}{46}

%e                                                          {13}{26}{45}

%e                                                          {14}{23}{56}

%e                                                          {14}{25}{36}

%e                                                          {14}{26}{35}

%e                                                          {15}{23}{46}

%e                                                          {15}{24}{36}

%e                                                          {15}{26}{34}

%e                                                          {16}{23}{45}

%e                                                          {16}{24}{35}

%e                                                          {16}{25}{34}

%e                                                          {1}{1}{1}{1}{1}{1}

%e                                                          {1}{1}{1}{2}{2}{2}

%e                                                          {1}{1}{2}{2}{3}{3}

%e                                                          {1}{2}{3}{4}{5}{6}

%t sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];

%t mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];

%t Table[Sum[Length[Select[mps[m],SameQ@@Length/@#&]],{m,Table[Join@@Table[Range[n/d],{d}],{d,Divisors[n]}]}],{n,8}]

%Y Row sums of A322788.

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

%K nonn

%O 0,3

%A _Gus Wiseman_, Dec 26 2018

%E More terms from _Alois P. Heinz_, Jan 30 2019

%E Terms a(14) and beyond from _Andrew Howroyd_, Feb 03 2022