A321745 Sum of coefficients of monomial symmetric functions in the homogeneous symmetric function of the integer partition with Heinz number n.

1, 1, 2, 3, 3, 6, 5, 10, 16, 12, 7, 27, 11, 20, 32, 47, 15, 76, 22, 56, 65, 35, 30, 136, 79, 54, 263, 114, 42, 191, 56, 246, 113, 86, 160, 476, 77, 128, 199, 344

Offset: 1

Author

Gus Wiseman, Nov 19 2018

Keywords

• nonn
• more

Comments

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

Also the number of size-preserving permutations of multiset partitions of a multiset (such as row n of A305936) whose multiplicities are the prime indices of n.

Examples

			The sum of coefficients of h(211) = m(4) + 4m(22) + 3m(31) + 7m(211) + 12m(1111) is a(12) = 27.
The a(3) = 2 through a(9) = 16 size-preserving permutations of multiset partitions:
  {11}    {12}    {111}      {112}      {1111}        {123}      {1122}
  {1}{1}  {1}{2}  {1}{11}    {1}{12}    {1}{111}      {1}{23}    {1}{122}
          {2}{1}  {1}{1}{1}  {2}{11}    {11}{11}      {2}{13}    {11}{22}
                             {1}{1}{2}  {1}{1}{11}    {3}{12}    {12}{12}
                             {1}{2}{1}  {1}{1}{1}{1}  {1}{2}{3}  {2}{112}
                             {2}{1}{1}                {1}{3}{2}  {22}{11}
                                                      {2}{1}{3}  {1}{1}{22}
                                                      {2}{3}{1}  {1}{2}{12}
                                                      {3}{1}{2}  {2}{1}{12}
                                                      {3}{2}{1}  {2}{2}{11}
                                                                 {1}{1}{2}{2}
                                                                 {1}{2}{1}{2}
                                                                 {1}{2}{2}{1}
                                                                 {2}{1}{1}{2}
                                                                 {2}{1}{2}{1}
                                                                 {2}{2}{1}{1}
		

Links

• Wikipedia, Symmetric polynomial

Crossrefs

Row sums of A321744.

Cf. A005651, A007716, A008480, A056239, A124794, A124795, A181821, A255906, A296150, A318284, A319193, A319225, A319226, A321742-A321765.

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]]]];
nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]]];
Table[Sum[Times@@Factorial/@Length/@Split[Sort[Length/@mtn,Greater]]/Times@@Factorial/@Length/@Split[mtn],{mtn,mps[nrmptn[n]]}],{n,30}]