A321743 Sum of coefficients of monomial symmetric functions in the elementary symmetric function of the integer partition with Heinz number n.

1, 1, 1, 3, 1, 4, 1, 10, 9, 5, 1, 20, 1, 6, 14, 47, 1, 50, 1, 30, 20, 7, 1, 110, 29, 8, 157, 42, 1, 97, 1, 246, 27, 9, 49, 338, 1, 10, 35, 206, 1, 159, 1, 56, 353, 11, 1, 732, 99, 224, 44, 72, 1, 1184, 76, 332, 54, 12, 1, 743, 1, 13, 677, 1602, 111, 242, 1, 90

Offset: 1

Author

Gus Wiseman, Nov 19 2018

Keywords

• nonn

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 set multipartitions (multisets of sets) of a multiset (such as row n of A305936) whose multiplicities are the prime indices of n.

Examples

			The sum of coefficients of e(211) = 2m(22) + m(31) + 5m(211) + 12m(1111) is a(12) = 20.
The a(2) = 1 through a(9) = 9 size-preserving permutations of set multipartitions:
  {1} {1}{1} {12}   {1}{1}{1} {1}{12}   {1}{1}{1}{1} {123}     {12}{12}
             {1}{2}           {1}{1}{2}              {1}{23}   {1}{2}{12}
             {2}{1}           {1}{2}{1}              {2}{13}   {2}{1}{12}
                              {2}{1}{1}              {3}{12}   {1}{1}{2}{2}
                                                     {1}{2}{3} {1}{2}{1}{2}
                                                     {1}{3}{2} {1}{2}{2}{1}
                                                     {2}{1}{3} {2}{1}{1}{2}
                                                     {2}{3}{1} {2}{1}{2}{1}
                                                     {3}{1}{2} {2}{2}{1}{1}
                                                     {3}{2}{1}
		

Crossrefs

Row sums of A321742.

Cf. A005651, A008480, A049311, A056239, A116540, A124794, A124795, A181821, A296150, A318360, A319193, A319225, A319226, A321738, 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,Select[mps[nrmptn[n]],And@@UnsameQ@@@#&]}],{n,30}]