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

1,4

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.

Links

Table of n, a(n) for n=1..68.

Example

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}

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}]

Crossrefs

Row sums of A321742.

Cf. A005651, A008480, A049311, A056239, A116540, A124794, A124795, A181821, A296150, A318360, A319193, A319225, A319226, A321738, A321742-A321765.

Sequence in context: A079546 A014413 A262072 * A341766 A131632 A051348

Adjacent sequences: A321740 A321741 A321742 * A321744 A321745 A321746

Keyword

nonn

Author

Gus Wiseman, Nov 19 2018

Status

approved