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

1,3

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.

Links

Table of n, a(n) for n=1..40.

Wikipedia, Symmetric polynomial

Example

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}

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}]

Crossrefs

Row sums of A321744.

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

Sequence in context: A187754 A347732 A075258 * A212629 A127779 A207634

Adjacent sequences: A321742 A321743 A321744 * A321746 A321747 A321748

Keyword

nonn,more

Author

Gus Wiseman, Nov 19 2018

Status

approved