cp's OEIS Frontend

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

%I A321745 #5 Nov 20 2018 12:21:39
%S A321745 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,
%T A321745 54,263,114,42,191,56,246,113,86,160,476,77,128,199,344
%N A321745 Sum of coefficients of monomial symmetric functions in the homogeneous symmetric function of the integer partition with Heinz number n.
%C A321745 The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).
%C A321745 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.
%H A321745 Wikipedia, <a href="https://en.wikipedia.org/wiki/Symmetric_polynomial">Symmetric polynomial</a>
%e A321745 The sum of coefficients of h(211) = m(4) + 4m(22) + 3m(31) + 7m(211) + 12m(1111) is a(12) = 27.
%e A321745 The a(3) = 2 through a(9) = 16 size-preserving permutations of multiset partitions:
%e A321745   {11}    {12}    {111}      {112}      {1111}        {123}      {1122}
%e A321745   {1}{1}  {1}{2}  {1}{11}    {1}{12}    {1}{111}      {1}{23}    {1}{122}
%e A321745           {2}{1}  {1}{1}{1}  {2}{11}    {11}{11}      {2}{13}    {11}{22}
%e A321745                              {1}{1}{2}  {1}{1}{11}    {3}{12}    {12}{12}
%e A321745                              {1}{2}{1}  {1}{1}{1}{1}  {1}{2}{3}  {2}{112}
%e A321745                              {2}{1}{1}                {1}{3}{2}  {22}{11}
%e A321745                                                       {2}{1}{3}  {1}{1}{22}
%e A321745                                                       {2}{3}{1}  {1}{2}{12}
%e A321745                                                       {3}{1}{2}  {2}{1}{12}
%e A321745                                                       {3}{2}{1}  {2}{2}{11}
%e A321745                                                                  {1}{1}{2}{2}
%e A321745                                                                  {1}{2}{1}{2}
%e A321745                                                                  {1}{2}{2}{1}
%e A321745                                                                  {2}{1}{1}{2}
%e A321745                                                                  {2}{1}{2}{1}
%e A321745                                                                  {2}{2}{1}{1}
%t A321745 sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t A321745 mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
%t A321745 nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]]];
%t A321745 Table[Sum[Times@@Factorial/@Length/@Split[Sort[Length/@mtn,Greater]]/Times@@Factorial/@Length/@Split[mtn],{mtn,mps[nrmptn[n]]}],{n,30}]
%Y A321745 Row sums of A321744.
%Y A321745 Cf. A005651, A007716, A008480, A056239, A124794, A124795, A181821, A255906, A296150, A318284, A319193, A319225, A319226, A321742-A321765.
%K A321745 nonn,more
%O A321745 1,3
%A A321745 _Gus Wiseman_, Nov 19 2018