cp's OEIS Frontend

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

%I A321743 #8 Nov 20 2018 12:21:25
%S A321743 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,
%T A321743 1,97,1,246,27,9,49,338,1,10,35,206,1,159,1,56,353,11,1,732,99,224,44,
%U A321743 72,1,1184,76,332,54,12,1,743,1,13,677,1602,111,242,1,90
%N A321743 Sum of coefficients of monomial symmetric functions in the elementary symmetric function of the integer partition with Heinz number n.
%C A321743 The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).
%C A321743 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.
%e A321743 The sum of coefficients of e(211) = 2m(22) + m(31) + 5m(211) + 12m(1111) is a(12) = 20.
%e A321743 The a(2) = 1 through a(9) = 9 size-preserving permutations of set multipartitions:
%e A321743   {1} {1}{1} {12}   {1}{1}{1} {1}{12}   {1}{1}{1}{1} {123}     {12}{12}
%e A321743              {1}{2}           {1}{1}{2}              {1}{23}   {1}{2}{12}
%e A321743              {2}{1}           {1}{2}{1}              {2}{13}   {2}{1}{12}
%e A321743                               {2}{1}{1}              {3}{12}   {1}{1}{2}{2}
%e A321743                                                      {1}{2}{3} {1}{2}{1}{2}
%e A321743                                                      {1}{3}{2} {1}{2}{2}{1}
%e A321743                                                      {2}{1}{3} {2}{1}{1}{2}
%e A321743                                                      {2}{3}{1} {2}{1}{2}{1}
%e A321743                                                      {3}{1}{2} {2}{2}{1}{1}
%e A321743                                                      {3}{2}{1}
%t A321743 sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t A321743 mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
%t A321743 nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]]];
%t A321743 Table[Sum[Times@@Factorial/@Length/@Split[Sort[Length/@mtn,Greater]]/Times@@Factorial/@Length/@Split[mtn],{mtn,Select[mps[nrmptn[n]],And@@UnsameQ@@@#&]}],{n,30}]
%Y A321743 Row sums of A321742.
%Y A321743 Cf. A005651, A008480, A049311, A056239, A116540, A124794, A124795, A181821, A296150, A318360, A319193, A319225, A319226, A321738, A321742-A321765.
%K A321743 nonn
%O A321743 1,4
%A A321743 _Gus Wiseman_, Nov 19 2018