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 A321743 Sum of coefficients of monomial symmetric functions in the elementary symmetric function of the integer partition with Heinz number n. 1
 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 (list; graph; refs; listen; history; text; internal format)
 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 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 * A131632 A051348 A253828 Adjacent sequences:  A321740 A321741 A321742 * A321744 A321745 A321746 KEYWORD nonn AUTHOR Gus Wiseman, Nov 19 2018 STATUS approved

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Last modified February 24 08:47 EST 2020. Contains 332203 sequences. (Running on oeis4.)