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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

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 * 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.)