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A321743
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Sum of coefficients of monomial symmetric functions in the elementary symmetric function of the integer partition with Heinz number n.
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1
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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
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OFFSET
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1,4
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COMMENTS
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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.
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LINKS
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EXAMPLE
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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}
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MATHEMATICA
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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}]
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CROSSREFS
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Cf. A005651, A008480, A049311, A056239, A116540, A124794, A124795, A181821, A296150, A318360, A319193, A319225, A319226, A321738, A321742-A321765.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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