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A321742
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Irregular triangle read by rows where T(H(u),H(v)) is the coefficient of m(v) in e(u), where H is Heinz number, m is monomial symmetric functions, and e is elementary symmetric functions.
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40
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1, 1, 0, 1, 1, 2, 0, 0, 1, 0, 1, 3, 0, 0, 0, 0, 1, 1, 3, 6, 0, 1, 0, 2, 6, 0, 0, 0, 1, 4, 0, 0, 0, 0, 0, 0, 1, 0, 2, 1, 5, 12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 5, 0, 0, 0, 1, 0, 3, 10, 1, 6, 4, 12, 24, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
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OFFSET
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1,6
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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).
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LINKS
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EXAMPLE
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Triangle begins:
1
1
0 1
1 2
0 0 1
0 1 3
0 0 0 0 1
1 3 6
0 1 0 2 6
0 0 0 1 4
0 0 0 0 0 0 1
0 2 1 5 12
0 0 0 0 0 0 0 0 0 0 1
0 0 0 0 0 1 5
0 0 0 1 0 3 10
1 6 4 12 24
0 0 0 0 0 0 0 0 0 0 0 0 0 0 1
0 0 1 5 2 12 30
For example, row 12 gives: e(211) = 2m(22) + m(31) + 5m(211) + 12m(1111).
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MATHEMATICA
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primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
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[Table[Sum[Times@@Factorial/@Length/@Split[Sort[Length/@mtn, Greater]]/Times@@Factorial/@Length/@Split[mtn], {mtn, Select[mps[nrmptn[n]], And[And@@UnsameQ@@@#, Sort[Length/@#]==primeMS[k]]&]}], {k, Sort[Times@@Prime/@#&/@IntegerPartitions[Total[primeMS[n]]]]}], {n, 18}]
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CROSSREFS
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Cf. A008480, A049311, A056239, A116540, A124794, A124795, A300121, A319193, A321738, A321742-A321765, A321854.
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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