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A321742
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.
40
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
OFFSET
1,6
COMMENTS
Row n has length A000041(A056239(n)).
The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).
EXAMPLE
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).
MATHEMATICA
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}]
KEYWORD
nonn,tabf
AUTHOR
Gus Wiseman, Nov 19 2018
STATUS
approved