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%I #7 Nov 20 2018 12:21:19
%S 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,
%T 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,
%U 1,6,4,12,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0
%N 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.
%C Row n has length A000041(A056239(n)).
%C The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Symmetric_polynomial">Symmetric polynomial</a>
%e Triangle begins:
%e 1
%e 1
%e 0 1
%e 1 2
%e 0 0 1
%e 0 1 3
%e 0 0 0 0 1
%e 1 3 6
%e 0 1 0 2 6
%e 0 0 0 1 4
%e 0 0 0 0 0 0 1
%e 0 2 1 5 12
%e 0 0 0 0 0 0 0 0 0 0 1
%e 0 0 0 0 0 1 5
%e 0 0 0 1 0 3 10
%e 1 6 4 12 24
%e 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1
%e 0 0 1 5 2 12 30
%e For example, row 12 gives: e(211) = 2m(22) + m(31) + 5m(211) + 12m(1111).
%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
%t nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]]];
%t 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}]
%Y Row sums are A321743.
%Y Cf. A008480, A049311, A056239, A116540, A124794, A124795, A300121, A319193, A321738, A321742-A321765, A321854.
%K nonn,tabf
%O 1,6
%A _Gus Wiseman_, Nov 19 2018
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