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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.

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