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A321744 Irregular triangle read by rows where T(H(u),H(v)) is the coefficient of m(v) in h(u), where H is Heinz number, m is monomial symmetric functions, and h is homogeneous symmetric functions.

%I A321744 #6 Nov 20 2018 12:21:32
%S A321744 1,1,1,1,1,2,1,1,1,1,2,3,1,1,1,1,1,1,3,6,1,3,2,4,6,1,2,2,3,4,1,1,1,1,
%T A321744 1,1,1,1,4,3,7,12,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,5,1,2,3,5,4,7,10,
%U A321744 1,6,4,12,24,1,1,1,1,1,1,1,1,1,1,1,1,1,1
%N A321744 Irregular triangle read by rows where T(H(u),H(v)) is the coefficient of m(v) in h(u), where H is Heinz number, m is monomial symmetric functions, and h is homogeneous symmetric functions.
%C A321744 Row n has length A000041(A056239(n)).
%C A321744 The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).
%C A321744 Also the number of size-preserving permutations of type-v multiset partitions of a multiset whose multiplicities are the parts of u.
%C A321744 Also the coefficient of f(v) in e(u), where e is elementary symmetric functions and f is forgotten symmetric functions.
%H A321744 Wikipedia, <a href="https://en.wikipedia.org/wiki/Symmetric_polynomial">Symmetric polynomial</a>
%e A321744 Triangle begins:
%e A321744    1
%e A321744    1
%e A321744    1   1
%e A321744    1   2
%e A321744    1   1   1
%e A321744    1   2   3
%e A321744    1   1   1   1   1
%e A321744    1   3   6
%e A321744    1   3   2   4   6
%e A321744    1   2   2   3   4
%e A321744    1   1   1   1   1   1   1
%e A321744    1   4   3   7  12
%e A321744    1   1   1   1   1   1   1   1   1   1   1
%e A321744    1   2   2   3   3   4   5
%e A321744    1   2   3   5   4   7  10
%e A321744    1   6   4  12  24
%e A321744    1   1   1   1   1   1   1   1   1   1   1   1   1   1   1
%e A321744    1   3   5  11   8  18  30
%e A321744 For example, row 12 gives: h(211) = m(4) + 4m(22) + 3m(31) + 7m(211) + 12m(1111).
%t A321744 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A321744 sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t A321744 mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
%t A321744 nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]]];
%t A321744 Table[Table[Sum[Times@@Factorial/@Length/@Split[Sort[Length/@mtn,Greater]]/Times@@Factorial/@Length/@Split[mtn],{mtn,Select[mps[nrmptn[n]],Sort[Length/@#]==primeMS[k]&]}],{k,Sort[Times@@Prime/@#&/@IntegerPartitions[Total[primeMS[n]]]]}],{n,18}]
%Y A321744 Row sums are A321745.
%Y A321744 Cf. A005651, A007716, A008480, A056239, A124794, A124795, A255906, A300121, A321742-A321765, A321854.
%K A321744 nonn,tabf
%O A321744 1,6
%A A321744 _Gus Wiseman_, Nov 19 2018