%I #4 Nov 21 2018 09:24:38
%S 1,1,1,0,-1,1,1,0,0,-1,1,0,1,0,0,0,0,2,-3,1,-1,1,0,0,0,-1,0,1,0,0,1,0,
%T 0,0,0,0,0,2,-1,-2,1,0,1,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,-1,0,1,0,
%U 0,0,0,-6,3,8,-6,1,1,0,0,0,0,0,0,0,0,0,0
%N Irregular triangle read by rows where T(H(u),H(v)) is the coefficient of p(v) in M(u), where H is Heinz number, M is augmented monomial symmetric functions, and p is power sum 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).
%C The augmented monomial symmetric functions are given by M(y) = c(y) * m(y) where c(y) = Product_i (y)_i! where (y)_i is the number of i's in y and m is monomial symmetric functions.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Symmetric_polynomial">Symmetric polynomial</a>
%e Triangle begins:
%e 1
%e 1
%e 1 0
%e -1 1
%e 1 0 0
%e -1 1 0
%e 1 0 0 0 0
%e 2 -3 1
%e -1 1 0 0 0
%e -1 0 1 0 0
%e 1 0 0 0 0 0 0
%e 2 -1 -2 1 0
%e 1 0 0 0 0 0 0 0 0 0 0
%e -1 1 0 0 0 0 0
%e -1 0 1 0 0 0 0
%e -6 3 8 -6 1
%e 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0
%e 2 -1 -2 1 0 0 0
%e For example, row 12 gives: M(211) = 2p(4) - p(22) - 2p(31) + p(211).
%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 Table[Sum[Product[(-1)^(Length[t]-1)*(Length[t]-1)!,{t,s}],{s,Select[sps[Range[PrimeOmega[n]]]/.Table[i->If[n==1,{},primeMS[n]][[i]],{i,PrimeOmega[n]}],Times@@Prime/@Total/@#==m&]}],{n,18},{m,Sort[Times@@Prime/@#&/@IntegerPartitions[Total[primeMS[n]]]]}]
%Y Row sums are A080339.
%Y Cf. A005651, A008277, A008480, A048994, A056239, A124794, A124795, A319193, A321742-A321765.
%K sign,tabf
%O 1,18
%A _Gus Wiseman_, Nov 20 2018