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A321899 Irregular triangle read by rows where T(H(u),H(v)) is the coefficient of p(v) in F(u), where H is Heinz number, F is augmented forgotten symmetric functions, and p is power sum symmetric functions.

%I A321899 #4 Nov 21 2018 09:25:07
%S A321899 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 A321899 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,
%U A321899 -1,0,0,0,0,6,3,8,6,1,1,0,0,0,0,0,0,0,0,0,0
%N A321899 Irregular triangle read by rows where T(H(u),H(v)) is the coefficient of p(v) in F(u), where H is Heinz number, F is augmented forgotten symmetric functions, and p is power sum symmetric functions.
%C A321899 Row n has length A000041(A056239(n)).
%C A321899 The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).
%C A321899 The augmented forgotten symmetric functions are given by F(y) = c(y) * f(y) where f is forgotten symmetric functions and c(y) = Product_i (y)_i!, where (y)_i is the number of i's in y.
%H A321899 Wikipedia, <a href="https://en.wikipedia.org/wiki/Symmetric_polynomial">Symmetric polynomial</a>
%e A321899 Triangle begins:
%e A321899    1
%e A321899    1
%e A321899   -1   0
%e A321899    1   1
%e A321899    1   0   0
%e A321899   -1  -1   0
%e A321899   -1   0   0   0   0
%e A321899    2   3   1
%e A321899    1   1   0   0   0
%e A321899    1   0   1   0   0
%e A321899    1   0   0   0   0   0   0
%e A321899   -2  -1  -2  -1   0
%e A321899   -1   0   0   0   0   0   0   0   0   0   0
%e A321899   -1  -1   0   0   0   0   0
%e A321899   -1   0  -1   0   0   0   0
%e A321899    6   3   8   6   1
%e A321899    1   0   0   0   0   0   0   0   0   0   0   0   0   0   0
%e A321899    2   1   2   1   0   0   0
%e A321899 For example, row 12 gives: F(211) = -2p(4) - p(22) - 2p(31) - p(211).
%t A321899 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A321899 sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t A321899 Table[Sum[(-1)^(Total[primeMS[m]]-PrimeOmega[m])*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 A321899 Row sums are A130675, up to sign. Same as A321895, up to sign.
%Y A321899 Cf. A005651, A008277, A008480, A048994, A056239, A124794, A124795, A319193, A321742-A321765.
%K A321899 sign,tabf
%O A321899 1,18
%A A321899 _Gus Wiseman_, Nov 20 2018