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A321895 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.

%I A321895 #4 Nov 21 2018 09:24:38
%S A321895 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 A321895 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 A321895 0,0,0,-6,3,8,-6,1,1,0,0,0,0,0,0,0,0,0,0
%N A321895 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 A321895 Row n has length A000041(A056239(n)).
%C A321895 The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).
%C A321895 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 A321895 Wikipedia, <a href="https://en.wikipedia.org/wiki/Symmetric_polynomial">Symmetric polynomial</a>
%e A321895 Triangle begins:
%e A321895    1
%e A321895    1
%e A321895    1   0
%e A321895   -1   1
%e A321895    1   0   0
%e A321895   -1   1   0
%e A321895    1   0   0   0   0
%e A321895    2  -3   1
%e A321895   -1   1   0   0   0
%e A321895   -1   0   1   0   0
%e A321895    1   0   0   0   0   0   0
%e A321895    2  -1  -2   1   0
%e A321895    1   0   0   0   0   0   0   0   0   0   0
%e A321895   -1   1   0   0   0   0   0
%e A321895   -1   0   1   0   0   0   0
%e A321895   -6   3   8  -6   1
%e A321895    1   0   0   0   0   0   0   0   0   0   0   0   0   0   0
%e A321895    2  -1  -2   1   0   0   0
%e A321895 For example, row 12 gives: M(211) = 2p(4) - p(22) - 2p(31) + p(211).
%t A321895 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A321895 sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t A321895 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 A321895 Row sums are A080339.
%Y A321895 Cf. A005651, A008277, A008480, A048994, A056239, A124794, A124795, A319193, A321742-A321765.
%K A321895 sign,tabf
%O A321895 1,18
%A A321895 _Gus Wiseman_, Nov 20 2018