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%I #7 Sep 15 2018 15:48:55
%S 1,-1,1,2,-3,1,-6,3,8,-6,1,24,-30,-20,15,20,-10,1,-120,90,144,40,-15,
%T -90,-120,45,40,-15,1,720,-840,-504,-420,630,504,210,280,-105,-210,
%U -420,105,70,-21,1,-5040,5760,3360,1260,-3360,2688,-1260,-4032,-3360,-1120
%N Irregular triangle where T(n,k) is the coefficient of p(y) in n! * Sum_{i1 < ... < in} (x_i1 * ... * x_in), where p is power-sum symmetric functions and y is the integer partition with Heinz number A215366(n,k).
%C A generalization of the triangle of Stirling numbers of the first kind, these are the coefficients appearing in the expansion of single-part augmented elementary symmetric functions in terms of power-sum symmetric functions.
%e Triangle begins:
%e 1
%e -1 1
%e 2 -3 1
%e -6 3 8 -6 1
%e 24 -30 -20 15 20 -10 1
%e The fourth row corresponds to the symmetric function identity: 24 e(4) = -6 p(4) + 3 p(22) + 8 p(31) - 6 p(211) + p(1111).
%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t numPermsOfType[ptn_]:=Total[ptn]!/Times@@ptn/Times@@Factorial/@Length/@Split[ptn];
%t Table[(-1)^(Total[primeMS[m]]-PrimeOmega[m])*numPermsOfType[primeMS[m]],{n,5},{m,Sort[Times@@Prime/@#&/@IntegerPartitions[n]]}]
%Y Other row orderings are A036039 and A102189.
%Y Cf. A000041, A005651, A008277, A008480, A048994, A056239, A124794, A124795, A215366, A318762, A319182, A319191.
%K sign,tabf
%O 1,4
%A _Gus Wiseman_, Sep 13 2018
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