A319192 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).

1, -1, 1, 2, -3, 1, -6, 3, 8, -6, 1, 24, -30, -20, 15, 20, -10, 1, -120, 90, 144, 40, -15, -90, -120, 45, 40, -15, 1, 720, -840, -504, -420, 630, 504, 210, 280, -105, -210, -420, 105, 70, -21, 1, -5040, 5760, 3360, 1260, -3360, 2688, -1260, -4032, -3360, -1120

Offset: 1

Author

Gus Wiseman, Sep 13 2018

Keywords

• sign
• tabf

Comments

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.

Examples

			Triangle begins:
   1
  -1   1
   2  -3   1
  -6   3   8  -6   1
  24 -30 -20  15  20 -10   1
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).
		

Crossrefs

Other row orderings are A036039 and A102189.

Cf. A000041, A005651, A008277, A008480, A048994, A056239, A124794, A124795, A215366, A318762, A319182, A319191.

Programs

Mathematica

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
numPermsOfType[ptn_]:=Total[ptn]!/Times@@ptn/Times@@Factorial/@Length/@Split[ptn];
Table[(-1)^(Total[primeMS[m]]-PrimeOmega[m])*numPermsOfType[primeMS[m]],{n,5},{m,Sort[Times@@Prime/@#&/@IntegerPartitions[n]]}]