OFFSET
1,4
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.
EXAMPLE
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).
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]]}]
CROSSREFS
KEYWORD
sign,tabf
AUTHOR
Gus Wiseman, Sep 13 2018
STATUS
approved