A321747 Sum of coefficients of elementary symmetric functions in the monomial symmetric function of the integer partition with Heinz number n.

1, 1, -1, 1, 1, -2, -1, 1, 1, 2, 1, -3, -1, -2, -2, 1, 1, 3, -1, 3, 2, 2, 1, -4, 1, -2, -1, -3, -1, -6, 1, 1, -2, 2, -2, 6, -1, -2, 2, 4, 1, 6, -1, 3, 3, 2, 1, -5, 1, 3, -2, -3, -1, -4, 2, -4, 2, -2, 1, -12, -1, 2, -3, 1, -2, -6, 1, 3, -2, -6, -1, 10, 1, -2

Offset: 1

Author

Gus Wiseman, Nov 19 2018

Keywords

• sign

Comments

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

Examples

			The sum of coefficients of m(2211) = 9e(6) + e(42) - 4e(51) is a(36) = 6.
		

Links

• Wikipedia, Symmetric polynomial

Crossrefs

Row sums of A321746. An unsigned version is A008480.

Cf. A005651, A056239, A124794, A124795, A296150, A321738, A321742-A321765.

Programs

Mathematica

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[(-1)^(Total[primeMS[n]]-PrimeOmega[n])*Length[Permutations[primeMS[n]]],{n,50}]

Formula

a(n) = (-1)^(A056239(n) - A001222(n)) * A008480(n).