A321751 Sum of coefficients of monomial symmetric functions in the power sum symmetric function of the integer partition with Heinz number n.

1, 1, 1, 3, 1, 2, 1, 10, 3, 2, 1, 7, 1, 2, 2, 47, 1, 6, 1, 6, 2, 2, 1, 26, 3, 2, 10, 6, 1, 6, 1, 246, 2, 2, 2, 26, 1, 2, 2, 24, 1, 5, 1, 6, 6, 2, 1, 138, 3, 6, 2, 6, 1, 23, 2, 23, 2, 2, 1, 20, 1, 2, 7, 1602, 2, 5, 1, 6, 2, 6, 1, 105, 1, 2, 6, 6, 2, 5, 1, 114

Offset: 1

Author

Gus Wiseman, Nov 20 2018

Keywords

• nonn

Comments

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

Also the number of ordered set partitions of {1, 2, ..., A001222(n)} whose blocks, when i is replaced by the i-th prime index of n, have weakly decreasing sums.

Examples

			The sum of coefficients of p(211) = m(4) + 2m(22) + 2m(31) + 2m(211) is a(12) = 7.
		

Links

• Wikipedia, Symmetric polynomial

Crossrefs

Row sums of A321750.

Cf. A005651, A008277, A008480, A056239, A124794, A124795, A296150, A319182, A319225, A319226, A321742-A321765.

Programs

Mathematica

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
Table[Sum[Times@@Factorial/@Length/@Split[Sort[Total/@s]],{s,sps[Range[PrimeOmega[n]]]/.Table[i->If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]][[i]],{i,PrimeOmega[n]}]}],{n,50}]