A321751 - OEIS

%I #5 Nov 20 2018 16:30:38

%S 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,

%T 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,

%U 2,7,1602,2,5,1,6,2,6,1,105,1,2,6,6,2,5,1,114

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

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

%C 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.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Symmetric_polynomial">Symmetric polynomial</a>

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

%t sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];

%t 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}]

%Y Row sums of A321750.

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

%K nonn

%O 1,4

%A _Gus Wiseman_, Nov 20 2018