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

1,4

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.

Links

Table of n, a(n) for n=1..80.

Wikipedia, Symmetric polynomial

Example

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

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}]

Crossrefs

Row sums of A321750.

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

Sequence in context: A340147 A011086 A321889 * A364906 A188584 A103514

Adjacent sequences: A321748 A321749 A321750 * A321752 A321753 A321754

Keyword

nonn

Author

Gus Wiseman, Nov 20 2018

Status

approved