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A321751
Sum of coefficients of monomial symmetric functions in the power sum symmetric function of the integer partition with Heinz number n.
4
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.
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}]
KEYWORD
nonn
AUTHOR
Gus Wiseman, Nov 20 2018
STATUS
approved