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A321745 Sum of coefficients of monomial symmetric functions in the homogeneous symmetric function of the integer partition with Heinz number n. 1
1, 1, 2, 3, 3, 6, 5, 10, 16, 12, 7, 27, 11, 20, 32, 47, 15, 76, 22, 56, 65, 35, 30, 136, 79, 54, 263, 114, 42, 191, 56, 246, 113, 86, 160, 476, 77, 128, 199, 344 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

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

Also the number of size-preserving permutations of multiset partitions of a multiset (such as row n of A305936) whose multiplicities are the prime indices of n.

LINKS

Table of n, a(n) for n=1..40.

Wikipedia, Symmetric polynomial

EXAMPLE

The sum of coefficients of h(211) = m(4) + 4m(22) + 3m(31) + 7m(211) + 12m(1111) is a(12) = 27.

The a(3) = 2 through a(9) = 16 size-preserving permutations of multiset partitions:

{11} {12} {111} {112} {1111} {123} {1122}

{1}{1} {1}{2} {1}{11} {1}{12} {1}{111} {1}{23} {1}{122}

{2}{1} {1}{1}{1} {2}{11} {11}{11} {2}{13} {11}{22}

{1}{1}{2} {1}{1}{11} {3}{12} {12}{12}

{1}{2}{1} {1}{1}{1}{1} {1}{2}{3} {2}{112}

{2}{1}{1} {1}{3}{2} {22}{11}

{2}{1}{3} {1}{1}{22}

{2}{3}{1} {1}{2}{12}

{3}{1}{2} {2}{1}{12}

{3}{2}{1} {2}{2}{11}

{1}{1}{2}{2}

{1}{2}{1}{2}

{1}{2}{2}{1}

{2}{1}{1}{2}

{2}{1}{2}{1}

{2}{2}{1}{1}

MATHEMATICA

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

mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];

nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]], {#1}]&, If[n==1, {}, Flatten[Cases[FactorInteger[n]//Reverse, {p_, k_}:>Table[PrimePi[p], {k}]]]]];

Table[Sum[Times@@Factorial/@Length/@Split[Sort[Length/@mtn, Greater]]/Times@@Factorial/@Length/@Split[mtn], {mtn, mps[nrmptn[n]]}], {n, 30}]

CROSSREFS

Row sums of A321744.

Cf. A005651, A007716, A008480, A056239, A124794, A124795, A181821, A255906, A296150, A318284, A319193, A319225, A319226, A321742-A321765.

Sequence in context: A187754 A347732 A075258 * A212629 A127779 A207634

Adjacent sequences: A321742 A321743 A321744 * A321746 A321747 A321748

KEYWORD

nonn,more

AUTHOR

Gus Wiseman, Nov 19 2018

STATUS

approved

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Last modified January 29 10:45 EST 2023. Contains 359921 sequences. (Running on oeis4.)