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 A317655 Number of free pure symmetric multifunctions with leaves a multiset whose multiplicities are the integer partition with Heinz number n. 7
 0, 1, 1, 2, 3, 8, 10, 15, 50, 35, 37, 96, 144, 160, 299, 184, 589, 840, 2483, 578, 1729, 750, 10746, 1627, 2246, 3578, 9357, 3367, 47420, 6397, 212668, 3155, 9818, 17280, 15666, 18250, 966324, 84232, 54990, 12471, 4439540, 45015 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,4 COMMENTS The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k). A free pure symmetric multifunction f in EPSM is either (case 1) a positive integer, or (case 2) an expression of the form h[g_1, ..., g_k] where k > 0, h is in EPSM, each of the g_i for i = 1, ..., k is in EPSM, and for i < j we have g_i <= g_j under a canonical total ordering of EPSM, such as the Mathematica ordering of expressions. LINKS Table of n, a(n) for n=1..42. EXAMPLE The a(6) = 8 free pure symmetric multifunctions: 1[1[2]] 1[2[1]] 2[1[1]] 1[1][2] 1[2][1] 2[1][1] 1[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]]]]; exprUsing[m_]:=exprUsing[m]=If[Length[m]==0, {}, If[Length[m]==1, {First[m]}, Join@@Cases[Union[Table[PR[m[[s]], m[[Complement[Range[Length[m]], s]]]], {s, Take[Subsets[Range[Length[m]]], {2, -2}]}]], PR[h_, g_]:>Join@@Table[Apply@@@Tuples[{exprUsing[h], Union[Sort/@Tuples[exprUsing/@p]]}], {p, mps[g]}]]]]; got[y_]:=Join@@Table[Table[i, {y[[i]]}], {i, Range[Length[y]]}]; primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]]; Table[Length[exprUsing[got[Reverse[primeMS[n]]]]], {n, 40}] CROSSREFS Cf. A001003, A052893, A053492, A255906, A277996, A279944, A280000. Cf. A317652, A317653, A317654, A317656, A317658. Sequence in context: A286092 A100317 A295030 * A060697 A326714 A240217 Adjacent sequences: A317652 A317653 A317654 * A317656 A317657 A317658 KEYWORD nonn AUTHOR Gus Wiseman, Aug 03 2018 STATUS approved

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Last modified July 20 15:17 EDT 2024. Contains 374459 sequences. (Running on oeis4.)