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A317655
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Number of free pure symmetric multifunctions with leaves a multiset whose multiplicities are the integer partition with Heinz number n.
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7
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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
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OFFSET
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1,4
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COMMENTS
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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.
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LINKS
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EXAMPLE
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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]
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MATHEMATICA
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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}]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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