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A free pure symmetric multifunction f in EPSM is either (case 1) f = the leaf symbol "o", a positive integer, or (case 2) f = 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.

A free pure symmetric multifunction f in PSM EPSM is either (case 1) f = the leaf symbol "o", or (case 2) f = an expression of the form h[g_1, ..., g_k] where k > 0, h is in PSM, EPSM, each of the g_i for i = 1, ..., k is in PSM, EPSM, and for i < j we have g_i <= g_j under a canonical total ordering of PSM, EPSM, such as the Mathematica ordering of expressions.

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

Cf. A001003, A052893, A053492, A255906, A277996, A279944, A280000, A317652, A317653, A317654.

Cf. A317652, A317653, A317654, A317656, A317658.

allocated for Gus WisemanNumber of free pure symmetric multifunctions with leaves a multiset whose multiplicities are the integer partition with Heinz number n.

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

1,4

A free pure symmetric multifunction f in PSM is either (case 1) f = the leaf symbol "o", or (case 2) f = an expression of the form h[g_1, ..., g_k] where k > 0, h is in PSM, each of the g_i for i = 1, ..., k is in PSM, and for i < j we have g_i <= g_j under a canonical total ordering of PSM, such as the Mathematica ordering of expressions.

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]

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}]

Cf. A001003, A052893, A053492, A255906, A277996, A279944, A280000, A317652, A317653, A317654.

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Gus Wiseman, Aug 03 2018

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editing

allocated for Gus Wiseman

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