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1, 3, 26, 375, 6696, 159837, 4389226, 144915350, 5377002075, 227624621051, 10632808475596, 550932945236121, 31062550998284221, 1907051034025848314, 126052420069459211076, 8956882232940915920404, 679298518935625486287703, 54868537321267493152151502, 4696952405203792017289469056

(PARI) \\ See links in A339645 for combinatorial species functions.

cycleIndexSeries(n)={my(p=O(x)); for(n=1, n, p = x*sv(1) + p*(sExp(p)-1)); p}

StronglyNormalLabelingsSeq(cycleIndexSeries(15)) \\ Andrew Howroyd, Jan 01 2021

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Terms a(8) and beyond from Andrew Howroyd, Jan 01 2021

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A multiset is strongly normal if it spans an initial interval of positive integers with weakly decreasing multiplicities. A free pure symmetric multifunction f in PSM 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 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 a(3) = 26 expressionsfree pure symmetric multifunctions:

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

Cf. A317652, A317653, A317655, A317656, A317658.

allocated for Gus WisemanNumber of free pure symmetric multifunctions whose leaves are a strongly normal multiset of size n.

1, 3, 26, 375, 6696, 159837, 4389226

1,2

A multiset is strongly normal if it spans an initial interval of positive integers with weakly decreasing multiplicities. A free pure symmetric multifunction f in PSM 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 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(3) = 26 expressions:

1[1[1]], 1[1,1], 1[1][1],

1[1[2]], 1[2[1]], 1[1,2], 2[1[1]], 2[1,1], 1[1][2], 1[2][1], 2[1][1],

1[2[3]], 1[3[2]], 1[2,3], 2[1[3]], 2[3[1]], 2[1,3], 3[1[2]], 3[2[1]], 3[1,2], 1[2][3], 2[1][3], 1[3][2], 3[1][2], 2[3][1], 3[2][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]]}];

Table[Sum[Length[exprUsing[got[y]]], {y, IntegerPartitions[n]}], {n, 6}]

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

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Gus Wiseman, Aug 03 2018

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