A317654 Number of free pure symmetric multifunctions whose leaves are a strongly normal multiset of size n.

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

Offset: 1

Author

Gus Wiseman, Aug 03 2018

Keywords

• nonn

Comments

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 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.

Examples

			The a(3) = 26 free pure symmetric multifunctions:
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].
		

Crossrefs

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

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

Programs

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

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

Extensions

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