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%I #9 Aug 28 2018 20:20:43
%S 1,1,2,6,22,93,421,2010,9926,50357,260728,1372436,7321982,39504181,
%T 215168221,1181540841,6534058589,36357935615,203414689462,
%U 1143589234086,6457159029573,36602333187792,208214459462774,1188252476400972,6801133579291811,39032172166792887
%N Number of free pure symmetric multifunctions whose leaves are an integer partition of n.
%C 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.
%H Andrew Howroyd, <a href="/A317652/b317652.txt">Table of n, a(n) for n = 0..200</a>
%e The a(4) = 22 free pure symmetric multifunctions:
%e 1[1[1[1]]] 1[1[2]] 1[3] 2[2] 4
%e 1[1[1][1]] 1[2[1]] 3[1]
%e 1[1][1[1]] 2[1[1]]
%e 1[1[1]][1] 1[1][2]
%e 1[1][1][1] 1[2][1]
%e 1[1[1,1]] 2[1][1]
%e 1[1,1[1]] 1[1,2]
%e 1[1][1,1] 2[1,1]
%e 1[1,1][1]
%e 1[1,1,1]
%t sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
%t 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]}]]]];
%t Table[Sum[Length[exprUsing[y]],{y,IntegerPartitions[n]}],{n,0,6}]
%o (PARI) EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
%o seq(n)={my(v=[]); for(n=1, n, my(t=EulerT(v)); v=concat(v, 1 + sum(k=1, n-1, v[k]*t[n-k]))); concat([1],v)} \\ _Andrew Howroyd_, Aug 28 2018
%Y Cf. A001003, A052893, A053492, A277996, A279944, A280000.
%Y Cf. A317653, A317654, A317655, A317656, A317658.
%K nonn
%O 0,3
%A _Gus Wiseman_, Aug 03 2018
%E Terms a(12) and beyond from _Andrew Howroyd_, Aug 28 2018
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