%I #11 Sep 15 2018 02:05:27

%S 1,3,34,602,14872,472138,18323359,840503724,44489123726,2668985463839,

%T 178960530393633,13263068003965046,1076580864432281157,

%U 94987639225399100006,9051397653144246683937,926407121115738135640677,101357200280211387377806719,11804887470887800839909147484

%N Number of free pure symmetric multifunctions whose leaves are a normal multiset of size n.

%C A multiset is normal if it spans an initial interval of positive integers. 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="/A317653/b317653.txt">Table of n, a(n) for n = 1..200</a>

%e The a(3) = 34 free pure symmetric multifunctions:

%e 1[1[1]], 1[1,1], 1[1][1],

%e 1[2[2]], 1[2,2], 2[1[2]], 2[2[1]], 2[1,2], 1[2][2], 2[1][2], 2[2][1],

%e 1[1[2]], 1[2[1]], 1[1,2], 2[1[1]], 2[1,1], 1[1][2], 1[2][1], 2[1][1],

%e 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].

%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 got[y_]:=Join@@Table[Table[i,{y[[i]]}],{i,Range[Length[y]]}];

%t Table[Sum[Length[exprUsing[got[y]]],{y,Join@@Permutations/@IntegerPartitions[n]}],{n,6}]

%o (PARI) \\ here R(n,1) is A052893.

%o EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}

%o R(n,k)={my(v=[k]); for(n=2, n, my(t=EulerT(v)); v=concat(v, sum(k=1, n-1, v[k]*t[n-k]))); v}

%o seq(n)={sum(k=1, n, R(n,k)*sum(r=k, n, binomial(r,k)*(-1)^(r-k)) )} \\ _Andrew Howroyd_, Sep 14 2018

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

%Y Cf. A317652, A317654, A317655, A317656, A317658.

%K nonn

%O 1,2

%A _Gus Wiseman_, Aug 03 2018

%E Terms a(8) and beyond from _Andrew Howroyd_, Sep 14 2018