OFFSET
1,2
COMMENTS
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.
LINKS
Andrew Howroyd, Table of n, a(n) for n = 1..200
EXAMPLE
The a(3) = 34 free pure symmetric multifunctions:
1[1[1]], 1[1,1], 1[1][1],
1[2[2]], 1[2,2], 2[1[2]], 2[2[1]], 2[1,2], 1[2][2], 2[1][2], 2[2][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].
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, Join@@Permutations/@IntegerPartitions[n]}], {n, 6}]
PROG
(PARI) \\ here R(n, 1) is A052893.
EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
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}
seq(n)={sum(k=1, n, R(n, k)*sum(r=k, n, binomial(r, k)*(-1)^(r-k)) )} \\ Andrew Howroyd, Sep 14 2018
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Aug 03 2018
EXTENSIONS
Terms a(8) and beyond from Andrew Howroyd, Sep 14 2018
STATUS
approved