OFFSET
0,3
COMMENTS
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 = 0..200
EXAMPLE
The a(4) = 22 free pure symmetric multifunctions:
1[1[1[1]]] 1[1[2]] 1[3] 2[2] 4
1[1[1][1]] 1[2[1]] 3[1]
1[1][1[1]] 2[1[1]]
1[1[1]][1] 1[1][2]
1[1][1][1] 1[2][1]
1[1[1,1]] 2[1][1]
1[1,1[1]] 1[1,2]
1[1][1,1] 2[1,1]
1[1,1][1]
1[1,1,1]
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]}]]]];
Table[Sum[Length[exprUsing[y]], {y, IntegerPartitions[n]}], {n, 0, 6}]
PROG
(PARI) EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
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
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Aug 03 2018
EXTENSIONS
Terms a(12) and beyond from Andrew Howroyd, Aug 28 2018
STATUS
approved