OFFSET
0,3
COMMENTS
Number of free pure symmetric multifunctions with n + 1 unlabeled leaves. A free pure symmetric multifunction f in PSM is either (case 1) f = the leaf symbol "o", or (case 2) f = an expression of the form h[g_1, ..., g_k] where k > 0, h is in PSM, each of the g_i for i = 1, ..., k is in PSM, and for i < j we have g_i <= g_j under a canonical total ordering of PSM, such as the Mathematica ordering of expressions. - Gus Wiseman, Aug 02 2018
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..1000
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 869
Maplesoft, Combstruct grammars.
Mathematica Reference, Orderless.
FORMULA
G.f.: 1/(1 - g(x)) where g(x) is the g.f. of A052891. - Andrew Howroyd, Aug 09 2020
EXAMPLE
From Gus Wiseman, Aug 02 2018: (Start)
The a(3) = 10 free pure symmetric multifunctions with 4 unlabeled leaves:
o[o[o[o]]]
o[o[o][o]]
o[o][o[o]]
o[o[o]][o]
o[o][o][o]
o[o[o,o]]
o[o,o[o]]
o[o][o,o]
o[o,o][o]
o[o,o,o]
(End)
MAPLE
spec := [S, {C = Set(B, 1 <= card), B=Prod(Z, S), S=Sequence(C)}, unlabeled]:
seq(combstruct[count](spec, size=n), n=0..20);
MATHEMATICA
multing[t_, n_]:=Array[(t+#-1)/#&, n, 1, Times];
a[n_]:=a[n]=If[n==1, 1, Sum[a[k]*Sum[Product[multing[a[First[s]], Length[s]], {s, Split[p]}], {p, IntegerPartitions[n-k]}], {k, 1, n-1}]];
Array[a, 30] (* Gus Wiseman, Aug 02 2018 *)
PROG
(PARI) EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
seq(n)={my(v=[1]); for(n=1, n, v=Vec(1/(1-x*Ser(EulerT(v))))); v} \\ Andrew Howroyd, Aug 09 2020
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
encyclopedia(AT)pommard.inria.fr, Jan 25 2000
EXTENSIONS
More terms from Gus Wiseman, Aug 02 2018
STATUS
approved