cp's OEIS Frontend

A316652 Number of series-reduced rooted trees whose leaves span an initial interval of positive integers with multiplicities an integer partition of n.

%I A316652 #15 Jan 22 2021 16:49:37
%S A316652 1,2,9,69,623,7793,110430,1906317,36833614,816101825,19925210834,
%T A316652 541363267613,15997458049946,515769374925576,17905023985615254,
%U A316652 669030297769291562,26689471638523499483,1134895275721374771655,51161002326406795249910,2440166138715867838359915
%N A316652 Number of series-reduced rooted trees whose leaves span an initial interval of positive integers with multiplicities an integer partition of n.
%C A316652 A rooted tree is series-reduced if every non-leaf node has at least two branches.
%e A316652 The a(3) = 9 trees:
%e A316652 (1(11)), (111),
%e A316652 (1(12)), (2(11)), (112),
%e A316652 (1(23)), (2(13)), (3(12)), (123).
%t A316652 sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t A316652 mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
%t A316652 gro[m_]:=If[Length[m]==1,m,Union[Sort/@Join@@(Tuples[gro/@#]&/@Select[mps[m],Length[#]>1&])]];
%t A316652 Table[Sum[Length[gro[m]],{m,Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n]}],{n,4}]
%o A316652 (PARI) \\ See A339645 for combinatorial species functions.
%o A316652 cycleIndexSeries(n)={my(v=vector(n)); v[1]=sv(1); for(n=2, #v, v[n] = polcoef( sExp(x*Ser(v[1..n])), n )); x*Ser(v)}
%o A316652 StronglyNormalLabelingsSeq(cycleIndexSeries(15)) \\ _Andrew Howroyd_, Jan 04 2021
%Y A316652 Cf. A000081, A000311, A000669, A001678, A005804, A141268, A181821, A292504, A304660.
%Y A316652 Cf. A316651, A316653, A316654, A316655, A316656.
%K A316652 nonn
%O A316652 1,2
%A A316652 _Gus Wiseman_, Jul 09 2018
%E A316652 Terms a(10) and beyond from _Andrew Howroyd_, Jan 04 2021