cp's OEIS Frontend

A316654 Number of series-reduced rooted identity trees whose leaves span an initial interval of positive integers with multiplicities an integer partition of n.

%I A316654 #11 Jan 22 2021 16:49:01
%S A316654 1,1,5,39,387,4960,74088,1312716,26239484,595023510,14908285892,
%T A316654 412903136867,12448252189622,407804188400373,14380454869464352,
%U A316654 544428684832123828,21991444994187529639,945234507638271696504,43042162953650721470752,2071216980365429970912347
%N A316654 Number of series-reduced rooted identity trees whose leaves span an initial interval of positive integers with multiplicities an integer partition of n.
%C A316654 A rooted tree is series-reduced if every non-leaf node has at least two branches. It is an identity tree if no branch appears multiple times under the same root.
%e A316654 The a(3) = 5 trees are (1(12)), (1(23)), (2(13)), (3(12)), (123).
%t A316654 sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t A316654 mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
%t A316654 gro[m_]:=If[Length[m]==1,m,Select[Union[Sort/@Join@@(Tuples[gro/@#]&/@Select[mps[m],Length[#]>1&])],UnsameQ@@#&]];
%t A316654 Table[Sum[Length[gro[m]],{m,Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n]}],{n,5}]
%o A316654 (PARI) \\ See links in A339645 for combinatorial species functions.
%o A316654 cycleIndexSeries(n)={my(v=vector(n)); v[1]=sv(1); for(n=2, #v, v[n]=polcoef(sWeighT(x*Ser(v[1..n])), n)); x*Ser(v)}
%o A316654 StronglyNormalLabelingsSeq(cycleIndexSeries(12)) \\ _Andrew Howroyd_, Jan 22 2021
%Y A316654 Cf. A000081, A000311, A000669, A001678, A004111, A005804, A141268, A181821, A292504, A300660, A304660.
%Y A316654 Cf. A316651, A316652, A316653, A316655, A316656.
%K A316654 nonn
%O A316654 1,3
%A A316654 _Gus Wiseman_, Jul 09 2018
%E A316654 Terms a(9) and beyond from _Andrew Howroyd_, Jan 22 2021