A316654 - OEIS

A316654

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

1, 1, 5, 39, 387, 4960, 74088, 1312716, 26239484, 595023510, 14908285892, 412903136867, 12448252189622, 407804188400373, 14380454869464352, 544428684832123828, 21991444994187529639, 945234507638271696504, 43042162953650721470752, 2071216980365429970912347

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OFFSET

1,3

COMMENTS

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.

LINKS

Table of n, a(n) for n=1..20.

EXAMPLE

The a(3) = 5 trees are (1(12)), (1(23)), (2(13)), (3(12)), (123).

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]]]];

gro[m_]:=If[Length[m]==1, m, Select[Union[Sort/@Join@@(Tuples[gro/@#]&/@Select[mps[m], Length[#]>1&])], UnsameQ@@#&]];

Table[Sum[Length[gro[m]], {m, Flatten[MapIndexed[Table[#2, {#1}]&, #]]&/@IntegerPartitions[n]}], {n, 5}]

PROG

(PARI) \\ See links in A339645 for combinatorial species functions.

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)}

StronglyNormalLabelingsSeq(cycleIndexSeries(12)) \\ Andrew Howroyd, Jan 22 2021

CROSSREFS

Cf. A000081, A000311, A000669, A001678, A004111, A005804, A141268, A181821, A292504, A300660, A304660.

Cf. A316651, A316652, A316653, A316655, A316656.

Sequence in context: A277424 A182954 A215506 * A070767 A124549 A308939

Adjacent sequences: A316651 A316652 A316653 * A316655 A316656 A316657

KEYWORD

nonn

AUTHOR

Gus Wiseman, Jul 09 2018

EXTENSIONS

Terms a(9) and beyond from Andrew Howroyd, Jan 22 2021

STATUS

approved