A330467 Number of series-reduced rooted trees whose leaves are multisets whose multiset union is a strongly normal multiset of size n.

1, 1, 4, 18, 154, 1614, 23733, 396190, 8066984, 183930948, 4811382339, 138718632336, 4451963556127, 155416836338920, 5920554613563841, 242873491536944706, 10725017764009207613, 505671090907469848248, 25415190929321149684700, 1354279188424092012064226

Offset: 0

Author

Gus Wiseman, Dec 22 2019

Keywords

• nonn

Comments

A multiset is strongly normal if it covers an initial interval of positive integers with weakly decreasing multiplicities.

Also the number of different colorings of phylogenetic trees with n labels using strongly normal multisets of colors. A phylogenetic tree is a series-reduced rooted tree whose leaves are (usually disjoint) sets.

Examples

			The a(3) = 18 trees:
  {1,1,1}          {1,1,2}          {1,2,3}
  {{1},{1,1}}      {{1},{1,2}}      {{1},{2,3}}
  {{1},{1},{1}}    {{2},{1,1}}      {{2},{1,3}}
  {{1},{{1},{1}}}  {{1},{1},{2}}    {{3},{1,2}}
                   {{1},{{1},{2}}}  {{1},{2},{3}}
                   {{2},{{1},{1}}}  {{1},{{2},{3}}}
                                    {{2},{{1},{3}}}
                                    {{3},{{1},{2}}}
		

Crossrefs

The singleton-reduced version is A316652.

The unlabeled version is A330465.

Not requiring weakly decreasing multiplicities gives A330469.

The case where the leaves are sets is A330625.

Cf. A000311, A000669, A004114, A005121, A005804, A141268, A292504, A292505, A318812, A318849, A319312, A330471, A330475.

Programs

Mathematica

strnorm[n_]:=Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n];
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]]]];
multing[t_,n_]:=Array[(t+#-1)/#&,n,1,Times];
amemo[m_]:=amemo[m]=1+Sum[Product[multing[amemo[s[[1]]],Length[s]],{s,Split[c]}],{c,Select[mps[m],Length[#]>1&]}];
Table[Sum[amemo[m],{m,strnorm[n]}],{n,0,5}]

PARI

\\ See links in A339645 for combinatorial species functions.
cycleIndexSeries(n)={my(v=vector(n), p=sExp(x*sv(1) + O(x*x^n))); v[1]=sv(1); for(n=2, #v, v[n] = polcoef( sExp(x*Ser(v[1..n])), n ) + polcoef(p, n)); 1 + x*Ser(v)}
StronglyNormalLabelingsSeq(cycleIndexSeries(15)) \\ Andrew Howroyd, Dec 28 2020

Extensions

Terms a(10) and beyond from Andrew Howroyd, Dec 28 2020