A319312 -id:A319312 - cp's OEIS frontend

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

Gus Wiseman, Dec 22 2019

nonn

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.

			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}}}

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.

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}]

\\ 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

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

A330469 Number of series-reduced rooted trees whose leaves are multisets with a total of n elements covering an initial interval of positive integers.

Original entry on oeis.org

1, 1, 4, 24, 250, 3744, 73408, 1768088, 50468854, 1664844040, 62304622320, 2607765903568, 120696071556230, 6120415124163512, 337440974546042416, 20096905939846645064, 1285779618228281270718, 87947859243850506008984, 6404472598196204610148232

Offset: 0

Gus Wiseman, Dec 22 2019

nonn

Also the number of different colorings of phylogenetic trees with n labels using a multiset of colors covering an initial interval of positive integers. A phylogenetic tree is a series-reduced rooted tree whose leaves are (usually disjoint) sets.

			The a(3) = 24 trees:
  (123)          (122)          (112)          (111)
  ((1)(23))      ((1)(22))      ((1)(12))      ((1)(11))
  ((2)(13))      ((2)(12))      ((2)(11))      ((1)(1)(1))
  ((3)(12))      ((1)(2)(2))    ((1)(1)(2))    ((1)((1)(1)))
  ((1)(2)(3))    ((1)((2)(2)))  ((1)((1)(2)))
  ((1)((2)(3)))  ((2)((1)(2)))  ((2)((1)(1)))
  ((2)((1)(3)))
  ((3)((1)(2)))

Andrew Howroyd, Table of n, a(n) for n = 0..200

The singleton-reduced version is A316651.
The unlabeled version is A330465.
The strongly normal case is A330467.
The case when leaves are sets is A330764.
Row sums of A330762.
Cf. A000311, A000669, A004114, A005121, A005804, A141268, A292504, A292505, A316652, A318812, A318849, A319312, A330625.

allnorm[n_]:=If[n<=0,{{}},Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1]];
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,allnorm[n]}],{n,0,5}]

EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
R(n, k)={my(v=[]); for(n=1, n, v=concat(v, EulerT(concat(v, [binomial(n+k-1, k-1)]))[n])); v}
seq(n)={concat([1], sum(k=1, n, R(n,k)*sum(r=k, n, binomial(r,k)*(-1)^(r-k))))} \\ Andrew Howroyd, Dec 29 2019

Terms a(9) and beyond from Andrew Howroyd, Dec 29 2019

A320169 Number of balanced enriched p-trees of weight n.

Original entry on oeis.org

1, 2, 3, 6, 9, 20, 31, 70, 114, 243, 415, 961, 1603, 3564, 6559, 14913, 26630, 60037, 110160, 248859, 458445, 1001190, 1882350, 4220358, 7765303, 16822107, 32307240, 70081784, 133716083, 291788153, 561823990, 1230204229, 2396185727, 5176454708, 10220127290

Offset: 1

Gus Wiseman, Oct 07 2018

nonn

An enriched p-tree of weight n is either the number n itself or a finite sequence of enriched p-trees whose weights are weakly decreasing and sum to n.

A tree is balanced if all leaves have the same height.

			The a(1) = 1 through a(6) = 20 balanced enriched p-trees:
  1  2     3      4           5            6
     (11)  (21)   (22)        (32)         (33)
           (111)  (31)        (41)         (42)
                  (211)       (221)        (51)
                  (1111)      (311)        (222)
                  ((11)(11))  (2111)       (321)
                              (11111)      (411)
                              ((21)(11))   (2211)
                              ((111)(11))  (3111)
                                           (21111)
                                           (111111)
                                           ((21)(21))
                                           ((22)(11))
                                           ((31)(11))
                                           ((111)(21))
                                           ((21)(111))
                                           ((211)(11))
                                           ((111)(111))
                                           ((1111)(11))
                                           ((11)(11)(11))

Andrew Howroyd, Table of n, a(n) for n = 1..500

Cf. A000311, A000669, A001678, A005804, A048816, A079500, A119262, A120803, A141268, A196545, A289501, A319312.

Cf. A316624, A320154, A320155, A320160, A320179.

eptrs[n_]:=Prepend[Join@@Table[Tuples[eptrs/@p],{p,Rest[IntegerPartitions[n]]}],n];
Table[Length[Select[eptrs[n],SameQ@@Length/@Position[#,_Integer]&]],{n,12}]

seq(n)={my(p=x/(1-x) + O(x*x^n), q=0); while(p, q+=p; p = 1/prod(k=1, n, 1 - polcoef(p,k)*x^k + O(x*x^n)) - 1 - p); Vec(q)} \\ Andrew Howroyd, Oct 26 2018

Terms a(16) and beyond from Andrew Howroyd, Oct 26 2018

A320179 Regular triangle where T(n,k) is the number of unlabeled series-reduced rooted trees with n leaves in which every leaf is at height k.

Original entry on oeis.org

1, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 3, 0, 0, 0, 0, 1, 3, 0, 0, 0, 0, 0, 1, 6, 1, 0, 0, 0, 0, 0, 1, 7, 1, 0, 0, 0, 0, 0, 0, 1, 11, 4, 0, 0, 0, 0, 0, 0, 0, 1, 13, 6, 0, 0, 0, 0, 0, 0, 0, 0, 1, 20, 16, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 23, 23, 0, 0, 0

Offset: 1

Gus Wiseman, Oct 07 2018

			Triangle begins:
  1
  0  1
  0  1  0
  0  1  1  0
  0  1  1  0  0
  0  1  3  0  0  0
  0  1  3  0  0  0  0
  0  1  6  1  0  0  0  0
  0  1  7  1  0  0  0  0  0
  0  1 11  4  0  0  0  0  0  0
  0  1 13  6  0  0  0  0  0  0  0
  0  1 20 16  0  0  0  0  0  0  0  0
  0  1 23 23  0  0  0  0  0  0  0  0  0
  0  1 33 46  0  0  0  0  0  0  0  0  0  0
The T(10,3) = 4 rooted trees:
   (((oo)(oo))((oo)(oooo)))
   (((oo)(oo))((ooo)(ooo)))
   (((oo)(ooo))((oo)(ooo)))
  (((oo)(oo))((oo)(oo)(oo)))

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (rows 1..50)

Row sums are A120803. Third column is A083751. An irregular version is A320221.
Cf. A000669, A001678, A048816, A079500, A119262, A244925, A316655, A319312.
Cf. A316624, A320154, A320155, A320160, A320172, A320173.

qurt[n_]:=If[n==1,{{}},Join@@Table[Union[Sort/@Tuples[qurt/@ptn]],{ptn,Select[IntegerPartitions[n],Length[#]>1&]}]];
Table[Length[Select[qurt[n],SameQ[##,k]&@@Length/@Position[#,{}]&]],{n,14},{k,0,n-1}]

EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
T(n)={my(u=vector(n), v=vector(n), h=1); u[1]=1; while(u, v+=u*h; h*=x; u=EulerT(u)-u); vector(n, n, Vecrev(v[n], n))}
{ my(A=T(15)); for(n=1, #A, print(A[n])) } \\ Andrew Howroyd, Dec 09 2020

A331678 Number of lone-child-avoiding locally disjoint rooted trees whose leaves are integer partitions whose multiset union is an integer partition of n.

Original entry on oeis.org

1, 3, 6, 18, 44, 149, 450, 1573, 5352, 19283, 69483, 257206

Offset: 1

Gus Wiseman, Jan 25 2020

Lone-child-avoiding means there are no unary branchings. Locally disjoint means no child of any vertex has branches overlapping the branches of any other unequal child of the same vertex.

			The a(1) = 1 through a(4) = 18 trees:
  (1)  (2)       (3)            (4)
       (11)      (12)           (13)
       ((1)(1))  (111)          (22)
                 ((1)(2))       (112)
                 ((1)(1)(1))    (1111)
                 ((1)((1)(1)))  ((1)(3))
                                ((2)(2))
                                ((2)(11))
                                ((11)(11))
                                ((1)(1)(2))
                                ((1)((1)(2)))
                                ((2)((1)(1)))
                                ((1)(1)(1)(1))
                                ((11)((1)(1)))
                                ((1)((1)(1)(1)))
                                ((1)(1)((1)(1)))
                                (((1)(1))((1)(1)))
                                ((1)((1)((1)(1))))

David Callan, A sign-reversing involution to count labeled lone-child-avoiding trees, arXiv:1406.7784 [math.CO], (30-June-2014).
Gus Wiseman, Sequences counting series-reduced and lone-child-avoiding trees by number of vertices.

The case where all leaves are singletons is A316696.
The case where all leaves are (1) is A316697.
The non-locally disjoint version is A319312.
The case with all atoms equal to 1 is A331679.
The identity tree case is A331686.
Cf. A000081, A000669, A001678, A005804, A060356, A141268, A196545, A300660, A316471, A316694, A316495, A330465, A331680, A331687.

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]]]];
disjointQ[u_]:=Apply[And,Outer[#1==#2||Intersection[#1,#2]=={}&,u,u,1],{0,1}];
mpti[m_]:=Prepend[Join@@Table[Select[Union[Sort/@Tuples[mpti/@p]],disjointQ],{p,Select[mps[m],Length[#]>1&]}],m];
Table[Sum[Length[mpti[m]],{m,Sort/@IntegerPartitions[n]}],{n,8}]

A331687 Number of locally disjoint enriched p-trees of weight n.

Original entry on oeis.org

1, 2, 4, 12, 29, 93, 249, 803, 2337, 7480, 23130, 77372, 247598, 834507, 2762222

Offset: 1

Gus Wiseman, Jan 31 2020

A locally disjoint enriched p-tree of weight n is either the number n itself or a finite sequence of non-overlapping locally disjoint enriched p-trees whose weights are weakly decreasing and sum to n.

			The a(1) = 1 through a(4) = 12 enriched p-trees:
  1  2     3        4
     (11)  (21)     (22)
           (111)    (31)
           ((11)1)  (211)
                    (1111)
                    ((11)2)
                    ((21)1)
                    (2(11))
                    ((11)11)
                    ((111)1)
                    (((11)1)1)
                    ((11)(11))

The orderless version is A316696.
The identity case is A331684.
P-trees are A196545.
Enriched p-trees are A289501.
Locally disjoint identity trees are A316471.
Enriched identity p-trees are A331875.
Cf. A000669, A141268, A316473, A316495, A316694, A316697, A319312, A331678, A331679, A331680, A331686, A331871, A331874.

disjointQ[u_]:=Apply[And,Outer[#1==#2||Intersection[#1,#2]=={}&,u,u,1],{0,1}];
ldep[n_]:=Prepend[Select[Join@@Table[Tuples[ldep/@p],{p,Rest[IntegerPartitions[n]]}],disjointQ[DeleteCases[#,_Integer]]&],n];
Table[Length[ldep[n]],{n,10}]

A320155 Number of series-reduced balanced rooted trees with n labeled leaves.

Original entry on oeis.org

1, 1, 1, 4, 11, 41, 162, 1030, 7205, 55522, 442443, 3810852, 35272030, 351697516, 3735838550, 42719792640, 529195988635, 7128835815387, 103651381499810, 1610812109555323, 26489497655582729, 457497408108551450, 8248899117402701046, 154624472715479106919

Offset: 1

Gus Wiseman, Oct 06 2018

nonn

A rooted tree is series-reduced if every non-leaf node has at least two branches, and balanced if all leaves are the same distance from the root.

			The a(1) = 1 through a(5) = 11 rooted trees:
  1  (12)  (123)    (1234)      (12345)
                  ((12)(34))  ((12)(345))
                  ((13)(24))  ((13)(245))
                  ((14)(23))  ((14)(235))
                              ((15)(234))
                              ((23)(145))
                              ((24)(135))
                              ((25)(134))
                              ((34)(125))
                              ((35)(124))
                              ((45)(123))

Andrew Howroyd, Table of n, a(n) for n = 1..200

Cf. A000081, A000311, A000669, A001678, A005804, A048816, A079500, A119262, A120803, A141268, A244925, A319312.

Cf. A320154, A320160, A316624, A320169, A320173, A320179.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
phy2[labs_]:=If[Length[labs]==1,labs,Union@@Table[Sort/@Tuples[phy2/@ptn],{ptn,Select[sps[Sort[labs]],Length[#1]>1&]}]];
Table[Length[Select[phy2[Range[n]],SameQ@@Length/@Position[#,_Integer]&]],{n,7}]

EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
b(n,k)={my(u=vector(n), v=vector(n)); u[1]=k; while(u, v+=u; u=EulerT(u)-u); v}
seq(n)={my(M=Mat(vectorv(n,k,b(n,k)))); vector(n, k, sum(i=1, k, binomial(k,i)*(-1)^(k-i)*M[i,k]))} \\ Andrew Howroyd, Oct 26 2018

E.g.f. A(x) satisfies A(x) = x + A(exp(x)-x-1). - Ira M. Gessel, Nov 17 2021

Terms a(10) and beyond from Andrew Howroyd, Oct 26 2018

A320173 Number of inequivalent colorings of series-reduced balanced rooted trees with n leaves.

Original entry on oeis.org

1, 2, 3, 12, 23, 84, 204, 830, 2940, 13397, 58794, 283132, 1377302, 7087164, 37654377, 209943842, 1226495407, 7579549767, 49541194089, 341964495985, 2476907459261, 18703210872343, 146284738788714, 1179199861398539, 9760466433602510, 82758834102114911, 717807201648148643

Offset: 1

Gus Wiseman, Oct 07 2018

nonn

A rooted tree is series-reduced if every non-leaf node has at least two branches, and balanced if all leaves are the same distance from the root.

			Inequivalent representatives of the a(1) = 1 through a(5) = 23 colorings:
  1  (11)  (111)    (1111)      (11111)
     (12)  (112)    (1112)      (11112)
           (123)    (1122)      (11122)
                    (1123)      (11123)
                    (1234)      (11223)
                  ((11)(11))    (11234)
                  ((11)(12))    (12345)
                  ((11)(22))  ((11)(111))
                  ((11)(23))  ((11)(112))
                  ((12)(12))  ((11)(122))
                  ((12)(13))  ((11)(123))
                  ((12)(34))  ((11)(223))
                              ((11)(234))
                              ((12)(111))
                              ((12)(112))
                              ((12)(113))
                              ((12)(123))
                              ((12)(134))
                              ((12)(345))
                              ((13)(122))
                              ((22)(111))
                              ((23)(111))
                              ((23)(114))

Cf. A000669, A005804, A048816, A079500, A119262, A120803, A141268, A244925, A319312.

Cf. A320154, A320155, A320160, A320174, A320175, A320179, A339645.

\\ See links in A339645 for combinatorial species functions.
cycleIndexSeries(n)={my(p=x*sv(1) + O(x*x^n), q=0); while(p, q+=p; p=sEulerT(p)-1-p); q}
InequivalentColoringsSeq(cycleIndexSeries(15)) \\ Andrew Howroyd, Dec 11 2020

Terms a(8) and beyond from Andrew Howroyd, Dec 11 2020

A320330 Number of T_0 multiset partitions of integer partitions of n.

Original entry on oeis.org

1, 1, 3, 5, 13, 25, 50, 100, 195, 366, 707, 1333, 2440

Offset: 0

Gus Wiseman, Oct 11 2018

The dual of a multiset partition has, for each vertex, one part consisting of the indices (or positions) of the parts containing that vertex, counted with multiplicity. For example, the dual of {{1,2},{2,2}} is {{1},{1,2,2}}. The T_0 condition means the dual is strict.

			The a(1) = 1 through a(5) = 25 multiset partitions:
  {{1}}  {{2}}      {{3}}          {{4}}              {{5}}
         {{1,1}}    {{1,1,1}}      {{2,2}}            {{1,1,3}}
         {{1},{1}}  {{1},{2}}      {{1,1,2}}          {{1,2,2}}
                    {{1},{1,1}}    {{1},{3}}          {{1},{4}}
                    {{1},{1},{1}}  {{2},{2}}          {{2},{3}}
                                   {{1,1,1,1}}        {{1,1,1,2}}
                                   {{1},{1,2}}        {{1},{1,3}}
                                   {{2},{1,1}}        {{1},{2,2}}
                                   {{1},{1,1,1}}      {{2},{1,2}}
                                   {{1,1},{1,1}}      {{3},{1,1}}
                                   {{1},{1},{2}}      {{1,1,1,1,1}}
                                   {{1},{1},{1,1}}    {{1},{1,1,2}}
                                   {{1},{1},{1},{1}}  {{1,1},{1,2}}
                                                      {{1},{1},{3}}
                                                      {{1},{2},{2}}
                                                      {{2},{1,1,1}}
                                                      {{1},{1,1,1,1}}
                                                      {{1,1},{1,1,1}}
                                                      {{1},{1},{1,2}}
                                                      {{1},{2},{1,1}}
                                                      {{1},{1},{1,1,1}}
                                                      {{1},{1,1},{1,1}}
                                                      {{1},{1},{1},{2}}
                                                      {{1},{1},{1},{1,1}}
                                                      {{1},{1},{1},{1},{1}}

Cf. A001970, A047968, A050342, A089259, A141268, A261049, A289501, A305551, A316983, A319066, A319312, A320328, A320331.

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]]]];
dual[eds_]:=Table[First/@Position[eds,x],{x,Union@@eds}];
Table[Length[Select[Join@@mps/@IntegerPartitions[n],UnsameQ@@dual[#]&]],{n,8}]

A330471 Number of series/singleton-reduced rooted trees on strongly normal multisets of size n.

Original entry on oeis.org

1, 1, 2, 9, 69, 623, 7803, 110476, 1907428

Offset: 0

Gus Wiseman, Dec 23 2019

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

A series/singleton-reduced rooted tree on a multiset m is either the multiset m itself or a sequence of series/singleton-reduced rooted trees, one on each part of a multiset partition of m that is neither minimal (all singletons) nor maximal (only one part). This is a multiset generalization of singleton-reduced phylogenetic trees (A000311).

			The a(0) = 1 through a(3) = 9 trees:
  ()  (1)  (11)  (111)
           (12)  (112)
                 (123)
                 ((1)(11))
                 ((1)(12))
                 ((1)(23))
                 ((2)(11))
                 ((2)(13))
                 ((3)(12))
The a(4) = 69 trees, with singleton leaves (x) replaced by just x:
  (1111)      (1112)      (1122)      (1123)      (1234)
  (1(111))    (1(112))    (1(122))    (1(123))    (1(234))
  (11(11))    (11(12))    (11(22))    (11(23))    (12(34))
  ((11)(11))  (12(11))    (12(12))    (12(13))    (13(24))
  (1(1(11)))  (2(111))    (2(112))    (13(12))    (14(23))
              ((11)(12))  (22(11))    (2(113))    (2(134))
              (1(1(12)))  ((11)(22))  (23(11))    (23(14))
              (1(2(11)))  (1(1(22)))  (3(112))    (24(13))
              (2(1(11)))  ((12)(12))  ((11)(23))  (3(124))
                          (1(2(12)))  (1(1(23)))  (34(12))
                          (2(1(12)))  ((12)(13))  (4(123))
                          (2(2(11)))  (1(2(13)))  ((12)(34))
                                      (1(3(12)))  (1(2(34)))
                                      (2(1(13)))  ((13)(24))
                                      (2(3(11)))  (1(3(24)))
                                      (3(1(12)))  ((14)(23))
                                      (3(2(11)))  (1(4(23)))
                                                  (2(1(34)))
                                                  (2(3(14)))
                                                  (2(4(13)))
                                                  (3(1(24)))
                                                  (3(2(14)))
                                                  (3(4(12)))
                                                  (4(1(23)))
                                                  (4(2(13)))
                                                  (4(3(12)))

The case with all atoms different is A000311.
The case with all atoms equal is A196545.
The orderless version is A316652.
The unlabeled version is A330470.
The case where the leaves are sets is A330628.
The version for just normal (not strongly normal) is A330654.
Cf. A000669, A004114, A005121, A005804, A281118, A318812, A318848, A319312, A330465, A330467, A330475.

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]]]];
strnorm[n_]:=Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n];
mtot[m_]:=Prepend[Join@@Table[Tuples[mtot/@p],{p,Select[mps[m],Length[#]>1&&Length[#]

cp's OEIS Frontend

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

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A330469 Number of series-reduced rooted trees whose leaves are multisets with a total of n elements covering an initial interval of positive integers.

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A320169 Number of balanced enriched p-trees of weight n.

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A320179 Regular triangle where T(n,k) is the number of unlabeled series-reduced rooted trees with n leaves in which every leaf is at height k.

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A331678 Number of lone-child-avoiding locally disjoint rooted trees whose leaves are integer partitions whose multiset union is an integer partition of n.

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A331687 Number of locally disjoint enriched p-trees of weight n.

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A320155 Number of series-reduced balanced rooted trees with n labeled leaves.

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A320173 Number of inequivalent colorings of series-reduced balanced rooted trees with n leaves.

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