A318812 -id:A318812 - cp's OEIS frontend

A339645 Triangle read by rows: T(n,k) is the number of inequivalent colorings of lone-child-avoiding rooted trees with n colored leaves using exactly k colors.

1, 1, 1, 2, 3, 2, 5, 17, 12, 5, 12, 73, 95, 44, 12, 33, 369, 721, 512, 168, 33, 90, 1795, 5487, 5480, 2556, 625, 90, 261, 9192, 41945, 58990, 36711, 12306, 2342, 261, 766, 47324, 321951, 625088, 516952, 224241, 57155, 8702, 766, 2312, 249164, 2483192, 6593103, 7141755, 3965673, 1283624, 258887, 32313, 2312

Offset: 1

Andrew Howroyd, Dec 11 2020

Only the leaves are colored. Equivalence is up to permutation of the colors.

Lone-child-avoiding rooted trees are also called planted series-reduced trees in some other sequences.

			Triangle begins:
    1;
    1,     1;
    2,     3,      2;
    5,    17,     12,      5;
   12,    73,     95,     44,     12;
   33,   369,    721,    512,    168,     33;
   90,  1795,   5487,   5480,   2556,    625,    90;
  261,  9192,  41945,  58990,  36711,  12306,  2342,  261;
  766, 47324, 321951, 625088, 516952, 224241, 57155, 8702, 766;
  ...
From _Gus Wiseman_, Jan 02 2021: (Start)
Non-isomorphic representatives of the 39 = 5 + 17 + 12 + 5 trees with four colored leaves:
  (1111)      (1112)      (1123)      (1234)
  (1(111))    (1122)      (1(123))    (1(234))
  (11(11))    (1(112))    (11(23))    (12(34))
  ((11)(11))  (11(12))    (12(13))    ((12)(34))
  (1(1(11)))  (1(122))    (2(113))    (1(2(34)))
              (11(22))    (23(11))
              (12(11))    ((11)(23))
              (12(12))    (1(1(23)))
              (2(111))    ((12)(13))
              ((11)(12))  (1(2(13)))
              (1(1(12)))  (2(1(13)))
              ((11)(22))  (2(3(11)))
              (1(1(22)))
              (1(2(11)))
              ((12)(12))
              (1(2(12)))
              (2(1(11)))
(End)

Andrew Howroyd, PARI Functions for Combinatorial Species, v2, Dec 2020.
Wikipedia, Combinatorial species

The case with only one color is A000669.
Counting by nodes gives A318231.
A labeled version is A319376.
Row sums are A330470.
A000311 counts singleton-reduced phylogenetic trees.
A001678 counts unlabeled lone-child-avoiding rooted trees.
A005121 counts chains of set partitions, with maximal case A002846.
A005804 counts phylogenetic rooted trees with n labels.
A060356 counts labeled lone-child-avoiding rooted trees.
A141268 counts lone-child-avoiding rooted trees with leaves summing to n.
A291636 lists Matula-Goebel numbers of lone-child-avoiding rooted trees.
A316651 counts lone-child-avoiding rooted trees with normal leaves.
A316652 counts lone-child-avoiding rooted trees with strongly normal leaves.
A330465 counts inequivalent leaf-colorings of phylogenetic rooted trees.
Cf. A196545, A213427, A281118, A289501, A292504, A318812, A319312, A330627.

\\ See link above for combinatorial species functions.
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)}
{my(A=InequivalentColoringsTriangle(cycleIndexSeries(10))); for(n=1, #A~, print(A[n,1..n]))}

A318813 Number of balanced reduced multisystems with n atoms all equal to 1.

Original entry on oeis.org

1, 1, 2, 6, 20, 90, 468, 2910, 20644, 165874, 1484344, 14653890, 158136988, 1852077284, 23394406084, 317018563806, 4587391330992, 70598570456104, 1151382852200680, 19835976878704628, 359963038816096924, 6863033015330999110, 137156667020252478684, 2867083618970831936826

Offset: 1

Gus Wiseman, Sep 04 2018

nonn

For n > 1, also the number of balanced reduced multisystems whose atoms are an integer partition of n with at least one part > 1. A balanced reduced multisystem is either a finite multiset, or a multiset partition with at least two parts, not all of which are singletons, of a balanced reduced multisystem. - Gus Wiseman, Dec 31 2019

			The a(5) = 20 balanced reduced multisystems (with n written in place of 1^n):
  5  (14)  (23)  (113)      (122)      (1112)
                 ((1)(13))  ((1)(22))  ((1)(112))
                 ((3)(11))  ((2)(12))  ((2)(111))
                                       ((11)(12))
                                       ((1)(1)(12))
                                       ((1)(2)(11))
                                       (((1))((1)(12)))
                                       (((1))((2)(11)))
                                       (((2))((1)(11)))
                                       (((12))((1)(1)))
                                       (((11))((1)(2)))

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

The maximum-depth case is A000111.

Cf. A000311, A001055, A002846, A005121, A213427, A281118, A281119, A317145, A318812, A318846, A320154, A330474, A330679.

normize[m_]:=m/.Rule@@@Table[{Union[m][[i]],i},{i,Length[Union[m]]}];
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
totfact[n_]:=totfact[n]=1+Sum[totfact[Times@@Prime/@normize[f]],{f,Select[facs[n],1

EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
seq(n)={my(v=vector(n, i, i==1), u=vector(n)); for(r=1, #v, u += v*sum(j=r, #v, (-1)^(j-r)*binomial(j-1, r-1)); v=EulerT(v)); u} \\ Andrew Howroyd, Dec 30 2019

a(n > 1) = A330679(n)/2. - Gus Wiseman, Dec 31 2019

Terms a(14) and beyond from Andrew Howroyd, Dec 30 2019

Terminology corrected by Gus Wiseman, Dec 31 2019

A330465 Number of non-isomorphic series-reduced rooted trees whose leaves are multisets with a total of n elements.

Original entry on oeis.org

1, 4, 14, 87, 608, 5573, 57876, 687938, 9058892, 130851823, 2048654450, 34488422057, 620046639452, 11839393796270, 238984150459124, 5079583100918338, 113299159314626360, 2644085918303683758, 64393240540265515110, 1632731130253043991252, 43013015553755764179000

Offset: 1

Gus Wiseman, Dec 21 2019

nonn

Also inequivalent leaf-colorings of phylogenetic rooted trees with n labels. A phylogenetic rooted tree is a series-reduced rooted tree whose leaves are (usually disjoint) sets.

			Non-isomorphic representatives of the a(3) = 14 trees:
  ((1)((1)(1)))  ((1)((1)(2)))  ((1)((2)(3)))  ((2)((1)(1)))
  ((1)(1)(1))    ((1)(1)(2))    ((1)(2)(3))    ((2)(1,1))
  ((1)(1,1))     ((1)(1,2))     ((1)(2,3))
  (1,1,1)        (1,1,2)        (1,2,3)

The version where leaves are atoms is A318231.
The case with sets as leaves is A330624.
The case with disjoint sets as leaves is A141268.
Labeled versions are A330467 (strongly normal) and A330469 (normal).
The singleton-reduced version is A330470.
Cf. A000311, A000669, A004114, A005804, A007716, A281118, A289501, A292504, A316651, A316652, A318812, A319312, A330471, A330474, A330625, A339645.

\\ See links in A339645 for combinatorial species functions.
cycleIndexSeries(n)={my(v=vector(n), p=sEulerT(x*sv(1) + O(x*x^n))); v[1]=sv(1); for(n=2, #v, v[n] = polcoef( sEulerT(x*Ser(v[1..n])), n ) + polcoef(p,n)); x*Ser(v)}
InequivalentColoringsSeq(cycleIndexSeries(15)) \\ Andrew Howroyd, Dec 13 2020

Terms a(7) and beyond from Andrew Howroyd, Dec 13 2020

A323787 Number of non-isomorphic multiset partitions of strict multiset partitions of weight n.

Original entry on oeis.org

1, 1, 4, 14, 56, 219, 1001, 4588

Offset: 0

Gus Wiseman, Jan 27 2019

The weight of an atom is 1, and the weight of a multiset is the sum of weights of its elements, counting multiplicity.

			Non-isomorphic representatives of the a(1) = 1 through a(3) = 14 multiset partitions:
  {{1}}  {{11}}      {{111}}
         {{12}}      {{112}}
         {{1}{2}}    {{123}}
         {{1}}{{2}}  {{1}{11}}
                     {{1}{12}}
                     {{1}{23}}
                     {{2}{11}}
                     {{1}}{{11}}
                     {{1}}{{12}}
                     {{1}}{{23}}
                     {{1}{2}{3}}
                     {{2}}{{11}}
                     {{1}}{{2}{3}}
                     {{1}}{{2}}{{3}}

Cf. A002846, A005121, A007716, A050343, A213427, A269134, A283877, A306186, A316980, A317791, A318564, A318565, A318566, A318812.

Cf. A323788, A323789, A323790, A323791, A323792, A323793, A323794, A323795.

A330475 Number of balanced reduced multisystems whose atoms constitute a strongly normal multiset of size n.

Original entry on oeis.org

1, 1, 2, 9, 85, 1143, 25270

Offset: 0

Gus Wiseman, Dec 27 2019

A balanced reduced multisystem is either a finite multiset, or a multiset partition with at least two parts, not all of which are singletons, of a balanced reduced multisystem.

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

			The a(0) = 1 through a(3) = 9 multisystems:
  {}  {1}  {1,1}  {1,1,1}
           {1,2}  {1,1,2}
                  {1,2,3}
                  {{1},{1,1}}
                  {{1},{1,2}}
                  {{1},{2,3}}
                  {{2},{1,1}}
                  {{2},{1,3}}
                  {{3},{1,2}}

The (weakly) normal version is A330655.
The maximum-depth case is A330675.
The case where the atoms are {1..n} is A005121.
The case where the atoms are all 1's is A318813.
The tree version is A330471.
Multiset partitions of strongly normal multisets are A035310.
Cf. A000311, A000669, A001678, A316652, A318812, A330467, A330474, A330628, A330663, A330679.

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];
totm[m_]:=Prepend[Join@@Table[totm[p],{p,Select[mps[m],1

A330679 Number of balanced reduced multisystems whose atoms constitute an integer partition of n.

Original entry on oeis.org

1, 1, 2, 4, 12, 40, 180, 936, 5820, 41288, 331748, 2968688, 29307780, 316273976, 3704154568, 46788812168, 634037127612, 9174782661984, 141197140912208, 2302765704401360, 39671953757409256, 719926077632193848, 13726066030661998220, 274313334040504957368

Offset: 0

Gus Wiseman, Dec 31 2019

nonn

A balanced reduced multisystem is either a finite multiset, or a multiset partition with at least two parts, not all of which are singletons, of a balanced reduced multisystem.

			The a(0) = 1 through a(4) = 12 multisystems:
  {}  {1}  {2}    {3}          {4}
           {1,1}  {1,2}        {1,3}
                  {1,1,1}      {2,2}
                  {{1},{1,1}}  {1,1,2}
                               {1,1,1,1}
                               {{1},{1,2}}
                               {{2},{1,1}}
                               {{1},{1,1,1}}
                               {{1,1},{1,1}}
                               {{1},{1},{1,1}}
                               {{{1}},{{1},{1,1}}}
                               {{{1,1}},{{1},{1}}}

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

The case where the atoms are all 1's is A318813 = a(n)/2.
The version where the atoms constitute a strongly normal multiset is A330475.
The version where the atoms cover an initial interval is A330655.
The maximum-depth version is A330726.
Cf. A000041, A000111, A000669, A001970, A002846, A005121, A141268, A196545, A213427, A318812, A320160, A330474.

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]]]];
totm[m_]:=Prepend[Join@@Table[totm[p],{p,Select[mps[m],1

a(n > 1) = 2 * A318813(n).

a(12) onwards from Andrew Howroyd, Jan 20 2024

A330470 Number of non-isomorphic series/singleton-reduced rooted trees on a multiset of size n.

Original entry on oeis.org

1, 1, 2, 7, 39, 236, 1836, 16123, 162008, 1802945, 22012335, 291290460, 4144907830, 62986968311, 1016584428612, 17344929138791, 311618472138440, 5875109147135658, 115894178676866576, 2385755803919949337, 51133201045333895149, 1138659323863266945177, 26296042933904490636133

Offset: 0

Gus Wiseman, Dec 22 2019

nonn

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).

			Non-isomorphic representatives of the a(4) = 39 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))    ((12)(34))
  (1(1(11)))  (2(111))    ((11)(22))  (2(113))    (1(2(34)))
              ((11)(12))  (1(1(22)))  (23(11))
              (1(1(12)))  ((12)(12))  ((11)(23))
              (1(2(11)))  (1(2(12)))  (1(1(23)))
              (2(1(11)))              ((12)(13))
                                      (1(2(13)))
                                      (2(1(13)))
                                      (2(3(11)))

The case with all atoms equal or all atoms different is A000669.
Not requiring singleton-reduction gives A330465.
Labeled versions are A316651 (normal orderless) and A330471 (strongly normal).
The case where the leaves are sets is A330626.
Row sums of A339645.
Cf. A000311, A005121, A005804, A141268, A213427, A292504, A292505, A318812, A318848, A318849, A330467, A330469, A330474, A330624.

\\ 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( sEulerT(x*Ser(v[1..n])), n )); x*Ser(v)}
InequivalentColoringsSeq(cycleIndexSeries(15)) \\ Andrew Howroyd, Dec 11 2020

Terms a(7) and beyond from Andrew Howroyd, Dec 11 2020

A323790 Number of non-isomorphic weight-n sets of sets of sets.

Original entry on oeis.org

1, 1, 3, 9, 33, 113, 474, 1985

Offset: 0

Gus Wiseman, Jan 27 2019

Non-isomorphic sets of sets are counted by A283877.
All sets and multisets must be finite, and only the outermost may be empty.
The weight of an atom is 1, and the weight of a multiset is the sum of weights of its elements, counting multiplicity.

			Non-isomorphic representatives of the a(1) = 1 through a(3) = 9 sets of sets of sets:
  {{1}}  {{12}}      {{123}}
         {{1}{2}}    {{1}{12}}
         {{1}}{{2}}  {{1}{23}}
                     {{1}}{{12}}
                     {{1}}{{23}}
                     {{1}{2}{3}}
                     {{1}}{{1}{2}}
                     {{1}}{{2}{3}}
                     {{1}}{{2}}{{3}}
Non-isomorphic representatives of the a(4) = 33 sets of sets of sets:
  {{1234}}             {{1}{123}}         {{1}{2}{12}}       {{1}}{{1}{12}}
  {{1}{234}}           {{12}{13}}         {{1}}{{2}{12}}
  {{12}{34}}           {{1}}{{123}}       {{12}}{{1}{2}}
  {{1}}{{234}}         {{1}{2}{13}}       {{1}}{{2}}{{12}}
  {{1}{2}{34}}         {{12}}{{13}}       {{1}}{{2}}{{1}{2}}
  {{12}}{{34}}         {{1}}{{1}{23}}
  {{1}}{{2}{34}}       {{1}}{{2}{13}}
  {{1}{2}{3}{4}}       {{12}}{{1}{3}}
  {{12}}{{3}{4}}       {{2}}{{1}{13}}
  {{1}}{{2}}{{34}}     {{1}}{{1}{2}{3}}
  {{1}}{{2}{3}{4}}     {{1}}{{2}}{{13}}
  {{1}{2}}{{3}{4}}     {{1}{2}}{{1}{3}}
  {{1}}{{2}}{{3}{4}}   {{1}}{{2}}{{1}{3}}
  {{1}}{{2}}{{3}}{{4}}

Cf. A004111, A007716, A049311, A050326, A050343, A283877, A306186, A316980, A318564, A318565, A318566, A318812.

Cf. A323787, A323788, A323789, A323791, A323792, A323793, A323794, A323795.

A330655 Number of balanced reduced multisystems of weight n whose atoms cover an initial interval of positive integers.

Original entry on oeis.org

1, 1, 2, 12, 138, 2652, 78106, 3256404, 182463296, 13219108288, 1202200963522, 134070195402644, 17989233145940910, 2858771262108762492, 530972857546678902490, 113965195745030648131036, 27991663753030583516229824, 7800669355870672032684666900, 2448021231611414334414904013956

Offset: 0

Gus Wiseman, Dec 27 2019

nonn

A balanced reduced multisystem is either a finite multiset, or a multiset partition with at least two parts, not all of which are singletons, of a balanced reduced multisystem. The weight of an atom is 1, while the weight of a multiset is the sum of weights of its elements.

			The a(0) = 1 through a(3) = 12 multisystems:
  {}  {1}  {1,1}  {1,1,1}
           {1,2}  {1,1,2}
                  {1,2,2}
                  {1,2,3}
                  {{1},{1,1}}
                  {{1},{1,2}}
                  {{1},{2,2}}
                  {{1},{2,3}}
                  {{2},{1,1}}
                  {{2},{1,2}}
                  {{2},{1,3}}
                  {{3},{1,2}}

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

Row sums of A330776.
The unlabeled version is A330474.
The strongly normal case is A330475.
The tree version is A330654.
The maximum-depth case is A330676.
The case where the atoms are all different is A005121.
The case where the atoms are all equal is A318813.
Multiset partitions of normal multisets are A255906.
Series-reduced rooted trees with normal leaves are A316651.
Cf. A000311, A000669, A001678, A213427, A318812, A330675, A330679.

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]]]];
totm[m_]:=Prepend[Join@@Table[totm[p],{p,Select[mps[m],1

EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
R(n,k)={my(v=vector(n), u=vector(n)); v[1]=k; for(n=1, #v, u += v*sum(j=n, #v, (-1)^(j-n)*binomial(j-1,n-1)); v=EulerT(v)); u}
seq(n)={concat([1], sum(k=1, n, R(n, k)*sum(r=k, n, binomial(r, k)*(-1)^(r-k))))} \\ Andrew Howroyd, Dec 30 2019

Terms a(7) and beyond from Andrew Howroyd, Dec 30 2019

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

Original entry on oeis.org

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

cp's OEIS Frontend

A339645 Triangle read by rows: T(n,k) is the number of inequivalent colorings of lone-child-avoiding rooted trees with n colored leaves using exactly k colors.

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A318813 Number of balanced reduced multisystems with n atoms all equal to 1.

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A330465 Number of non-isomorphic series-reduced rooted trees whose leaves are multisets with a total of n elements.

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A323787 Number of non-isomorphic multiset partitions of strict multiset partitions of weight n.

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A330475 Number of balanced reduced multisystems whose atoms constitute a strongly normal multiset of size n.

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Mathematica

A330679 Number of balanced reduced multisystems whose atoms constitute an integer partition of n.

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A330470 Number of non-isomorphic series/singleton-reduced rooted trees on a multiset of size n.

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A323790 Number of non-isomorphic weight-n sets of sets of sets.

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A330655 Number of balanced reduced multisystems of weight n whose atoms cover an initial interval of positive integers.