A330470 -id:A330470 - 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]))}

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

A330474 Number of non-isomorphic balanced reduced multisystems of weight n.

Original entry on oeis.org

1, 1, 2, 7, 48, 424

Offset: 0

Gus Wiseman, Dec 26 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. The weight of an atom is 1, while the weight of a multiset is the sum of weights of its elements.

			Non-isomorphic representatives of the a(3) = 7 multisystems:
  {1,1,1}
  {1,1,2}
  {1,2,3}
  {{1},{1,1}}
  {{1},{1,2}}
  {{1},{2,3}}
  {{2},{1,1}}
Non-isomorphic representatives of the a(4) = 48 multisystems:
  {1,1,1,1}  {{1},{1,1,1}}    {{{1}},{{1},{1,1}}}
  {1,1,1,2}  {{1,1},{1,1}}    {{{1,1}},{{1},{1}}}
  {1,1,2,2}  {{1},{1,1,2}}    {{{1}},{{1},{1,2}}}
  {1,1,2,3}  {{1,1},{1,2}}    {{{1,1}},{{1},{2}}}
  {1,2,3,4}  {{1},{1,2,2}}    {{{1}},{{1},{2,2}}}
             {{1,1},{2,2}}    {{{1,1}},{{2},{2}}}
             {{1},{1,2,3}}    {{{1}},{{1},{2,3}}}
             {{1,1},{2,3}}    {{{1,1}},{{2},{3}}}
             {{1,2},{1,2}}    {{{1}},{{2},{1,1}}}
             {{1,2},{1,3}}    {{{1,2}},{{1},{1}}}
             {{1},{2,3,4}}    {{{1}},{{2},{1,2}}}
             {{1,2},{3,4}}    {{{1,2}},{{1},{2}}}
             {{2},{1,1,1}}    {{{1}},{{2},{1,3}}}
             {{2},{1,1,3}}    {{{1,2}},{{1},{3}}}
             {{1},{1},{1,1}}  {{{1}},{{2},{3,4}}}
             {{1},{1},{1,2}}  {{{1,2}},{{3},{4}}}
             {{1},{1},{2,2}}  {{{2}},{{1},{1,1}}}
             {{1},{1},{2,3}}  {{{2}},{{1},{1,3}}}
             {{1},{2},{1,1}}  {{{2}},{{3},{1,1}}}
             {{1},{2},{1,2}}  {{{2,3}},{{1},{1}}}
             {{1},{2},{1,3}}
             {{1},{2},{3,4}}
             {{2},{3},{1,1}}

Labeled versions are A330475 (strongly normal) and A330655 (normal).
The case where the atoms are all different is A318813.
The case where the atoms are all equal is (also) A318813.
The labeled case of set partitions is A005121.
The labeled case of integer partitions is A330679.
The case of maximal depth is A330663.
The version where leaves are sets (as opposed to multisets) is A330668.
Cf. A000311, A000669, A001678, A002846, A004114, A007716, A048816, A213427, A306186, A320154, A320160, A330470, A330666.

A330663 Number of non-isomorphic balanced reduced multisystems of weight n and maximum depth.

Original entry on oeis.org

1, 1, 2, 4, 20, 140, 1411

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

			Non-isomorphic representatives of the a(2) = 2 through a(4) = 20 multisystems:
  {1,1}  {{1},{1,1}}  {{{1}},{{1},{1,1}}}
  {1,2}  {{1},{1,2}}  {{{1,1}},{{1},{1}}}
         {{1},{2,3}}  {{{1}},{{1},{1,2}}}
         {{2},{1,1}}  {{{1,1}},{{1},{2}}}
                      {{{1}},{{1},{2,2}}}
                      {{{1,1}},{{2},{2}}}
                      {{{1}},{{1},{2,3}}}
                      {{{1,1}},{{2},{3}}}
                      {{{1}},{{2},{1,1}}}
                      {{{1,2}},{{1},{1}}}
                      {{{1}},{{2},{1,2}}}
                      {{{1,2}},{{1},{2}}}
                      {{{1}},{{2},{1,3}}}
                      {{{1,2}},{{1},{3}}}
                      {{{1}},{{2},{3,4}}}
                      {{{1,2}},{{3},{4}}}
                      {{{2}},{{1},{1,1}}}
                      {{{2}},{{1},{1,3}}}
                      {{{2}},{{3},{1,1}}}
                      {{{2,3}},{{1},{1}}}

The non-maximal version is A330474.
Labeled versions are A330675 (strongly normal) and A330676 (normal).
The case where the leaves are sets (as opposed to multisets) is A330677.
The case with all atoms distinct is A000111.
The case with all atoms equal is (also) A000111.
Cf. A000311, A004114, A005121, A006472, A007716, A048816, A141268, A306186, A330470, A330655, A330664.

A330624 Number of non-isomorphic series-reduced rooted trees whose leaves are sets (not necessarily disjoint) with a total of n elements.

Original entry on oeis.org

1, 1, 3, 10, 61, 410, 3630

Offset: 0

Gus Wiseman, Dec 25 2019

A rooted tree is series-reduced if it has no unary branchings, so every non-leaf node covers at least two other nodes.

			Non-isomorphic representatives of the a(1) = 1 through a(3) = 10 trees:
  {1}  {1,2}      {1,2,3}
       {{1},{1}}  {{1},{1,2}}
       {{1},{2}}  {{1},{2,3}}
                  {{1},{1},{1}}
                  {{1},{1},{2}}
                  {{1},{2},{3}}
                  {{1},{{1},{1}}}
                  {{1},{{1},{2}}}
                  {{1},{{2},{3}}}
                  {{2},{{1},{1}}}

The version with multisets as leaves is A330465.
The singleton-reduced case is A330626.
A labeled version is A330625 (strongly normal).
The case with all atoms distinct is A141268.
The case where all leaves are singletons is A330470.
Cf. A000669, A004111, A005804, A007716, A273873, A300660, A320296, A330628, A330668, A330677.

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[#]

A330626 Number of non-isomorphic series/singleton-reduced rooted trees whose leaves are sets (not necessarily disjoint) with a total of n atoms.

Original entry on oeis.org

1, 1, 1, 3, 17, 100, 755

Offset: 0

Gus Wiseman, Dec 26 2019

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(1) = 1 through a(4) = 17 trees:
  {1}  {1,2}  {1,2,3}      {1,2,3,4}
              {{1},{1,2}}  {{1},{1,2,3}}
              {{1},{2,3}}  {{1,2},{1,2}}
                           {{1,2},{1,3}}
                           {{1},{2,3,4}}
                           {{1,2},{3,4}}
                           {{1},{1},{1,2}}
                           {{1},{1},{2,3}}
                           {{1},{2},{1,2}}
                           {{1},{2},{1,3}}
                           {{1},{2},{3,4}}
                           {{1},{{1},{1,2}}}
                           {{1},{{1},{2,3}}}
                           {{1},{{2},{1,2}}}
                           {{1},{{2},{1,3}}}
                           {{1},{{2},{3,4}}}
                           {{2},{{1},{1,3}}}

The non-singleton-reduced version is A330624.
The generalization where leaves are multisets is A330470.
A labeled version is A330628 (strongly normal).
The case with all atoms distinct is A004114.
The balanced version is A330668.
Cf. A000669, A004111, A005804, A007716, A141268, A330465, A330625, A330627, A330654, A330663, A330677.

A330664 Number of non-isomorphic balanced reduced multisystems of maximum depth whose degrees (atom multiplicities) are the weakly decreasing prime indices of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 1, 4, 5, 5, 7, 16, 16, 27, 2, 61, 33, 272, 27, 123, 61, 1385, 27, 78, 272, 95, 123, 7936, 362

Offset: 1

Gus Wiseman, Dec 28 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 prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798. A multiset whose multiplicities are the prime indices of n (such as row n of A305936) is generally not the same as the multiset of prime indices of n. For example, the prime indices of 12 are {1,1,2}, while a multiset whose multiplicities are {1,1,2} is {1,1,2,3}.

			Non-isomorphic representatives of the a(n) multisystems for n = 2, 3, 6, 9, 10, 12 (commas and outer brackets elided):
  1  11  {1}{12}  {{1}}{{1}{22}}  {{1}}{{1}{12}}  {{1}}{{1}{23}}
         {2}{11}  {{11}}{{2}{2}}  {{11}}{{1}{2}}  {{11}}{{2}{3}}
                  {{1}}{{2}{12}}  {{1}}{{2}{11}}  {{1}}{{2}{13}}
                  {{12}}{{1}{2}}  {{12}}{{1}{1}}  {{12}}{{1}{3}}
                                  {{2}}{{1}{11}}  {{2}}{{1}{13}}
                                                  {{2}}{{3}{11}}
                                                  {{23}}{{1}{1}}

The non-maximal version is A330666.
The case of constant or strict atoms is A000111.
Labeled versions are A330728, A330665 (prime indices), and A330675 (strongly normal).
Non-isomorphic multiset partitions whose degrees are the prime indices of n are A318285.
Cf. A004114, A005121, A007716, A048816, A141268, A306186, A318846, A318848, A330470, A330474, A330663.

For n > 1, a(2^n) = a(prime(n)) = A000111(n - 1).

A330668 Number of non-isomorphic balanced reduced multisystems of weight n whose leaves (which are multisets of atoms) are all sets.

Original entry on oeis.org

1, 1, 1, 3, 22, 204, 2953

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

			Non-isomorphic representatives of the a(1) = 1 through a(4) = 22 multisystems:
  {1}  {1,2}  {1,2,3}      {1,2,3,4}
              {{1},{1,2}}  {{1},{1,2,3}}
              {{1},{2,3}}  {{1,2},{1,2}}
                           {{1,2},{1,3}}
                           {{1},{2,3,4}}
                           {{1,2},{3,4}}
                           {{1},{1},{1,2}}
                           {{1},{1},{2,3}}
                           {{1},{2},{1,2}}
                           {{1},{2},{1,3}}
                           {{1},{2},{3,4}}
                           {{{1}},{{1},{1,2}}}
                           {{{1}},{{1},{2,3}}}
                           {{{1,2}},{{1},{1}}}
                           {{{1}},{{2},{1,2}}}
                           {{{1,2}},{{1},{2}}}
                           {{{1}},{{2},{1,3}}}
                           {{{1,2}},{{1},{3}}}
                           {{{1}},{{2},{3,4}}}
                           {{{1,2}},{{3},{4}}}
                           {{{2}},{{1},{1,3}}}
                           {{{2,3}},{{1},{1}}}

The case with all atoms different is A318813.
The version where the leaves are multisets is A330474.
The tree version is A330626.
The maximum-depth case is A330677.
Unlabeled series-reduced rooted trees whose leaves are sets are A330624.
Cf. A000311, A004114, A005121, A005804, A007716, A048816, A141268, A283877, A306186, A318812, A320154, A330470, A330628, A330663.

A330677 Number of non-isomorphic balanced reduced multisystems of weight n and maximum depth whose leaves (which are multisets of atoms) are sets.

Original entry on oeis.org

1, 1, 1, 2, 11, 81, 859

Offset: 0

Gus Wiseman, Dec 30 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. The weight of an atom is 1, while the weight of a multiset is the sum of weights of its elements.

			Non-isomorphic representatives of the a(0) = 1 through a(4) = 11 multisystems:
  {}  {1}  {1,2}  {{1},{1,2}}  {{{1}},{{1},{1,2}}}
                  {{1},{2,3}}  {{{1}},{{1},{2,3}}}
                               {{{1,2}},{{1},{1}}}
                               {{{1}},{{2},{1,2}}}
                               {{{1,2}},{{1},{2}}}
                               {{{1}},{{2},{1,3}}}
                               {{{1,2}},{{1},{3}}}
                               {{{1}},{{2},{3,4}}}
                               {{{1,2}},{{3},{4}}}
                               {{{2}},{{1},{1,3}}}
                               {{{2,3}},{{1},{1}}}

The version with all distinct atoms is A000111.
Non-isomorphic set multipartitions are A049311.
The (non-maximal) tree version is A330626.
Allowing leaves to be multisets gives A330663.
The case with prescribed degrees is A330664.
The version allowing all depths is A330668.
Cf. A000669, A001678, A004114, A005121, A007716, A141268, A283877, A306186, A330465, A330470, A330624.

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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PARI

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

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A330474 Number of non-isomorphic balanced reduced multisystems of weight n.

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A330663 Number of non-isomorphic balanced reduced multisystems of weight n and maximum depth.

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A330624 Number of non-isomorphic series-reduced rooted trees whose leaves are sets (not necessarily disjoint) with a total of n elements.

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

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Mathematica

A330626 Number of non-isomorphic series/singleton-reduced rooted trees whose leaves are sets (not necessarily disjoint) with a total of n atoms.

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A330664 Number of non-isomorphic balanced reduced multisystems of maximum depth whose degrees (atom multiplicities) are the weakly decreasing prime indices of n.

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A330668 Number of non-isomorphic balanced reduced multisystems of weight n whose leaves (which are multisets of atoms) are all sets.

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A330677 Number of non-isomorphic balanced reduced multisystems of weight n and maximum depth whose leaves (which are multisets of atoms) are sets.

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