A292504 -id:A292504 - cp's OEIS frontend

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

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

A292505 Number of complete orderless tree-factorizations of n >= 2.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 5, 1, 3, 1, 3, 1, 1, 1, 9, 1, 1, 2, 3, 1, 4, 1, 12, 1, 1, 1, 12, 1, 1, 1, 9, 1, 4, 1, 3, 3, 1, 1, 29, 1, 3, 1, 3, 1, 9, 1, 9, 1, 1, 1, 17, 1, 1, 3, 33, 1, 4, 1, 3, 1, 4, 1, 44, 1, 1, 3, 3, 1, 4, 1, 29, 5, 1, 1, 17, 1

Offset: 2

Gus Wiseman, Sep 17 2017

nonn

An orderless tree-factorization (see A292504 for definition) is complete if all leaves are prime numbers. This sequence first differs from A281119 at a(64)=33.

a(n) depends only on the prime signature of n. - Andrew Howroyd, Nov 18 2018

			The a(60)=17 complete orderless tree-factorizations are: (2(2(35))), (2(3(25))), (2(5(23))), (2(235)), (3(2(25))), (3(5(22))), (3(225)), (5(2(23))), (5(3(22))), (5(223)), ((22)(35)), ((23)(25)), (22(35)), (23(25)), (25(23)), (35(22)), (2235).

Andrew Howroyd, Table of n, a(n) for n = 2..10000

Cf. A000311, A000669, A001055, A050336, A281119, A292504.

postfacs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[postfacs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
oltfacs[n_]:=If[n<=1,{{}},Prepend[Union@@Function[q,Sort/@Tuples[oltfacs/@q]]/@DeleteCases[postfacs[n],{n}],n]];
Table[Length[Select[oltfacs[n],FreeQ[#,_Integer?(!PrimeQ[#]&)]&]],{n,2,100}]

seq(n)={my(v=vector(n), w=vector(n)); v[1]=1; for(k=2, n, w[k]=v[k]+isprime(k); forstep(j=n\k*k, k, -k, my(i=j, e=0); while(i%k==0, i/=k; e++; v[j]+=binomial(e+w[k]-1, e)*v[i]))); w[2..n]} \\ Andrew Howroyd, Nov 18 2018

a(p^n) = A000669(n) for prime p. - Andrew Howroyd, Nov 18 2018

A320154 Number of series-reduced balanced rooted trees whose leaves form a set partition of {1,...,n}.

Original entry on oeis.org

1, 2, 5, 18, 92, 588, 4328, 35920, 338437, 3654751, 45105744, 625582147, 9539374171, 157031052142, 2757275781918, 51293875591794, 1007329489077804, 20840741773898303, 453654220906310222, 10380640686263467204, 249559854371799622350, 6301679967177242849680

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.

Also the number of balanced phylogenetic rooted trees on n distinct labels.

			The a(1) = 1 through a(4) = 18 rooted trees:
  (1)  (12)      (123)        (1234)
       ((1)(2))  ((1)(23))    ((1)(234))
                 ((2)(13))    ((12)(34))
                 ((3)(12))    ((13)(24))
                 ((1)(2)(3))  ((14)(23))
                              ((2)(134))
                              ((3)(124))
                              ((4)(123))
                              ((1)(2)(34))
                              ((1)(3)(24))
                              ((1)(4)(23))
                              ((2)(3)(14))
                              ((2)(4)(13))
                              ((3)(4)(12))
                              ((1)(2)(3)(4))
                              (((1)(2))((3)(4)))
                              (((1)(3))((2)(4)))
                              (((1)(4))((2)(3)))

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

Cf. A000081, A000311, A000669, A001678, A005804, A048816, A079500, A119262, A120803, A141268, A244925, A292504, A300660, A319312.

Cf. A320155, A320160, A316624, A320169, A320173, A320176, A320179.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
gug[m_]:=Prepend[Join@@Table[Union[Sort/@Tuples[gug/@mtn]],{mtn,Select[sps[m],Length[#]>1&]}],m];
Table[Length[Select[gug[Range[n]],SameQ@@Length/@Position[#,_Integer]&]],{n,9}]

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; u=EulerT(u); 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

Terms a(9) and beyond from Andrew Howroyd, Oct 26 2018

A316655 Number of series-reduced rooted trees whose leaves span an initial interval of positive integers with multiplicities the integer partition with Heinz number n.

Original entry on oeis.org

0, 1, 1, 1, 2, 3, 5, 4, 12, 9, 12, 17, 33, 29, 44, 26, 90, 90, 261, 68, 168, 93, 766, 144, 197, 307, 575, 269, 2312, 428, 7068, 236, 625, 1017, 863, 954, 21965, 3409, 2342, 712

Offset: 1

Gus Wiseman, Jul 09 2018

A rooted tree is series-reduced if every non-leaf node has at least two branches.

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			Sequence of sets of trees begins:
1:
2: 1
3: (11)
4: (12)
5: (1(11)), (111)
6: (1(12)), (2(11)), (112)
7: (1(1(11))), (1(111)), ((11)(11)), (11(11)), (1111)
8: (1(23)), (2(13)), (3(12)), (123)
9: (1(1(22))), (1(2(12))), (1(122)), (2(1(12))), (2(2(11))), (2(112)), ((11)(22)), ((12)(12)), (11(22)), (12(12)), (22(11)), (1122)

Cf. A000081, A000311, A000669, A001678, A005804, A056239, A141268, A181821, A292504, A296150, A300660, A304660.

Cf. A316651, A316652, A316653, A316654, A316656.

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,Union[Sort/@Join@@(Tuples[gro/@#]&/@Select[mps[m],Length[#]>1&])]];
Table[Length[gro[Flatten[MapIndexed[Table[#2,{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]]]]],{n,20}]

a(prime(n)) = A000669(n).

a(2^n) = A000311(n).

a(37)-a(40) from Robert Price, Sep 13 2018

A316694 Number of lone-child-avoiding locally disjoint rooted identity trees whose leaves form an integer partition of n.

Original entry on oeis.org

1, 1, 2, 3, 6, 13, 28, 62, 143, 338, 804, 1948, 4789, 11886, 29796, 75316, 191702, 491040, 1264926, 3274594, 8514784, 22229481, 58243870

Offset: 1

Gus Wiseman, Jul 10 2018

A rooted tree is lone-child-avoiding if every non-leaf node has at least two branches. It is locally disjoint if no branch overlaps any other (unequal) branch of the same root. It is an identity tree if no branch appears multiple times under the same root.

			The a(7) = 28 rooted trees:
  7,
  (16),
  (25),
  (1(15)),
  (34),
  (1(24)), (2(14)), (4(12)), (124),
  (1(1(14))),
  (3(13)),
  (2(23)),
  (1(1(23))), (1(2(13))), (1(3(12))), (1(123)), (2(1(13))), (3(1(12))), (12(13)), (13(12)),
  (1(1(1(13)))),
  (2(2(12))),
  (1(1(2(12)))), (1(2(1(12)))), (1(12(12))), (2(1(1(12)))), (12(1(12))),
  (1(1(1(1(12))))).
Missing from this list but counted by A300660 are ((12)(13)) and ((12)(1(12))).

Gus Wiseman, Sequences counting series-reduced and lone-child-avoiding trees by number of vertices.

Cf. A000081, A000669, A001678, A141268, A292504, A316653, A316654, A316656.
The semi-identity tree version is A212804.
Not requiring local disjointness gives A300660.
The non-identity tree version is A316696.
This is the case of A331686 where all leaves are singletons.
Rooted identity trees are A004111.
Locally disjoint rooted identity trees are A316471.
Lone-child-avoiding locally disjoint rooted trees are A331680.
Locally disjoint enriched identity p-trees are A331684.
Cf. A306200, A316697, A331678, A331679, A331681, A331683, A331783, A331874.

disjointQ[u_]:=Apply[And,Outer[#1==#2||Intersection[#1,#2]=={}&,u,u,1],{0,1}];
nms[n_]:=nms[n]=Prepend[Join@@Table[Select[Union[Sort/@Tuples[nms/@ptn]],And[UnsameQ@@#,disjointQ[#]]&],{ptn,Rest[IntegerPartitions[n]]}],{n}];
Table[Length[nms[n]],{n,10}]

a(21)-a(23) from Robert Price, Sep 16 2018

Updated with corrected terminology by Gus Wiseman, Feb 06 2020

A316782 Number of achiral tree-factorizations of n.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1

Offset: 1

Gus Wiseman, Jul 13 2018

nonn

A factorization of n is a finite nonempty multiset of positive integers greater than 1 with product n. An achiral tree-factorization of n is either (case 1) the number n itself or (case 2) a finite constant multiset of two or more achiral tree-factorizations, one of each factor in a factorization of n.
a(n) is also the number of ways to write n as a left-nested power-tower ((a^b)^c)^... of positive integers greater than one. For example, the a(64) = 6 ways are 64, 8^2, 4^3, 2^6, (2^3)^2, (2^2)^3.
a(n) depends only on the prime signature of n. - Andrew Howroyd, Nov 18 2018

			The a(1296) = 4 achiral tree-factorizations are 1296, (36*36), (6*6*6*6), ((6*6)*(6*6)).

Andrew Howroyd, Table of n, a(n) for n = 1..10000

Cf. A001055, A001597, A003238, A067824, A089723, A214577, A281118, A289078, A292504, A294336.

a[n_]:=1+Sum[a[d],{d,n^(1/Rest[Divisors[GCD@@FactorInteger[n][[All,2]]]])}];
Array[a,100]

a(n)={my(z, e=ispower(n,,&z)); 1 + if(e, sumdiv(e, d, if(dAndrew Howroyd, Nov 18 2018

a(n) = 1 + Sum_{n = d^k, k>1} a(d).

a(p^n) = A067824(n) for prime p. - Andrew Howroyd, Nov 18 2018

A295279 Number of strict tree-factorizations of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 4, 1, 2, 2, 3, 1, 4, 1, 4, 2, 2, 1, 10, 1, 2, 2, 4, 1, 8, 1, 6, 2, 2, 2, 12, 1, 2, 2, 10, 1, 8, 1, 4, 4, 2, 1, 26, 1, 4, 2, 4, 1, 10, 2, 10, 2, 2, 1, 28, 1, 2, 4, 12, 2, 8, 1, 4, 2, 8, 1, 44, 1, 2, 4, 4, 2, 8, 1, 26, 3, 2, 1

Offset: 1

Gus Wiseman, Nov 19 2017

nonn

A strict tree-factorization of n is either (case 1) the number n itself or (case 2) a set of two or more strict tree-factorizations, one of each factor in a factorization of n into distinct factors greater than one.

a(n) depends only on the prime signature of n. - Andrew Howroyd, Nov 18 2018

			The a(30) = 8 strict tree-factorizations are: 30, (2*3*5), (2*15), (2*(3*5)), (3*10), (3*(2*5)), (5*6), (5*(2*3)).
The a(36) = 12 strict tree-factorizations are: 36, (2*3*6), (2*3*(2*3)), (2*18), (2*(2*9)), (2*(3*6)), (2*(3*(2*3))), (3*12), (3*(2*6)), (3*(2*(2*3))), (3*(3*4)), (4*9).

Andrew Howroyd, Table of n, a(n) for n = 1..10000

Cf. A005804, A045778, A273873, A281113 A281118, A292504, A294786, A295281.

sfs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[sfs[n/d],Min@@#>d&]],{d,Rest[Divisors[n]]}]];
sft[n_]:=1+Total[Function[fac,Times@@sft/@fac]/@Select[sfs[n],Length[#]>1&]];
Array[sft,100]

seq(n)={my(v=vector(n), w=vector(n)); w[1]=v[1]=1; for(k=2, n, w[k]=v[k]+1; forstep(j=n\k*k, k, -k, v[j]+=w[k]*v[j/k])); w} \\ Andrew Howroyd, Nov 18 2018

a(product of n distinct primes) = A005804(n).
a(prime^n) = A273873(n).
Dirichlet g.f.: (Zeta(s) + Product_{n >= 2}(1 + a(n)/n^s))/2.

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

A316653 Number of series-reduced rooted identity trees with n leaves spanning an initial interval of positive integers.

Original entry on oeis.org

1, 1, 6, 58, 774, 13171, 272700, 6655962, 187172762, 5959665653, 211947272186, 8327259067439, 358211528524432, 16744766791743136, 845195057333580332, 45814333121920927067, 2654330505021077873594, 163687811930206581162063, 10705203621191765328300832

Offset: 1

Gus Wiseman, Jul 09 2018

nonn

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.

			The a(3) = 6 trees are (1(12)), (2(12)), (1(23)), (2(13)), (3(12)), (123).

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

Cf. A000081, A000311, A000669, A001678, A004111, A005804, A034691, A141268, A292504, A300660.

Cf. A316651, A316652, A316654, A316655, A316656.

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@@#&]];
allnorm[n_Integer]:=Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1];
Table[Sum[Length[gro[m]],{m,allnorm[n]}],{n,5}]

\\ here R(n,2) is A031148.
WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v,n,(-1)^(n-1)/n))))-1,-#v)}
R(n,k)={my(v=[k]); for(n=2, n, v=concat(v, WeighT(concat(v,[0]))[n])); v}
seq(n)={sum(k=1, n, R(n,k)*sum(r=k, n, binomial(r,k)*(-1)^(r-k)) )} \\ Andrew Howroyd, Sep 14 2018

Terms a(9) and beyond from Andrew Howroyd, Sep 14 2018

A316656 Number of series-reduced rooted identity trees whose leaves span an initial interval of positive integers with multiplicities the integer partition with Heinz number n.

Original entry on oeis.org

0, 1, 0, 1, 0, 1, 0, 4, 3, 1, 0, 9, 0, 1, 6, 26, 0, 36, 0, 16, 10, 1, 0, 92, 21, 1, 197, 25, 0, 100, 0, 236, 15, 1, 53, 474

Offset: 1

Gus Wiseman, Jul 09 2018

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.

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			Sequence of sets of trees begins:
   1:
   2: 1
   3:
   4: (12)
   5:
   6: (1(12))
   7:
   8: (1(23)), (2(13)), (3(12)), (123)
   9: (1(2(12))), (2(1(12))), (12(12))
  10: (1(1(12)))
  11:
  12: (1(1(23))), (1(2(13))), (1(3(12))), (1(123)), (2(1(13))), (3(1(12))), ((12)(13)), (12(13)), (13(12))

Cf. A000081, A000311, A000669, A001678, A004111, A005804, A056239, A141268, A181821, A292504, A296150, A300660, A304660.

Cf. A316651, A316652, A316653. A316654, A316655.

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[Length[gro[Flatten[MapIndexed[Table[#2,{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]]]]],{n,30}]

a(prime(n>1)) = 0.

a(2^n) = A000311(n).

cp's OEIS Frontend

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

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A292505 Number of complete orderless tree-factorizations of n >= 2.

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A320154 Number of series-reduced balanced rooted trees whose leaves form a set partition of {1,...,n}.

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A316655 Number of series-reduced rooted trees whose leaves span an initial interval of positive integers with multiplicities the integer partition with Heinz number n.

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A316694 Number of lone-child-avoiding locally disjoint rooted identity trees whose leaves form an integer partition of n.

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Mathematica

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A316782 Number of achiral tree-factorizations of n.

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A295279 Number of strict tree-factorizations of n.

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