A281118 -id:A281118 - cp's OEIS frontend

A292504 Number of orderless tree-factorizations of n.

1, 1, 1, 2, 1, 2, 1, 4, 2, 2, 1, 6, 1, 2, 2, 11, 1, 6, 1, 6, 2, 2, 1, 20, 2, 2, 4, 6, 1, 8, 1, 30, 2, 2, 2, 27, 1, 2, 2, 20, 1, 8, 1, 6, 6, 2, 1, 74, 2, 6, 2, 6, 1, 20, 2, 20, 2, 2, 1, 38, 1, 2, 6, 96, 2, 8, 1, 6, 2, 8, 1, 114, 1, 2, 6, 6, 2, 8, 1, 74, 11, 2, 1

Offset: 1

Gus Wiseman, Sep 17 2017

nonn

A factorization of n is a finite multiset of positive integers greater than 1 with product n. An orderless tree-factorization of n is either (case 1) the number n itself or (case 2) a finite multiset of two or more orderless tree-factorizations, one of each factor in a factorization of n.

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

			The a(16)=11 orderless tree-factorizations are: 16, (28), (2(24)), (2(2(22))), (2(222)), (44), (4(22)), ((22)(22)), (224), (22(22)), (2222).

Michael De Vlieger, Table of n, a(n) for n = 1..16384
Michael De Vlieger, Records and indices of records.

Cf. A000311, A001055, A050336, A141268, A281118, A292505.

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[oltfacs[n]],{n,83}]

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, my(i=j, e=0); while(i%k==0, i/=k; e++; v[j] += binomial(e+w[k]-1, e)*v[i]))); w} \\ Andrew Howroyd, Nov 18 2018

a(p^n) = A141268(n) for prime p. - Andrew Howroyd, Nov 18 2018

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.

Original entry on oeis.org

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]))}

A318812 Number of total multiset partitions of the multiset of prime indices of n. Number of total factorizations of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 6, 1, 3, 1, 3, 1, 1, 1, 11, 1, 1, 2, 3, 1, 4, 1, 20, 1, 1, 1, 15, 1, 1, 1, 11, 1, 4, 1, 3, 3, 1, 1, 51, 1, 3, 1, 3, 1, 11, 1, 11, 1, 1, 1, 21, 1, 1, 3, 90, 1, 4, 1, 3, 1, 4, 1, 80, 1, 1, 3, 3, 1, 4, 1, 51, 6, 1, 1

Offset: 1

Gus Wiseman, Sep 04 2018

nonn

A total multiset partition of m is either m itself or a total multiset partition of a multiset partition of m that is neither minimal nor maximal.

a(n) depends only on the prime signature of n. - Andrew Howroyd, Dec 30 2019

			The a(24) = 11 total multiset partitions:
  {1,1,1,2}
  {{1},{1,1,2}}
  {{2},{1,1,1}}
  {{1,1},{1,2}}
  {{1},{1},{1,2}}
  {{1},{2},{1,1}}
  {{{1}},{{1},{1,2}}}
  {{{1}},{{2},{1,1}}}
  {{{2}},{{1},{1,1}}}
  {{{1,2}},{{1},{1}}}
  {{{1,1}},{{1},{2}}}
The a(24) = 11 total factorizations:
  24,
  (2*12), (3*8), (4*6),
  (2*2*6), (2*3*4),
  ((2)*(2*6)), ((6)*(2*2)), ((2)*(3*4)), ((3)*(2*4)), ((4)*(2*3)).

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

Cf. A000110, A001055, A002846, A005121, A213427, A281113, A281118, A281119, A317145, A318813.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
totfac[n_]:=1+Sum[totfac[Times@@Prime/@f],{f,Select[facs[n],1

MultEulerT(u)={my(v=vector(#u)); v[1]=1; for(k=2, #u, forstep(j=#v\k*k, k, -k, my(i=j, e=0); while(i%k==0, i/=k; e++; v[j]+=binomial(e+u[k]-1, e)*v[i]))); v}
seq(n)={my(v=vector(n, i, isprime(i)), u=vector(n), m=logint(n,2)+1); for(r=1, m, u += v*sum(j=r, m, (-1)^(j-r)*binomial(j-1, r-1)); v=MultEulerT(v)); u[1]=1; u} \\ Andrew Howroyd, Dec 30 2019

a(product of n distinct primes) = A005121(n).

a(prime^n) = A318813(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

A281119 Number of complete 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, 34, 1, 4, 1, 3, 1, 4, 1, 44, 1, 1, 3, 3, 1, 4, 1, 29, 5, 1, 1

Offset: 2

Gus Wiseman, Jan 15 2017

nonn

A tree-factorization of n>=2 is either (case 1) the number n or (case 2) a sequence of two or more tree-factorizations, one of each part of a weakly increasing factorization of n into factors greater than 1. A complete (or total) tree-factorization is a tree-factorization whose leaves are all prime numbers.

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

			The a(36)=12 complete tree-factorizations of 36 are:
(2(2(33))), (2(3(23))), (2(233)),   (3(2(23))),
(3(3(22))), (3(223)),   ((22)(33)), ((23)(23)),
(22(33)),   (23(23)),   (33(22)),   (2233).

Michael De Vlieger, Table of n, a(n) for n = 2..10000
A. Knopfmacher, M. Mays, Ordered and Unordered Factorizations of Integers, The Mathematica Journal 10(1), 2006.

Cf. A000311, A001055, A063834, A196545, A262673, A273873, A281113, A281118.

postfacs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[postfacs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
treefacs[n_]:=If[n<=1,{{}},Prepend[Join@@Function[q,Tuples[treefacs/@q]]/@DeleteCases[postfacs[n],{n}],n]];
Table[Length[Select[treefacs[n],FreeQ[#,_Integer?(!PrimeQ[#]&)]&]],{n,2,83}]

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]+=w[k]^e*v[i]))); w[2..n]} \\ Andrew Howroyd, Nov 18 2018

a(p^n) = A196545(n) for prime p. - Andrew Howroyd, Nov 18 2018

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.

A319122 Number of phylogenetic plane trees on n labels.

Original entry on oeis.org

1, 3, 25, 387, 8521, 241683, 8383705, 343826787, 16273985641, 873119718963, 52360707915385, 3470858539699587, 252000934472119561, 19888355652445884243, 1695252683833578455065, 155208762048402360698787, 15190477481877333732410281, 1582657042668691276257233523

Offset: 1

Gus Wiseman, Sep 11 2018

nonn

A phylogenetic plane tree on n labels is either the set of labels itself or a finite sequence of at least two phylogenetic plane trees, one on each block of an ordered set partition of the labels.

			The a(2) = 3 phylogenetic plane trees are {1,2}, ({1},{2}), ({2},{1}).

Cf. A000108, A000670, A001003, A005804, A074206, A118376, A277130, A281118, A281119, A319137.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
t[n_]:=t[n]=1+Sum[Times@@t/@f,{f,Join@@Permutations/@Select[sps[Range[n]],Length[#]>1&]}];
Array[t,8]

A330935 Irregular triangle read by rows where T(n,k) is the number of length-k chains from minimum to maximum in the poset of factorizations of n into factors > 1, ordered by refinement.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 0, 1, 0, 1, 3, 2, 1, 0, 1, 2, 1, 0, 1, 2, 0, 1, 0, 1, 1, 0, 1, 5, 5, 0, 1, 0, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 1, 0, 1, 5, 8, 4, 0, 1, 0, 1, 0, 1, 0, 1, 7, 7, 1, 0, 1, 0, 1, 0, 1, 5, 5, 1, 0, 1

Offset: 1

Gus Wiseman, Jan 04 2020

This poset is equivalent to the poset of multiset partitions of the prime indices of n, ordered by refinement.

			Triangle begins:
   1:          16: 0 1 3 2    31: 1            46: 0 1
   2: 1        17: 1          32: 0 1 5 8 4    47: 1
   3: 1        18: 0 1 2      33: 0 1          48: 0 1 10 23 15
   4: 0 1      19: 1          34: 0 1          49: 0 1
   5: 1        20: 0 1 2      35: 0 1          50: 0 1 2
   6: 0 1      21: 0 1        36: 0 1 7 7      51: 0 1
   7: 1        22: 0 1        37: 1            52: 0 1 2
   8: 0 1 1    23: 1          38: 0 1          53: 1
   9: 0 1      24: 0 1 5 5    39: 0 1          54: 0 1 5 5
  10: 0 1      25: 0 1        40: 0 1 5 5      55: 0 1
  11: 1        26: 0 1        41: 1            56: 0 1 5 5
  12: 0 1 2    27: 0 1 1      42: 0 1 3        57: 0 1
  13: 1        28: 0 1 2      43: 1            58: 0 1
  14: 0 1      29: 1          44: 0 1 2        59: 1
  15: 0 1      30: 0 1 3      45: 0 1 2        60: 0 1 9 11
Row n = 48 counts the following chains (minimum and maximum not shown):
  ()  (6*8)      (2*3*8)->(6*8)       (2*2*2*6)->(2*4*6)->(6*8)
      (2*24)     (2*4*6)->(6*8)       (2*2*3*4)->(2*3*8)->(6*8)
      (3*16)     (2*3*8)->(2*24)      (2*2*3*4)->(2*4*6)->(6*8)
      (4*12)     (2*3*8)->(3*16)      (2*2*2*6)->(2*4*6)->(2*24)
      (2*3*8)    (2*4*6)->(2*24)      (2*2*2*6)->(2*4*6)->(4*12)
      (2*4*6)    (2*4*6)->(4*12)      (2*2*3*4)->(2*3*8)->(2*24)
      (3*4*4)    (3*4*4)->(3*16)      (2*2*3*4)->(2*3*8)->(3*16)
      (2*2*12)   (3*4*4)->(4*12)      (2*2*3*4)->(2*4*6)->(2*24)
      (2*2*2*6)  (2*2*12)->(2*24)     (2*2*3*4)->(2*4*6)->(4*12)
      (2*2*3*4)  (2*2*12)->(4*12)     (2*2*3*4)->(3*4*4)->(3*16)
                 (2*2*2*6)->(6*8)     (2*2*3*4)->(3*4*4)->(4*12)
                 (2*2*3*4)->(6*8)     (2*2*2*6)->(2*2*12)->(2*24)
                 (2*2*2*6)->(2*24)    (2*2*2*6)->(2*2*12)->(4*12)
                 (2*2*2*6)->(4*12)    (2*2*3*4)->(2*2*12)->(2*24)
                 (2*2*3*4)->(2*24)    (2*2*3*4)->(2*2*12)->(4*12)
                 (2*2*3*4)->(3*16)
                 (2*2*3*4)->(4*12)
                 (2*2*2*6)->(2*4*6)
                 (2*2*3*4)->(2*3*8)
                 (2*2*3*4)->(2*4*6)
                 (2*2*3*4)->(3*4*4)
                 (2*2*2*6)->(2*2*12)
                 (2*2*3*4)->(2*2*12)

Row lengths are A001222.
Row sums are A317176.
Column k = 1 is A010051.
Column k = 2 is A066247.
Column k = 3 is A330936.
Final terms of each row are A317145.
The version for set partitions is A008826, with row sums A005121.
The version for integer partitions is A330785, with row sums A213427.
Cf. A001055, A002846, A003238, A007716, A281118, A292504, A292505, A318812, A330665, A330727.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
upfacs[q_]:=Union[Sort/@Join@@@Tuples[facs/@q]];
paths[eds_,start_,end_]:=If[start==end,Prepend[#,{}],#]&[Join@@Table[Prepend[#,e]&/@paths[eds,Last[e],end],{e,Select[eds,First[#]==start&]}]];
Table[Length[Select[paths[Join@@Table[{y,#}&/@DeleteCases[upfacs[y],y],{y,facs[n]}],{n},First[facs[n]]],Length[#]==k-1&]],{n,100},{k,PrimeOmega[n]}]

T(2^n,k) = A330785(n,k).

T(n,1) + T(n,2) = 1.

cp's OEIS Frontend

A292504 Number of orderless tree-factorizations of n.

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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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A318812 Number of total multiset partitions of the multiset of prime indices of n. Number of total factorizations of n.

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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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A281119 Number of complete tree-factorizations of n >= 2.

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A316782 Number of achiral tree-factorizations of n.

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