A120803 -id:A120803 - cp's OEIS frontend

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

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

A320172 Number of series-reduced balanced rooted identity trees whose leaves are integer partitions whose multiset union is an integer partition of n.

Original entry on oeis.org

1, 2, 5, 9, 19, 38, 79, 163, 352, 750, 1633, 3558, 7783, 17020, 37338, 81920, 180399, 398600, 885101, 1975638, 4435741, 10013855, 22726109, 51807432, 118545425, 272024659, 625488420, 1440067761, 3317675261, 7644488052, 17610215982, 40547552277, 93298838972, 214516498359, 492844378878

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. In an identity tree, all branches directly under any given node are different.

			The a(1) = 1 through a(5) = 19 rooted identity trees:
  (1)  (2)   (3)        (4)         (5)
       (11)  (21)       (22)        (32)
             (111)      (31)        (41)
             ((1)(2))   (211)       (221)
             ((1)(11))  (1111)      (311)
                        ((1)(3))    (2111)
                        ((1)(21))   (11111)
                        ((2)(11))   ((1)(4))
                        ((1)(111))  ((2)(3))
                                    ((1)(31))
                                    ((1)(22))
                                    ((2)(21))
                                    ((3)(11))
                                    ((1)(211))
                                    ((11)(21))
                                    ((2)(111))
                                    ((1)(1111))
                                    ((11)(111))
                                    ((1)(2)(11))

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

Cf. A000669, A005804, A048816, A079500, A119262, A120803, A141268, A292504, A300660, A319312.

Cf. A320154, A320155, A320160, A320171, A320173, A320177, A320178, A320179.

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]]]];
gig[m_]:=Prepend[Join@@Table[Union[Sort/@Select[Sort/@Tuples[gig/@mtn],UnsameQ@@#&]],{mtn,Select[mps[m],Length[#]>1&]}],m];
Table[Sum[Length[Select[gig[y],SameQ@@Length/@Position[#,_Integer]&]],{y,Sort /@IntegerPartitions[n]}],{n,8}]

WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v,n,(-1)^(n-1)/n))))-1,-#v)}
seq(n)={my(u=vector(n, n, numbpart(n)), v=vector(n)); while(u, v+=u; u=WeighT(u)-u); v} \\ Andrew Howroyd, Oct 25 2018

Terms a(13) and beyond from Andrew Howroyd, Oct 25 2018

A306269 Regular triangle read by rows where T(n,k) is the number of unlabeled balanced rooted semi-identity trees with n >= 1 nodes and depth 0 <= k < n.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 2, 1, 1, 1, 0, 1, 2, 2, 1, 1, 1, 0, 1, 3, 3, 2, 1, 1, 1, 0, 1, 3, 4, 3, 2, 1, 1, 1, 0, 1, 5, 6, 5, 3, 2, 1, 1, 1, 0, 1, 5, 9, 7, 5, 3, 2, 1, 1, 1, 0, 1, 7, 12, 12, 8, 5, 3, 2, 1, 1, 1, 0, 1, 8, 17, 17, 13, 8, 5, 3, 2, 1, 1, 1

Offset: 1

Gus Wiseman, Feb 01 2019

A rooted tree is a semi-identity tree if the non-leaf branches of the root are all distinct and are themselves semi-identity trees. It is balanced if all leaves are the same distance from the root.

			Triangle begins:
  1
  0  1
  0  1  1
  0  1  1  1
  0  1  1  1  1
  0  1  2  1  1  1
  0  1  2  2  1  1  1
  0  1  3  3  2  1  1  1
  0  1  3  4  3  2  1  1  1
  0  1  5  6  5  3  2  1  1  1
  0  1  5  9  7  5  3  2  1  1  1
  0  1  7 12 12  8  5  3  2  1  1  1
  0  1  8 17 17 13  8  5  3  2  1  1  1
  0  1 10 25 26 20 14  8  5  3  2  1  1  1
  0  1 12 34 39 31 21 14  8  5  3  2  1  1  1
The postpositive terms of row 9 {3, 4, 3, 2} count the following trees:
  ((ooooooo))   (((oooooo)))    ((((ooooo))))    (((((oooo)))))
  ((o)(ooooo))  (((o)(oooo)))   ((((o)(ooo))))   (((((o)(oo)))))
  ((oo)(oooo))  (((oo)(ooo)))   ((((o))((oo))))
                (((o))((ooo)))

Alois P. Heinz, Rows n = 1..200, flattened
Gus Wiseman, Unlabeled balanced rooted semi-identity trees with 12 nodes, organized by depth.

Row sums give A306201.
T(2n-1,n) gives A306274.
Cf. A000081, A004111, A048816, A079500, A120803, A184155, A276625, A306200, A306202, A306203, A320222, A320270.

ubk[n_,k_]:=Select[Join@@Table[Select[Union[Sort/@Tuples[ubk[#,k-1]&/@ptn]],UnsameQ@@DeleteCases[#,{}]&],{ptn,IntegerPartitions[n-1]}],SameQ[k,##]&@@Length/@Position[#,{}]&];
Table[Length[ubk[n,k]],{n,1,10},{k,0,n-1}]

A320221 Irregular 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, (n>=1, min(1,n-1) <= k <= log_2(n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 3, 1, 6, 1, 1, 7, 1, 1, 11, 4, 1, 13, 6, 1, 20, 16, 1, 23, 23, 1, 33, 46, 1, 40, 70, 1, 54, 127, 1, 1, 65, 189, 1, 1, 87, 320, 5, 1, 104, 476, 10, 1, 136, 771, 32, 1, 164, 1145, 63, 1, 209, 1795, 154, 1, 252, 2657, 304, 1, 319, 4091, 656

Offset: 1

Gus Wiseman, Oct 07 2018

			Triangle begins:
  1
  1
  1
  1  1
  1  1
  1  3
  1  3
  1  6  1
  1  7  1
  1 11  4
  1 13  6
  1 20 16
  1 23 23
  1 33 46
  1 40 70
The T(11,3) = 6 rooted trees:
   (((oo)(oo))((oo)(ooooo)))
   (((oo)(oo))((ooo)(oooo)))
   (((oo)(ooo))((oo)(oooo)))
   (((oo)(ooo))((ooo)(ooo)))
  (((oo)(oo))((oo)(oo)(ooo)))
  (((oo)(ooo))((oo)(oo)(oo)))

Andrew Howroyd, Table of n, a(n) for n = 1..1154 (rows 1..200)

Row sums are A120803. Second column is A083751. A regular version is A320179.
Cf. A000669, A005804, A048816, A119262, A120803, A141268, A244925, 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&]}]];
DeleteCases[Table[Length[Select[qurt[n],SameQ[##,k]&@@Length/@Position[#,{}]&]],{n,10},{k,0,n-1}],0,{2}]

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); v[1]=x; [Vecrev(p/x) | p<-v]}
{ my(A=T(15)); for(n=1, #A, print(A[n])) } \\ Andrew Howroyd, Dec 09 2020

Terms a(36) and beyond from Andrew Howroyd, Dec 09 2020

Name clarified by Andrew Howroyd, Dec 09 2020

A320266 Number of balanced orderless tree-factorizations of n.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 4, 1, 2, 2, 6, 1, 4, 1, 4, 2, 2, 1, 8, 2, 2, 3, 4, 1, 5, 1, 9, 2, 2, 2, 11, 1, 2, 2, 8, 1, 5, 1, 4, 4, 2, 1, 17, 2, 4, 2, 4, 1, 8, 2, 8, 2, 2, 1, 13, 1, 2, 4, 19, 2, 5, 1, 4, 2, 5, 1, 24, 1, 2, 4, 4, 2, 5, 1, 17, 6, 2, 1, 13, 2

Offset: 1

Gus Wiseman, Oct 08 2018

nonn

A rooted tree is balanced if all leaves are the same distance from the root.
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(36) = 11 balanced orderless tree-factorizations:
  36,
  (2*18), (3*12), (4*9), (6*6),
  (2*2*9), (2*3*6), (3*3*4),
  (2*2*3*3), ((2*2)*(3*3)), ((2*3)*(2*3)).

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

Cf. A000669, A001055, A048816, A050336, A281118, A292505, A119262, A120803, A141268, A292504, A319312, A320160, A320267.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
oltfacs[n_]:=If[n<=1,{{}},Prepend[Union@@Function[q,Sort/@Tuples[oltfacs/@q]]/@DeleteCases[facs[n],{n}],n]];
Table[Length[Select[oltfacs[n],SameQ@@Length/@Position[#,_Integer]&]],{n,100}]

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(u=vector(n, i, 1), v=vector(n)); while(u, v+=u; u[1]=1; u=MultEulerT(u)-u); v} \\ Andrew Howroyd, Nov 18 2018

a(p^n) = A320160(n) for prime p. - Andrew Howroyd, Nov 18 2018

A320267 Number of balanced complete orderless tree-factorizations of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 3, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 3, 2, 1, 1, 3, 1, 1, 1

Offset: 1

Gus Wiseman, Oct 08 2018

nonn

a(1) = 1 by convention.
A rooted tree is balanced if all leaves are the same distance from the root.
An orderless tree-factorization (see A292504 for definition) is complete if all leaves are prime numbers.
a(n) depends only on the prime signature of n. - Andrew Howroyd, Nov 18 2018

			The a(96) = 5 balanced complete orderless tree-factorizations:
     (2*2*2*2*2*3)
   ((2*2)*(2*2*2*3))
   ((2*3)*(2*2*2*2))
   ((2*2*2)*(2*2*3))
  ((2*2)*(2*2)*(2*3))

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

Cf. A001055, A048816, A050336, A281118, A292505, A119262, A120803, A292504, A320160, A320266.

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

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(u=vector(n, i, i==1 || isprime(i)), v=vector(n)); while(u, v+=u; u[1]=1; u=MultEulerT(u)-u); v} \\ Andrew Howroyd, Nov 18 2018

a(p^n) = A120803(n) for prime p. - Andrew Howroyd, Nov 18 2018

cp's OEIS Frontend

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

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A320172 Number of series-reduced balanced rooted identity trees whose leaves are integer partitions whose multiset union is an integer partition of n.

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A306269 Regular triangle read by rows where T(n,k) is the number of unlabeled balanced rooted semi-identity trees with n >= 1 nodes and depth 0 <= k < n.

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A320221 Irregular 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, (n>=1, min(1,n-1) <= k <= log_2(n)).

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A320266 Number of balanced orderless tree-factorizations of n.

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A320267 Number of balanced complete orderless tree-factorizations of n.

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