A319312 -id:A319312 - cp's OEIS frontend

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

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

A320176 Number of series-reduced rooted trees whose leaves are strict integer partitions whose multiset union is a strict integer partition of n.

Original entry on oeis.org

1, 1, 3, 3, 5, 13, 15, 23, 33, 99, 109, 183, 251, 383, 1071, 1261, 2007, 2875, 4291, 5829, 16297, 18563, 30313, 42243, 63707, 85351, 125465, 297843, 356657, 556729, 783637, 1151803, 1564173, 2249885, 2988729, 6803577, 8026109, 12465665, 17124495, 25272841, 33657209

Offset: 1

Gus Wiseman, Oct 07 2018

nonn

Also the number of orderless tree-factorizations of Heinz numbers of strict integer partitions of n.

Also the number of phylogenetic trees on a set of distinct labels summing to n.

			The a(1) = 1 through a(7) = 15 rooted trees:
  (1)  (2)  (3)       (4)       (5)       (6)            (7)
            (21)      (31)      (32)      (42)           (43)
            ((1)(2))  ((1)(3))  (41)      (51)           (52)
                                ((1)(4))  (321)          (61)
                                ((2)(3))  ((1)(5))       (421)
                                          ((2)(4))       ((1)(6))
                                          ((1)(23))      ((2)(5))
                                          ((2)(13))      ((3)(4))
                                          ((3)(12))      ((1)(24))
                                          ((1)(2)(3))    ((2)(14))
                                          ((1)((2)(3)))  ((4)(12))
                                          ((2)((1)(3)))  ((1)(2)(4))
                                          ((3)((1)(2)))  ((1)((2)(4)))
                                                         ((2)((1)(4)))
                                                         ((4)((1)(2)))

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Cf. A000669, A005804, A008289, A141268, A292504, A300660, A319312, A320171, A320174, A320175, A320177, A320178.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
got[m_]:=Prepend[Join@@Table[Union[Sort/@Tuples[got/@p]],{p,Select[sps[m],Length[#]>1&]}],m];
Table[Length[Join@@Table[got[m],{m,Select[IntegerPartitions[n],UnsameQ@@#&]}]],{n,20}]

\\ here S(n) is first n terms of A005804.
EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
b(n,k)={my(v=vector(n)); for(n=1, n, v[n]=binomial(n+k-1, n) + EulerT(v[1..n])[n]); v}
S(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]))}
seq(n)={my(u=S((sqrtint(8*n+1)-1)\2)); [sum(i=1, poldegree(p), polcoef(p,i)*u[i]) | p <- Vec(prod(k=1, n, 1 + x^k*y + O(x*x^n))-1)]} \\ Andrew Howroyd, Oct 26 2018

a(n) = Sum_{k>0} A008289(n, k)*A005804(k). - Andrew Howroyd, Oct 26 2018

Terms a(31) and beyond from Andrew Howroyd, Oct 26 2018

A320293 Number of series-reduced rooted trees whose leaves are integer partitions whose multiset union is an integer partition of n with no 1's.

Original entry on oeis.org

0, 1, 1, 3, 3, 9, 11, 30, 45, 112, 195, 475, 901, 2136, 4349, 10156, 21565, 50003, 109325, 252761, 563785, 1303296, 2948555, 6826494, 15604053, 36210591, 83415487, 194094257, 449813607, 1049555795, 2444027917, 5718195984, 13367881473, 31357008065, 73546933115

Offset: 1

Gus Wiseman, Oct 09 2018

nonn

Also phylogenetic trees on integer partitions of n with no 1's.

			The a(2) = 1 through a(7) = 11 trees:
  (2)  (3)  (4)       (5)       (6)            (7)
            (22)      (32)      (33)           (43)
            ((2)(2))  ((2)(3))  (42)           (52)
                                (222)          (322)
                                ((2)(4))       ((2)(5))
                                ((3)(3))       ((3)(4))
                                ((2)(22))      ((2)(23))
                                ((2)(2)(2))    ((3)(22))
                                ((2)((2)(2)))  ((2)(2)(3))
                                               ((2)((2)(3)))
                                               ((3)((2)(2)))

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

Cf. A000045, A000311, A000669, A002865, A141268, A292504, A300660, A304967, A319312, A320289, A320294, A320295, A320296.

EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
seq(n)={my(p=1/prod(k=2, n, 1 - x^k + O(x*x^n)), v=vector(n)); for(n=1, n, v[n]=polcoef(p, n) + EulerT(v[1..n])[n]); v} \\ Andrew Howroyd, Oct 25 2018

Terms a(23) and beyond from Andrew Howroyd, Oct 25 2018

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

A331685 Number of tree-factorizations of Heinz numbers of integer partitions of n.

Original entry on oeis.org

1, 3, 7, 23, 69, 261, 943, 3815, 15107, 63219, 262791, 1130953, 4838813, 21185125, 92593943, 411160627, 1823656199, 8186105099, 36728532951, 166310761655

Offset: 1

Gus Wiseman, Jan 31 2020

A tree-factorization of n > 1 is either (case 1) the number n itself, or (case 2) a sequence of two or more tree-factorizations, one of each part of a weakly increasing factorization of n into factors > 1.

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

			The a(1) = 1 through a(4) = 23 tree-factorizations:
  2  3      5          7
     4      6          9
     (2*2)  8          10
            (2*3)      12
            (2*4)      16
            (2*2*2)    (2*5)
            (2*(2*2))  (2*6)
                       (2*8)
                       (3*3)
                       (3*4)
                       (4*4)
                       (2*2*3)
                       (2*2*4)
                       (2*2*2*2)
                       (2*(2*3))
                       ((2*2)*4)
                       (2*(2*4))
                       (3*(2*2))
                       (4*(2*2))
                       (2*(2*2*2))
                       (2*2*(2*2))
                       ((2*2)*(2*2))
                       (2*(2*(2*2)))

The orderless version is A319312.
Factorizations are A001055.
P-trees are A196545.
Twice-factorizations are A281113.
Tree-factorizations are A281118.
Enriched p-trees are A289501.
Cf. A000081 A000669, A000720, A001678, A005804, A056239, A112798, A141268, A273873, A330465.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
physemi[n_]:=Prepend[Join@@Table[Tuples[physemi/@f],{f,Select[facs[n],Length[#]>1&]}],n];
Table[Sum[Length[physemi[Times@@Prime/@m]],{m,IntegerPartitions[n]}],{n,8}]

\\ here TF(n) is n terms of A281118 as vector.
TF(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] += w[k]^e*v[i]))); w}
a(n)={my(v=[prod(i=1, #p, prime(p[i])) | p<-partitions(n)], tf=TF(vecmax(v))); sum(i=1, #v, tf[v[i]])} \\ Andrew Howroyd, Dec 09 2020

a(n) = Sum_i A281118(A215366(n,i)).

a(13)-a(20) from Andrew Howroyd, Dec 09 2020

cp's OEIS Frontend

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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A320176 Number of series-reduced rooted trees whose leaves are strict integer partitions whose multiset union is a strict integer partition of n.

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A320293 Number of series-reduced rooted trees whose leaves are integer partitions whose multiset union is an integer partition of n with no 1's.

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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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A331685 Number of tree-factorizations of Heinz numbers of integer partitions of n.

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