A300660 -id:A300660 - cp's OEIS frontend

A331993 Number of semi-lone-child-avoiding rooted semi-identity trees with n unlabeled vertices.

1, 1, 1, 2, 3, 6, 11, 22, 43, 90, 185, 393, 835, 1802, 3904, 8540, 18756, 41463, 92022, 205179, 459086, 1030917, 2321949, 5245104, 11878750, 26967957, 61359917, 139902251, 319591669, 731385621, 1676573854, 3849288924, 8850674950, 20378544752, 46982414535

Offset: 1

Gus Wiseman, Feb 05 2020

nonn

Semi-lone-child-avoiding means there are no vertices with exactly one child unless that child is an endpoint/leaf.

In a semi-identity tree, the non-leaf branches of any given vertex are distinct.

			The a(1) = 1 through a(7) = 11 trees:
  o  (o)  (oo)  (ooo)   (oooo)   (ooooo)    (oooooo)
                (o(o))  (o(oo))  (o(ooo))   (o(oooo))
                        (oo(o))  (oo(oo))   (oo(ooo))
                                 (ooo(o))   (ooo(oo))
                                 ((o)(oo))  (oooo(o))
                                 (o(o(o)))  ((o)(ooo))
                                            (o(o)(oo))
                                            (o(o(oo)))
                                            (o(oo(o)))
                                            (oo(o(o)))
                                            ((o)(o(o)))

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

Not requiring any lone-child-avoidance gives A306200.
The locally disjoint case is A324969 (essentially A000045).
Matula-Goebel numbers of these trees are A331994.
Lone-child-avoiding rooted identity trees are A000007.
Semi-lone-child-avoiding rooted trees are A331934.
Semi-lone-child-avoiding rooted identity trees are A331964.
Lone-child-avoiding rooted semi-identity trees are A331966.
Cf. A001678, A004111, A300660, A316694, A331683, A331686, A331783, A331875, A331933, A331963, A331965.

sssb[n_]:=Switch[n,1,{{}},2,{{{}}},_,Join@@Function[c,Select[Union[Sort/@Tuples[sssb/@c]],UnsameQ@@DeleteCases[#,{}]&]]/@Rest[IntegerPartitions[n-1]]];
Table[Length[sssb[n]],{n,10}]

WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v,n,(-1)^(n-1)/n))))-1,-#v)}
seq(n)={my(v=[0]); for(n=1, n-1, v=concat(v, 1 + vecsum(WeighT(v)) - v[n])); v[1]=1; v} \\ Andrew Howroyd, Feb 09 2020

Terms a(26) and beyond from Andrew Howroyd, Feb 09 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

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

A316784 Number of orderless identity 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, 13, 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, 13, 2, 8, 1, 4, 2, 8, 1, 46, 1, 2, 4, 4, 2, 8, 1, 26, 3, 2, 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 orderless identity tree-factorization of n is either (case 1) the number n itself or (case 2) a finite set of two or more distinct orderless identity 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(24)=10 orderless identity tree-factorizations:
  24
  (4*6)
  (3*8)
  (2*12)
  (2*3*4)
  (4*(2*3))
  (3*(2*4))
  (2*(2*6))
  (2*(3*4))
  (2*(2*(2*3)))

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

Cf. A001055, A001678, A004111, A292504, A292505, A295279, A300660, A316782.

postfacs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[postfacs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
oltsfacs[n_]:=If[n<=1,{{}},Prepend[Select[Union@@Function[q,Sort/@Tuples[oltsfacs/@q]]/@DeleteCases[postfacs[n],{n}],UnsameQ@@#&],n]];
Table[Length[oltsfacs[n]],{n,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, my(i=j, e=0); while(i%k==0, i/=k; e++; v[j] += binomial(w[k], e)*v[i]))); w} \\ Andrew Howroyd, Nov 18 2018

a(p^n) = A300660(n) for prime p. - Andrew Howroyd, Nov 18 2018

A316766 Number of series-reduced locally stable rooted identity trees whose leaves form an integer partition of n.

Original entry on oeis.org

1, 1, 2, 3, 6, 13, 30, 72, 180, 458, 1194, 3160, 8459, 22881, 62417, 171526, 474405, 1319395, 3687711, 10352696, 29178988

Offset: 1

Gus Wiseman, Jul 12 2018

A rooted tree is series-reduced if every non-leaf node has at least two branches. It is locally stable if no branch is a submultiset of any other branch of the same root. It is an identity tree if no branch appears multiple times under the same root.

			The a(6) = 13 trees:
6,
(15),
(1(14)),
(1(1(13))),
(1(1(1(12)))),
(1(23)), (2(13)), (3(12)), (123),
(1(2(12))), (2(1(12))), (12(12)),
(24).
Example of non-stable trees are ((12)(123)) and ((12)(12(12))).

Cf. A000081, A000669, A001678, A004111, A141268, A292504, A300660, A316467, A316474, A316653, A316654, A316656.

submultisetQ[M_,N_]:=Or[Length[M]==0,MatchQ[{Sort[List@@M],Sort[List@@N]},{{x_,Z___},{_,x_,W___}}/;submultisetQ[{Z},{W}]]];
stableQ[u_]:=Apply[And,Outer[#1==#2||!submultisetQ[#1,#2]&&!submultisetQ[#2,#1]&,u,u,1],{0,1}];
nms[n_]:=nms[n]=Prepend[Join@@Table[Select[Union[Sort/@Tuples[nms/@ptn]],And[UnsameQ@@#,stableQ[#]]&],{ptn,Rest[IntegerPartitions[n]]}],{n}];
Table[Length[nms[n]],{n,10}]

a(18)-a(21) from Robert Price, Sep 14 2018

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A331993 Number of semi-lone-child-avoiding rooted semi-identity trees with n unlabeled vertices.

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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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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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A316784 Number of orderless identity tree-factorizations of n.

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A316766 Number of series-reduced locally stable rooted identity trees whose leaves form an integer partition of n.

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