A316471 -id:A316471 - cp's OEIS frontend

A316494 Matula-Goebel numbers of locally disjoint rooted identity trees, meaning no branch overlaps any other branch of the same root.

1, 2, 3, 5, 6, 10, 11, 13, 15, 22, 26, 29, 30, 31, 33, 41, 47, 55, 58, 62, 66, 79, 82, 93, 94, 101, 109, 110, 113, 123, 127, 137, 141, 143, 145, 155, 158, 165, 179, 186, 202, 205, 211, 218, 226, 246, 254, 257, 271, 274, 282, 286, 290, 293, 310, 317, 327, 330

Offset: 1

Gus Wiseman, Jul 04 2018

nonn

A prime index of n is a number m such that prime(m) divides n. A number is in the sequence iff either it is equal to 1, it is a prime number whose prime index already belongs to the sequence, or its prime indices are pairwise coprime, distinct, and already belong to the sequence.

			The sequence of all locally disjoint rooted identity trees preceded by their Matula-Goebel numbers begins:
   1: o
   2: (o)
   3: ((o))
   5: (((o)))
   6: (o(o))
  10: (o((o)))
  11: ((((o))))
  13: ((o(o)))
  15: ((o)((o)))
  22: (o(((o))))
  26: (o(o(o)))
  29: ((o((o))))
  30: (o(o)((o)))
  31: (((((o)))))

Cf. A000081, A004111, A007097, A276625, A277098, A302696, A303362, A304713, A316467, A316471, A316474, A316495.

primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[1000],Or[#==1,And[SquareFreeQ[#],Or[PrimeQ[#],CoprimeQ@@primeMS[#]],And@@#0/@primeMS[#]]]&]

A331678 Number of lone-child-avoiding locally disjoint rooted trees whose leaves are integer partitions whose multiset union is an integer partition of n.

Original entry on oeis.org

1, 3, 6, 18, 44, 149, 450, 1573, 5352, 19283, 69483, 257206

Offset: 1

Gus Wiseman, Jan 25 2020

Lone-child-avoiding means there are no unary branchings. Locally disjoint means no child of any vertex has branches overlapping the branches of any other unequal child of the same vertex.

			The a(1) = 1 through a(4) = 18 trees:
  (1)  (2)       (3)            (4)
       (11)      (12)           (13)
       ((1)(1))  (111)          (22)
                 ((1)(2))       (112)
                 ((1)(1)(1))    (1111)
                 ((1)((1)(1)))  ((1)(3))
                                ((2)(2))
                                ((2)(11))
                                ((11)(11))
                                ((1)(1)(2))
                                ((1)((1)(2)))
                                ((2)((1)(1)))
                                ((1)(1)(1)(1))
                                ((11)((1)(1)))
                                ((1)((1)(1)(1)))
                                ((1)(1)((1)(1)))
                                (((1)(1))((1)(1)))
                                ((1)((1)((1)(1))))

David Callan, A sign-reversing involution to count labeled lone-child-avoiding trees, arXiv:1406.7784 [math.CO], (30-June-2014).
Gus Wiseman, Sequences counting series-reduced and lone-child-avoiding trees by number of vertices.

The case where all leaves are singletons is A316696.
The case where all leaves are (1) is A316697.
The non-locally disjoint version is A319312.
The case with all atoms equal to 1 is A331679.
The identity tree case is A331686.
Cf. A000081, A000669, A001678, A005804, A060356, A141268, A196545, A300660, A316471, A316694, A316495, A330465, A331680, A331687.

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]]]];
disjointQ[u_]:=Apply[And,Outer[#1==#2||Intersection[#1,#2]=={}&,u,u,1],{0,1}];
mpti[m_]:=Prepend[Join@@Table[Select[Union[Sort/@Tuples[mpti/@p]],disjointQ],{p,Select[mps[m],Length[#]>1&]}],m];
Table[Sum[Length[mpti[m]],{m,Sort/@IntegerPartitions[n]}],{n,8}]

A331687 Number of locally disjoint enriched p-trees of weight n.

Original entry on oeis.org

1, 2, 4, 12, 29, 93, 249, 803, 2337, 7480, 23130, 77372, 247598, 834507, 2762222

Offset: 1

Gus Wiseman, Jan 31 2020

A locally disjoint enriched p-tree of weight n is either the number n itself or a finite sequence of non-overlapping locally disjoint enriched p-trees whose weights are weakly decreasing and sum to n.

			The a(1) = 1 through a(4) = 12 enriched p-trees:
  1  2     3        4
     (11)  (21)     (22)
           (111)    (31)
           ((11)1)  (211)
                    (1111)
                    ((11)2)
                    ((21)1)
                    (2(11))
                    ((11)11)
                    ((111)1)
                    (((11)1)1)
                    ((11)(11))

The orderless version is A316696.
The identity case is A331684.
P-trees are A196545.
Enriched p-trees are A289501.
Locally disjoint identity trees are A316471.
Enriched identity p-trees are A331875.
Cf. A000669, A141268, A316473, A316495, A316694, A316697, A319312, A331678, A331679, A331680, A331686, A331871, A331874.

disjointQ[u_]:=Apply[And,Outer[#1==#2||Intersection[#1,#2]=={}&,u,u,1],{0,1}];
ldep[n_]:=Prepend[Select[Join@@Table[Tuples[ldep/@p],{p,Rest[IntegerPartitions[n]]}],disjointQ[DeleteCases[#,_Integer]]&],n];
Table[Length[ldep[n]],{n,10}]

A331783 Number of locally disjoint rooted semi-identity trees with n unlabeled vertices.

Original entry on oeis.org

1, 1, 2, 4, 8, 17, 37, 83, 191, 450, 1076, 2610, 6404, 15875, 39676, 99880, 253016, 644524, 1649918, 4242226

Offset: 1

Gus Wiseman, Jan 31 2020

Locally disjoint means no branch of any vertex overlaps a different (unequal) branch of the same vertex. In a semi-identity tree, all non-leaf branches of any given vertex are distinct.

			The a(1) = 1 through a(6) = 17 trees:
  o  (o)  (oo)   (ooo)    (oooo)     (ooooo)
          ((o))  ((oo))   ((ooo))    ((oooo))
                 (o(o))   (o(oo))    (o(ooo))
                 (((o)))  (oo(o))    (oo(oo))
                          (((oo)))   (ooo(o))
                          ((o(o)))   (((ooo)))
                          (o((o)))   ((o(oo)))
                          ((((o))))  ((oo(o)))
                                     (o((oo)))
                                     (o(o(o)))
                                     (oo((o)))
                                     ((((oo))))
                                     (((o(o))))
                                     ((o)((o)))
                                     ((o((o))))
                                     (o(((o))))
                                     (((((o)))))

The lone-child-avoiding case is A212804.
The identity tree version is A316471.
The Matula-Goebel numbers of these trees are given by A331682.
Identity trees are A004111.
Semi-identity trees are A306200.
Locally disjoint rooted trees are A316473.
Matula-Goebel numbers of locally disjoint semi-identity trees are A316494.
Cf. A000081, A306202, A316475, A316495, A316694, A331683, A331684, A331686.

disjunsQ[u_]:=Length[u]==1||UnsameQ@@DeleteCases[u,{}]&&Apply[And,Outer[#1==#2||Intersection[#1,#2]=={}&,u,u,1],{0,1}];
ldrsi[n_]:=If[n==1,{{}},Select[Join@@Function[c,Union[Sort/@Tuples[ldrsi/@c]]]/@IntegerPartitions[n-1],disjunsQ]];
Table[Length[ldrsi[n]],{n,10}]

A316500 Number of unlabeled rooted identity trees with n nodes in which the branches of any node with more than one branch have empty intersection.

Original entry on oeis.org

1, 1, 1, 2, 3, 6, 11, 22, 46, 96, 205, 442, 976, 2146, 4789, 10719, 24202, 54841, 124967, 285724, 656011, 1510929, 3491151, 8088692, 18790084

Offset: 1

Gus Wiseman, Jul 05 2018

			The a(7) = 11 rooted identity trees:
  ((((((o))))))
  ((((o(o)))))
  (((o((o)))))
  ((o(((o)))))
  ((o(o(o))))
  (((o)((o))))
  (o((((o)))))
  (o((o(o))))
  (o(o((o))))
  ((o)(((o))))
  (o(o)((o)))

Cf. A000081, A004111, A007562, A078374, A289509, A316469, A316471, A316474, A316501, A316503.

strut[n_]:=strut[n]=If[n===1,{{}},Select[Join@@Function[c,Union[Sort/@Tuples[strut/@c]]]/@IntegerPartitions[n-1],UnsameQ@@#&&Or[Length[#]==1,Intersection@@#=={}]&]];
Table[Length[strut[n]],{n,20}]

A331937 a(1) = 1; a(2) = 2; a(n + 1) = 2 * prime(a(n)).

Original entry on oeis.org

1, 2, 6, 26, 202, 2462, 43954, 1063462, 33076174, 1270908802, 58596709306, 3170266564862, 197764800466826, 14024066291995502, 1117378164606478094

Offset: 1

Gus Wiseman, Feb 07 2020

Also Matula-Goebel numbers of semi-lone-child-avoiding locally disjoint rooted identity trees. A rooted tree is locally disjoint if no child of any vertex has branches overlapping the branches of any other (inequivalent) child of the same vertex. It is semi-lone-child-avoiding if there are no vertices with exactly one child unless that child is an endpoint/leaf. In an identity tree, the branches of any given vertex are all distinct. The Matula-Goebel number of a rooted tree is the product of primes indexed by the Matula-Goebel numbers of the branches of its root, which gives a bijective correspondence between positive integers and unlabeled rooted trees.

			The sequence of terms together with their associated trees begins:
     1: o
     2: (o)
     6: (o(o))
    26: (o(o(o)))
   202: (o(o(o(o))))
  2462: (o(o(o(o(o)))))

The semi-identity tree version is A331681.
Not requiring an identity tree gives A331873.
Not requiring local disjointness gives A331963.
Not requiring lone-child-avoidance gives A316494.
MG-numbers of semi-lone-child-avoiding rooted trees are A331935.
Cf. A007097, A061775, A276625, A316471, A316495, A316694, A331679, A331683, A331686, A331872, A331934, A331936, A331964, A331965.

msiQ[n_]:=n==1||n==2||!PrimeQ[n]&&SquareFreeQ[n]&&(PrimePowerQ[n]||CoprimeQ@@PrimePi/@First/@FactorInteger[n])&&And@@msiQ/@PrimePi/@First/@FactorInteger[n];
Select[Range[1000],msiQ]

Intersection of A276625 (identity), A316495 (locally disjoint), and A331935 (semi-lone-child-avoiding).

a(14)-a(15) from Giovanni Resta, Feb 10 2020

A331684 Number of locally disjoint enriched identity p-trees of weight n.

Original entry on oeis.org

1, 1, 2, 3, 6, 14, 30, 68, 157, 379, 901, 2229, 5488, 13846, 34801, 89368, 228186, 592943, 1533511, 4026833

Offset: 1

Gus Wiseman, Jan 31 2020

A locally disjoint enriched identity p-tree of weight n is either the number n itself or a finite sequence of distinct non-overlapping locally disjoint enriched identity p-trees whose weights are weakly decreasing and sum to n.

			The a(1) = 1 through a(6) = 14 enriched p-trees:
  1  2  3     4        5           6
        (21)  (31)     (32)        (42)
              ((21)1)  (41)        (51)
                       ((21)2)     (321)
                       ((31)1)     ((21)3)
                       (((21)1)1)  ((31)2)
                                   ((32)1)
                                   (3(21))
                                   ((41)1)
                                   ((21)21)
                                   (((21)1)2)
                                   (((21)2)1)
                                   (((31)1)1)
                                   ((((21)1)1)1)

The orderless version is A316694.
The non-identity version is A331687.
Identity trees are A004111.
P-trees are A196545.
Enriched p-trees are A289501.
Locally disjoint identity trees are A316471.
Enriched identity p-trees are A331875, with locally disjoint case A331687.
Cf. A000669, A005804, A141268, A300660, A316696, A316697, A331678, A331679, A331680, A331683, A331686, A331783, A331874.

disjointQ[u_]:=Apply[And,Outer[#1==#2||Intersection[#1,#2]=={}&,u,u,1],{0,1}];
ldeip[n_]:=Prepend[Select[Join@@Table[Tuples[ldeip/@p],{p,Rest[IntegerPartitions[n]]}],UnsameQ@@#&&disjointQ[DeleteCases[#,_Integer]]&],n];
Table[Length[ldeip[n]],{n,12}]

A316469 Matula-Goebel numbers of unlabeled rooted identity RPMG-trees, meaning the Matula-Goebel numbers of the branches of any non-leaf node are relatively prime.

Original entry on oeis.org

1, 2, 6, 26, 78, 202, 606, 794, 2382, 2462, 2626, 7386, 7878, 8914, 10322, 12178, 26742, 30966, 32006, 36534, 42374, 43954, 47206, 80194, 96018, 115882, 127122, 131862, 141618, 149782, 158314, 160978, 184622, 217058, 240582, 248662, 260422, 347646, 449346

Offset: 1

Gus Wiseman, Jul 04 2018

nonn

A prime index of n is a number m such that prime(m) divides n. A number is in the sequence iff it is 1 or its prime indices are distinct, relatively prime, and already belong to the sequence.

			78 = prime(1)*prime(2)*prime(6) belongs to the sequence because the indices {1,2,6} are relatively prime, distinct, and already belong to the sequence.
The sequence of all identity RPMG-trees preceded by their Matula-Goebel numbers begins:
     1: o
     2: (o)
     6: (o(o))
    26: (o(o(o)))
    78: (o(o)(o(o)))
   202: (o(o(o(o))))
   606: (o(o)(o(o(o))))
   794: (o(o(o)(o(o))))
  2382: (o(o)(o(o)(o(o))))
  2462: (o(o(o(o(o)))))
  2626: (o(o(o))(o(o(o))))
  7386: (o(o)(o(o(o(o)))))
  7878: (o(o)(o(o))(o(o(o))))

Cf. A000081, A000837, A004111, A007097, A078374, A276625, A289509, A302696, A302796, A316467, A316470, A316471, A316474, A316494.

primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[1000],Or[#==1,And[SquareFreeQ[#],GCD@@primeMS[#]==1,And@@#0/@primeMS[#]]]&]

A358460 Number of locally disjoint ordered rooted trees with n nodes.

Original entry on oeis.org

1, 1, 2, 5, 13, 36, 103, 301, 902, 2767, 8637, 27324, 87409, 282319, 919352

Offset: 1

Gus Wiseman, Nov 19 2022

Locally disjoint means no branch of any vertex overlaps a different (unequal) branch of the same vertex.

			The a(1) = 1 through a(5) = 13 trees:
  o  (o)  (oo)   (ooo)    (oooo)
          ((o))  ((o)o)   ((o)oo)
                 ((oo))   ((oo)o)
                 (o(o))   ((ooo))
                 (((o)))  (o(o)o)
                          (o(oo))
                          (oo(o))
                          (((o))o)
                          (((o)o))
                          (((oo)))
                          ((o(o)))
                          (o((o)))
                          ((((o))))

The locally non-intersecting version is A143363, unordered A007562.
The unordered version is A316473, ranked by A316495.
A000108 counts ordered rooted trees, unordered A000081.
A358453 counts transitive ordered trees, unordered A290689.
Cf. A006013, A302696, A316471, A316694, A318185, A319378, A324768, A324844.

aot[n_]:=If[n==1,{{}},Join @@ Table[Tuples[aot/@c],{c,Join@@Permutations/@IntegerPartitions[n-1]}]];
Table[Length[Select[aot[n],FreeQ[#,{_,{_,x_,_},_,{_,x_,_},_}]&]],{n,10}]

A316695 Number of series-reduced locally disjoint rooted trees whose leaves form the integer partition with Heinz number n.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 5, 1, 3, 1, 3, 1, 1, 1, 8, 1, 1, 2, 3, 1, 4, 1, 10, 1, 1, 1, 12, 1, 1, 1, 8, 1, 4, 1, 3, 3, 1, 1, 23, 1, 3, 1, 3, 1, 8, 1, 8, 1, 1, 1, 16, 1, 1, 3, 24, 1, 4, 1, 3, 1, 4, 1, 37, 1, 1, 3, 3, 1, 4, 1, 23, 5, 1, 1, 16

Offset: 1

Gus Wiseman, Jul 10 2018

nonn

A rooted tree is series-reduced 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.

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

			The a(24) = 8 trees:
  (1(1(12)))
  (1(2(11)))
  (2(1(11)))
  (1(112))
  (2(111))
  (11(12))
  (12(11))
  (1112)

Cf. A000081, A000669, A001678, A056239, A141268, A292504, A296150, A316471, A316651, A316652, 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]]]];
disjointQ[u_]:=Apply[And,Outer[#1==#2||Intersection[#1,#2]=={}&,u,u,1],{0,1}];
gro[m_]:=gro[m]=If[Length[m]==1,List/@m,Union[Sort/@Join@@(Tuples[gro/@#]&/@Select[mps[m],Length[#]>1&])]];
Table[Length[Select[gro[If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]],And@@Cases[#,q:{__List}:>disjointQ[q],{0,Infinity}]&]],{n,100}]

cp's OEIS Frontend

A316494 Matula-Goebel numbers of locally disjoint rooted identity trees, meaning no branch overlaps any other branch of the same root.

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A331678 Number of lone-child-avoiding locally disjoint rooted trees whose leaves are integer partitions whose multiset union is an integer partition of n.

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A331687 Number of locally disjoint enriched p-trees of weight n.

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A331783 Number of locally disjoint rooted semi-identity trees with n unlabeled vertices.

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A316500 Number of unlabeled rooted identity trees with n nodes in which the branches of any node with more than one branch have empty intersection.

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A331937 a(1) = 1; a(2) = 2; a(n + 1) = 2 * prime(a(n)).

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A331684 Number of locally disjoint enriched identity p-trees of weight n.

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A316469 Matula-Goebel numbers of unlabeled rooted identity RPMG-trees, meaning the Matula-Goebel numbers of the branches of any non-leaf node are relatively prime.

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A358460 Number of locally disjoint ordered rooted trees with n nodes.

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