A316697 -id:A316697 - cp's OEIS frontend

A331874 Number of semi-lone-child-avoiding locally disjoint rooted trees with n unlabeled leaves.

2, 3, 8, 24, 67, 214, 687, 2406, 8672, 32641, 125431, 493039, 1964611

Offset: 1

Gus Wiseman, Feb 02 2020

A rooted tree is semi-lone-child-avoiding if there are no vertices with exactly one child unless the child is an endpoint/leaf.

Locally disjoint means no child of any vertex has branches overlapping the branches of any other (inequivalent) child of the same vertex.

			The a(1) = 2 through a(4) = 24 trees:
  o    (oo)      (ooo)          (oooo)
  (o)  (o(o))    (o(oo))        (o(ooo))
       ((o)(o))  (oo(o))        (oo(oo))
                 (o(o)(o))      (ooo(o))
                 (o(o(o)))      ((oo)(oo))
                 ((o)(o)(o))    (o(o(oo)))
                 (o((o)(o)))    (o(oo(o)))
                 ((o)((o)(o)))  (oo(o)(o))
                                (oo(o(o)))
                                (o(o)(o)(o))
                                (o(o(o)(o)))
                                (o(o(o(o))))
                                (oo((o)(o)))
                                ((o)(o)(o)(o))
                                ((o(o))(o(o)))
                                ((oo)((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)(o))))
                                (((o)(o))((o)(o)))
                                ((o)((o)((o)(o))))

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.

Not requiring local disjointness gives A050381.
The non-semi version is A316697.
The same trees counted by number of vertices are A331872.
The Matula-Goebel numbers of these trees are A331873.
Lone-child-avoiding rooted trees counted  by leaves are A000669.
Semi-lone-child-avoiding rooted trees counted by vertices are A331934.
Cf. A000081, A001678, A300660, A316473, A316495, A316696, A331678, A331679, A331680, A331682, A331687, A331871, A331935.

disjointQ[u_]:=Apply[And,Outer[#1==#2||Intersection[#1,#2]=={}&,u,u,1],{0,1}];
slaurt[n_]:=If[n==1,{o,{o}},Join@@Table[Select[Union[Sort/@Tuples[slaurt/@ptn]],disjointQ[Select[#,!AtomQ[#]&]]&],{ptn,Rest[IntegerPartitions[n]]}]];
Table[Length[slaurt[n]],{n,8}]

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}]

A331871 Matula-Goebel numbers of lone-child-avoiding locally disjoint rooted trees.

Original entry on oeis.org

1, 4, 8, 14, 16, 28, 32, 38, 49, 56, 64, 76, 86, 98, 106, 112, 128, 152, 172, 196, 212, 214, 224, 256, 262, 304, 326, 343, 344, 361, 392, 424, 428, 448, 454, 512, 524, 526, 608, 622, 652, 686, 688, 722, 766, 784, 848, 856, 886, 896, 908, 1024, 1042, 1048, 1052

Offset: 1

Gus Wiseman, Feb 02 2020

nonn

First differs from A320269 in having 1589, the Matula-Goebel number of the tree ((oo)((oo)(oo))).
First differs from A331683 in having 49.
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.
Lone-child-avoiding means there are no unary branchings.
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.
Consists of one and all nonprime numbers whose distinct prime indices are pairwise coprime and already belong to the sequence, where a singleton is always considered to be pairwise coprime. A prime index of n is a number m such that prime(m) divides n.

			The sequence of all lone-child-avoiding locally disjoint rooted trees together with their Matula-Goebel numbers begins:
    1: o
    4: (oo)
    8: (ooo)
   14: (o(oo))
   16: (oooo)
   28: (oo(oo))
   32: (ooooo)
   38: (o(ooo))
   49: ((oo)(oo))
   56: (ooo(oo))
   64: (oooooo)
   76: (oo(ooo))
   86: (o(o(oo)))
   98: (o(oo)(oo))
  106: (o(oooo))
  112: (oooo(oo))
  128: (ooooooo)
  152: (ooo(ooo))
  172: (oo(o(oo)))
  196: (oo(oo)(oo))
The sequence of terms together with their prime indices begins:
     1: {}                  212: {1,1,16}
     4: {1,1}               214: {1,28}
     8: {1,1,1}             224: {1,1,1,1,1,4}
    14: {1,4}               256: {1,1,1,1,1,1,1,1}
    16: {1,1,1,1}           262: {1,32}
    28: {1,1,4}             304: {1,1,1,1,8}
    32: {1,1,1,1,1}         326: {1,38}
    38: {1,8}               343: {4,4,4}
    49: {4,4}               344: {1,1,1,14}
    56: {1,1,1,4}           361: {8,8}
    64: {1,1,1,1,1,1}       392: {1,1,1,4,4}
    76: {1,1,8}             424: {1,1,1,16}
    86: {1,14}              428: {1,1,28}
    98: {1,4,4}             448: {1,1,1,1,1,1,4}
   106: {1,16}              454: {1,49}
   112: {1,1,1,1,4}         512: {1,1,1,1,1,1,1,1,1}
   128: {1,1,1,1,1,1,1}     524: {1,1,32}
   152: {1,1,1,8}           526: {1,56}
   172: {1,1,14}            608: {1,1,1,1,1,8}
   196: {1,1,4,4}           622: {1,64}

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.

Not requiring local disjointness gives A291636.
Not requiring lone-child avoidance gives A316495.
A superset of A320269.
These trees are counted by A331680.
The semi-identity tree version is A331683.
The version containing 2 is A331873.
Cf. A001678, A007097, A050381, A061775, A196050, A302569, A302696, A316473, A316694, A316696, A316697, A331682, A331686, A331687, A331872, A331935.

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

Intersection of A291636 and A316495.

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}]

A319291 Number of series-reduced locally disjoint rooted trees with n leaves spanning an initial interval of positive integers.

Original entry on oeis.org

1, 2, 12, 107, 1299, 20764, 412957, 9817743

Offset: 1

Gus Wiseman, Sep 16 2018

			The a(3) = 12 series-reduced locally disjoint rooted trees:
  (1(11))
   (111)
  (1(22))
  (2(12))
   (122)
  (1(12))
  (2(11))
   (112)
  (1(23))
  (2(13))
  (3(12))
   (123)
The trees counted by A316651(4) but not by a(4):
  ((11)(12))
  ((12)(13))
  ((12)(22))
  ((12)(23))
  ((13)(23))

Cf. A000081, A007562, A301700, A316473, A316475, A316495, A316651, A316694, A316695, A316696, A316697, A319286.

disjointQ[u_]:=Apply[And,Outer[#1==#2||Intersection[#1,#2]=={}&,u,u,1],{0,1}];
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]]]];
gro[m_]:=gro[m]=If[Length[m]==1,{m},Select[Union[Sort/@Join@@(Tuples[gro/@#]&/@Select[mps[m],Length[#]>1&])],disjointQ]];
allnorm[n_Integer]:=Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1];
Table[Sum[Length[gro[m]],{m,allnorm[n]}],{n,5}]

cp's OEIS Frontend

A331874 Number of semi-lone-child-avoiding locally disjoint rooted trees with n unlabeled leaves.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A331871 Matula-Goebel numbers of lone-child-avoiding locally disjoint rooted trees.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A319291 Number of series-reduced locally disjoint rooted trees with n leaves spanning an initial interval of positive integers.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica