cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A324764 Number of anti-transitive rooted identity trees with n nodes.

Original entry on oeis.org

1, 1, 1, 1, 3, 4, 9, 20, 41, 89, 196, 443, 987, 2246, 5114, 11757, 27122, 62898, 146392, 342204, 802429, 1887882
Offset: 1

Views

Author

Gus Wiseman, Mar 17 2019

Keywords

Comments

A rooted identity tree is an unlabeled rooted tree with no repeated branches directly under the same root. It is anti-transitive if the branches of the branches of the root are disjoint from the branches of the root.
Also the number of finitary sets S with n brackets where no element of an element of S is also an element of S. For example, the a(8) = 20 finitary sets are (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}}}}}
{o,{{o,{{o}}}}}
{o,{{o},{{o}}}}
{{o},{{{{o}}}}}
{{o},{{o,{o}}}}
{{o},{o,{{o}}}}
{{{o}},{o,{o}}}

Examples

			The a(1) = 1 through a(7) = 9 anti-transitive 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)((o))))
                                                  (((o((o)))))
                                                  ((o)(((o))))
                                                  ((o(((o)))))
                                                  (o((((o)))))
                                                  ((((((o))))))

Links

Crossrefs

Cf. A000081, A004111, A276625, A279861, A290689, A304360, A306844 (non-identity version), A316500, A317787, A318185.
Cf. A324694, A324751, A324756, A324758, A324765, A324767, A324768, A324770, A324839, A324840, A324844.

Programs

  • Mathematica
    idall[n_]:=If[n==1,{{}},Select[Union[Sort/@Join@@(Tuples[idall/@#]&/@IntegerPartitions[n-1])],UnsameQ@@#&]];
Table[Length[Select[idall[n],Intersection[Union@@#,#]=={}&]],{n,10}]

Extensions

a(21)-a(22) from Jinyuan Wang, Jun 20 2020

A324838 Number of unlabeled rooted trees with n nodes where the branches of no branch of the root form a submultiset of the branches of the root.

Original entry on oeis.org

1, 0, 1, 2, 5, 10, 28, 64, 169, 422, 1108, 2872, 7627, 20202, 54216, 145867, 395288
Offset: 1

Views

Author

Gus Wiseman, Mar 18 2019

Keywords

Examples

			The a(1) = 1 through a(6) = 10 rooted trees:
  o  ((o))  ((oo))   ((ooo))    ((oooo))
            (((o)))  (((oo)))   (((ooo)))
                     ((o)(o))   ((o)(oo))
                     ((o(o)))   ((o(oo)))
                     ((((o))))  ((oo(o)))
                                ((((oo))))
                                (((o)(o)))
                                (((o(o))))
                                ((o((o))))
                                (((((o)))))

Crossrefs

Cf. A000081, A290689, A304360, A306844, A317787, A318185.
Cf. A324694, A324696, A324704, A324738, A324744, A324758, A324759, A324765, A324768, A324771, A324839, A324840, A324844, A324846.

Programs

  • Mathematica
    submultQ[cap_,fat_]:=And@@Function[i,Count[fat,i]>=Count[cap,i]]/@Union[List@@cap];
rtall[n_]:=Union[Sort/@Join@@(Tuples[rtall/@#]&/@IntegerPartitions[n-1])];
Table[Length[Select[rtall[n],And@@Table[!submultQ[b,#],{b,#}]&]],{n,10}]

A324767 Number of recursively anti-transitive rooted identity trees with n nodes.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 5, 9, 17, 33, 63, 126, 254, 511, 1039, 2124, 4371, 9059, 18839, 39339, 82385, 173111, 364829, 771010, 1633313
Offset: 1

Views

Author

Gus Wiseman, Mar 17 2019

Keywords

Comments

An unlabeled rooted tree is recursively anti-transitive if no branch of a branch of any terminal subtree is a branch of the same subtree. It is an identity tree if there are no repeated branches directly under a common root.
Also the number of finitary sets with n brackets where, at any level, no element of an element of a set is an element of the same set. For example, the a(8) = 9 finitary sets are (o = {}):
{{{{{{{o}}}}}}}
{{{{o,{{o}}}}}}
{{{o,{{{o}}}}}}
{{o,{{{{o}}}}}}
{{{o},{{{o}}}}}
{o,{{{{{o}}}}}}
{o,{{o,{{o}}}}}
{{o},{{{{o}}}}}
{{o},{o,{{o}}}}
The Matula-Goebel numbers of these trees are given by A324766.

Examples

			The a(4) = 1 through a(8) = 9 recursively anti-transitive 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)))))   (((o)(((o)))))
                                   ((((((o))))))  (((o(((o))))))
                                                  ((o)((((o)))))
                                                  ((o((((o))))))
                                                  (o(((((o))))))
                                                  (((((((o)))))))

Crossrefs

Cf. A000081, A004111, A276625, A279861, A290689, A290760, A304360, A306844, A316500.
Cf. A324695, A324751, A324758, A324764 (non-recursive version), A324765 (non-identity version), A324766, A324770, A324839, A324840, A324844.

Programs

  • Mathematica
    iallt[n_]:=Select[Union[Sort/@Join@@(Tuples[iallt/@#]&/@IntegerPartitions[n-1])],UnsameQ@@#&&Intersection[Union@@#,#]=={}&];
Table[Length[iallt[n]],{n,10}]

A324770 Number of fully anti-transitive rooted identity trees with n nodes.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 6, 13, 27, 58, 128, 286, 640, 1452, 3308, 7594, 17512, 40591, 94449, 220672
Offset: 1

Views

Author

Gus Wiseman, Mar 17 2019

Keywords

Comments

An unlabeled rooted tree is fully anti-transitive if no proper terminal subtree of any branch of the root is a branch of the root. It is an identity tree if there are no repeated branches directly under the same root.

Examples

			The a(1) = 1 through a(7) = 6 fully anti-transitive 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))))))

Links

Crossrefs

Cf. A000081, A004111, A276625, A279861, A290760, A304360, A306844.
Cf. A324695, A324751, A324763, A324764, A324765, A324767, A324769.
Cf. A324839, A324840, A324843, A324844, A324846.

Programs

  • Mathematica
    idall[n_]:=If[n==1,{{}},Select[Union[Sort/@Join@@(Tuples[idall/@#]&/@IntegerPartitions[n-1])],UnsameQ@@#&]];
Table[Length[Select[idall[n],Intersection[Union@@Rest[FixedPointList[Union@@#&,#]],#]=={}&]],{n,10}]
Showing 1-4 of 4 results.

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