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 1-10 of 11 results. Next

A080320 a(n) = A057118(A080295(n)) = A080300(A080318(n)). Positions of A080318 in A014486.

Original entry on oeis.org

1, 8, 625, 82461, 2414517826, 465696894874, 19586243923520645, 911881322544255111111, 2344958374133795816706574529598, 540549352213084909964319555106232
Offset: 0

Views

Author

Antti Karttunen, Mar 02 2003

Keywords

Crossrefs

Cf. A080321.

A057117 Permutation of nonnegative integers obtained by mapping each forest of A000108[n] rooted binary plane trees from breadth-first to depth-first encoding.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 7, 8, 6, 9, 10, 12, 13, 11, 17, 18, 21, 22, 20, 14, 15, 16, 19, 23, 24, 26, 27, 25, 31, 32, 35, 36, 34, 28, 29, 30, 33, 45, 46, 49, 50, 48, 58, 59, 63, 64, 62, 54, 55, 57, 61, 37, 38, 40, 41, 39, 44, 47, 42, 43, 56, 60, 51, 52, 53, 65, 66, 68, 69, 67, 73, 74
Offset: 0

Views

Author

Antti Karttunen, Aug 11 2000

Keywords

Links

Crossrefs

Restriction of the automorphism A072088 to the plane binary trees.
Add one to each term and "overlay" each successive subpermutation of A000108[n] terms and one obtains A038776. Inverse permutation is A057118.

Programs

  • Maple
    a(n) = CatalanRankGlobal(btbf2df(binrev(A014486[n]),0,1)/2)
Maple procedure CatalanRank is adapted from the algorithm 3.23 of the CAGES book, see A014486
CatalanRank := proc(n,aa) local x,y,lo,a; a := binrev(aa); y := 0; lo := 0; for x from 1 to (2*n)-1 do lo := lo + (1-(a mod 2))*Mn(n,x,y+1); y := y - ((-1)^a); a := floor(a/2); od; RETURN((binomial(2*n,n)/(n+1))-(lo+1)); end;
CatalanRankGlobal := proc(a) local n; n := floor(binwidth(a)/2); RETURN(add((binomial(2*j,j)/(j+1)),j=0..(n-1))+CatalanRank(n,a)); end;

A083927 Inverse function of N -> N injection A057123.

Original entry on oeis.org

0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 5, 0
Offset: 0

Views

Author

Antti Karttunen, May 13 2003

Keywords

Comments

a(0)=0 because A057123(0)=0, but a(n) = 0 also for those n which do not occur as values of A057123. All positive natural numbers occur here once.
If g(n) = A083927(f(A057123(n))) then we can say that Catalan bijection g embeds into Catalan bijection f in scale n:2n, using the obvious binary tree -> general tree embedding. E.g. we have: A057163 = A083927(A057164(A057123(n))), A057117 = A083927(A072088(A057123(n))), A057118 = A083927(A072089(A057123(n))), A069770 = A083927(A072796(A057123(n))), A072797 = A083927(A072797(A057123(n))).

Links

Crossrefs

a(A057123(n)) = n for all n. Cf. A083925-A083926, A083928-A083929, A083935.

A082355 Permutation of natural numbers induced by Catalan Automorphism *A082355 acting on the parenthesizations encoded by A014486/A063171.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 8, 6, 7, 9, 10, 13, 11, 12, 19, 20, 21, 14, 15, 22, 18, 16, 17, 23, 24, 27, 25, 26, 33, 34, 35, 28, 29, 36, 32, 30, 31, 51, 52, 55, 53, 54, 56, 57, 58, 37, 38, 59, 41, 39, 40, 60, 61, 64, 47, 48, 62, 49, 42, 43, 63, 50, 46, 44, 45, 65, 66, 69, 67, 68
Offset: 0

Views

Author

Antti Karttunen, Apr 17 2003

Keywords

Comments

This bijection maps between the "standard" ordering of binary trees as encoded by A014486 and "variant B quaternary encoding" as explained in the sequence A085184.

Links

Crossrefs

Inverse of A082356. a(n) = A082357(A057163(n)). a(n) = A082363(A082853(n))+A082852(n). Cf. also A082351-A082352, A082357-A082358.
Differs from A057118 first time at n=42: a(42)=56, while A057118(42)=58.

A072089 Permutation of natural numbers induced by the automorphism gt-df->bf (switch from the Depth First to the Breadth First coding for the general trees/parenthesizations) acting on the parenthesizations encoded by A014486.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 19, 16, 17, 18, 15, 20, 21, 22, 23, 24, 25, 26, 27, 28, 33, 30, 31, 32, 29, 34, 35, 36, 37, 39, 51, 56, 52, 42, 53, 44, 45, 46, 43, 48, 49, 50, 38, 60, 47, 54, 61, 40, 57, 58, 59, 41, 55, 62, 63, 64, 65, 66, 67, 68, 69, 70, 75
Offset: 0

Views

Author

Antti Karttunen, Jun 25 2002

Keywords

Comments

When restricted to the subset of plane binary trees, produces the automorphism A057118 and being "self-embeddable", this allows us also to construct the permutation A072620 (A072622). [See comment at A072088.]

Links

Crossrefs

Inverse permutation: A072088. Cf. also A014486, A057118, A072620.

A070041 Permutation of natural numbers induced by the automorphism df->bf (from the depth-first to breadth-first encoding) acting on the rooted planar binary trees encoded by A014486. (with one-based indexing).

Original entry on oeis.org

1, 2, 5, 3, 4, 11, 12, 13, 6, 7, 14, 10, 8, 9, 29, 30, 33, 31, 32, 36, 37, 34, 15, 16, 35, 19, 17, 18, 40, 41, 42, 25, 26, 38, 27, 20, 21, 39, 28, 24, 22, 23, 85, 86, 89, 87, 88, 95, 96, 97, 90, 91, 98, 94, 92, 93, 104, 105, 112, 108, 109, 106, 110, 99, 43, 44, 100, 47, 45
Offset: 1

Views

Author

Antti Karttunen, Apr 16 2002

Keywords

Links

Crossrefs

Compare to the plot of A082363 and A072620.
Inverse of A038776. Cf. also A057118.

A080311 Orbit size of each tree encoded by A014486(n) under Meeussen's bf<->df map on binary trees.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 3, 3, 3, 1, 1, 3, 3, 3, 4, 3, 2, 4, 3, 4, 3, 2, 4, 1, 1, 3, 3, 3, 4, 3, 2, 4, 3, 4, 3, 2, 4, 16, 16, 8, 16, 8, 2, 2, 16, 16, 16, 8, 8, 8, 16, 16, 16, 16, 8, 8, 16, 8, 2, 2, 16, 16, 16, 16, 16, 1, 1, 3, 3, 3, 4, 3, 2, 4, 3, 4, 3, 2, 4, 16, 16, 8, 16, 8, 2, 2, 16, 16, 16, 8, 8, 8, 16
Offset: 0

Views

Author

Antti Karttunen, Mar 02 2003

Keywords

Comments

This is the size of the cycle containing n in the permutations A057117/A057118.

Crossrefs

Cf. A080967, A080273, A080274, A080297.

A080270 Positions of A080268 in A014486.

Original entry on oeis.org

1, 8, 62, 6213, 72665, 11517948
Offset: 0

Views

Author

Antti Karttunen, Mar 02 2003

Keywords

Crossrefs

a(n) = A080300(A080268(n)). Cf. A080271.

Formula

a(n) = A057118(A080265(n)).

A080273 Orbit size of each tree A080263(n) under Meeussen's bf<->df map on binary trees.

Original entry on oeis.org

1, 3, 16, 1441, 41888, 3376173
Offset: 0

Views

Author

Antti Karttunen, Mar 02 2003

Keywords

Comments

This is the size of the cycle containing A080265(n) in the permutations A057117/A057118.

Crossrefs

Cf. A080274, A080297.

Formula

a(n) = A080311(A080265(n)).

A080297 Orbit size of each tree A080293(n) under Meeussen's bf<->df map on binary trees.

Original entry on oeis.org

1, 3, 202, 41888
Offset: 0

Views

Author

Antti Karttunen, Mar 02 2003

Keywords

Comments

This is the size of the cycle containing A080295(n) in the permutations A057117/A057118.

Crossrefs

Cf. A080273, A080274.

Formula

a(n) = A080311(A080295(n)).
Showing 1-10 of 11 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.