A072088 -id:A072088 - cp's OEIS frontend

A072619 Permutation of natural numbers obtained from the permutation A072088 (zero-based).

0, 1, 2, 3, 4, 5, 10, 7, 8, 9, 6, 11, 12, 13, 14, 28, 15, 33, 37, 19, 24, 21, 22, 23, 30, 25, 26, 27, 16, 18, 20, 31, 38, 17, 34, 35, 36, 29, 32, 39, 40, 41, 42, 84, 43, 98, 112, 44, 46, 45, 103, 107, 85, 117, 121, 126, 56, 70, 89, 75, 79, 61, 66, 63, 64, 65, 72, 67, 68, 69

Offset: 0

Antti Karttunen, Jun 25 2002

See the comment at A072088.

A. Karttunen, Table of n, a(n) for n = 0..1429
A. Karttunen, Gatomorphisms (Includes the complete Scheme program for computing this sequence)
Index entries for sequences that are permutations of the natural numbers

Compare to the plot of A038776.

A072619(n) = A072621(n+1)-1. Inverse permutation: A072620. Cf. also A072088, A038776.

A072621 Permutation of natural numbers obtained from the permutation A072088 (one-based).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 11, 8, 9, 10, 7, 12, 13, 14, 15, 29, 16, 34, 38, 20, 25, 22, 23, 24, 31, 26, 27, 28, 17, 19, 21, 32, 39, 18, 35, 36, 37, 30, 33, 40, 41, 42, 43, 85, 44, 99, 113, 45, 47, 46, 104, 108, 86, 118, 122, 127, 57, 71, 90, 76, 80, 62, 67, 64, 65, 66, 73, 68, 69, 70

Offset: 1

Antti Karttunen, Jun 25 2002

nonn

See the comment at A072088.

A. Karttunen, Gatomorphisms (Includes the complete Scheme program for computing this sequence)
Index entries for sequences that are permutations of the natural numbers

A072621(n) = A072619(n-1)+1. Inverse permutation: A072622. Cf. also A038776.

A086588 Number of cycles in range [A014137(n-1)..A014138(n-1)] of permutations A072088/A072089.

Original entry on oeis.org

1, 1, 2, 5, 13, 33, 84, 204, 485, 1147, 2622, 5864, 12870

Offset: 0

Antti Karttunen, Jun 23 2003

A086589 Number of fixed points in range [A014137(n-1)..A014138(n-1)] of permutations A072088/A072089.

Original entry on oeis.org

1, 1, 2, 5, 12, 27, 60, 127, 264, 541, 1104, 2229, 4492

Offset: 0

Antti Karttunen, Jun 23 2003

A086590 Maximum cycle size in range [A014137(n-1)..A014138(n-1)] of permutations A072088/A072089.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 8, 12, 45, 78, 252, 500, 2458

Offset: 0

Antti Karttunen, Jun 23 2003

A086591 Least common multiple of cycle sizes in range [A014137(n-1)..A014138(n-1)] of permutations A072088/A072089.

Original entry on oeis.org

1, 1, 1, 1, 2, 6, 24, 2520, 18378360, 6906955255200, 5987137029832314231356979603166508647200, 1000156941827668764767034398758639477404906113721462819980559968765060862592344000

Offset: 0

Antti Karttunen, Jun 23 2003

nonn

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

Antti Karttunen, Aug 11 2000

nonn

A. Karttunen, Gatomorphisms (Includes the complete Scheme program for computing this sequence)
Index entries for sequences that are permutations of the natural numbers

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.

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

Antti Karttunen, May 13 2003

nonn

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))).

Antti Karttunen, Gatomorphisms

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

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

Antti Karttunen, Jun 25 2002

nonn

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.]

A. Karttunen, Gatomorphisms (Includes the complete Scheme program for computing this sequence)
Index entries for sequences that are permutations of the natural numbers

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

A072620 Permutation of natural numbers obtained from the permutation A072089 (zero-based).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 10, 7, 8, 9, 6, 11, 12, 13, 14, 16, 28, 33, 29, 19, 30, 21, 22, 23, 20, 25, 26, 27, 15, 37, 24, 31, 38, 17, 34, 35, 36, 18, 32, 39, 40, 41, 42, 44, 47, 49, 48, 84, 112, 98, 103, 99, 85, 107, 87, 88, 56, 70, 89, 100, 90, 61, 91, 63, 64, 65, 62, 67, 68, 69, 57

Offset: 0

Antti Karttunen, Jun 25 2002

nonn

See the comment at A072088.

A. Karttunen, Table of n, a(n) for n = 0..1429
A. Karttunen, Gatomorphisms (Includes the complete Scheme program for computing this sequence)
Index entries for sequences that are permutations of the natural numbers

Compare to the plot of A070041.

A072620(n) = A072622(n+1)-1. Inverse permutation: A072619. Cf. also A072089, A070041.

cp's OEIS Frontend

A072619 Permutation of natural numbers obtained from the permutation A072088 (zero-based).

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A072621 Permutation of natural numbers obtained from the permutation A072088 (one-based).

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A086588 Number of cycles in range [A014137(n-1)..A014138(n-1)] of permutations A072088/A072089.

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A086589 Number of fixed points in range [A014137(n-1)..A014138(n-1)] of permutations A072088/A072089.

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A086590 Maximum cycle size in range [A014137(n-1)..A014138(n-1)] of permutations A072088/A072089.

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A086591 Least common multiple of cycle sizes in range [A014137(n-1)..A014138(n-1)] of permutations A072088/A072089.

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

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Maple

A083927 Inverse function of N -> N injection A057123.

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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.

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A072620 Permutation of natural numbers obtained from the permutation A072089 (zero-based).

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