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 12 results. Next

A057502 Permutation of natural numbers: rotations of non-crossing handshakes encoded by A014486 (to opposite direction of A057501).

Original entry on oeis.org

0, 1, 3, 2, 7, 6, 8, 4, 5, 17, 16, 18, 14, 15, 20, 19, 21, 9, 10, 22, 11, 12, 13, 45, 44, 46, 42, 43, 48, 47, 49, 37, 38, 50, 39, 40, 41, 54, 53, 55, 51, 52, 57, 56, 58, 23, 24, 59, 25, 26, 27, 61, 60, 62, 28, 29, 63, 30, 31, 32, 64, 33, 34, 35, 36, 129, 128, 130, 126, 127
Offset: 0

Views

Author

Antti Karttunen, Sep 03 2000

Keywords

Comments

In A057501 and A057502, the cycles between (A014138(n-1)+1)-th and (A014138(n))-th term partition A000108(n) objects encoded by the corresponding terms of A014486 into A002995(n+1) equivalence classes of planar trees, thus the latter sequence can be produced also with Maple procedure RotHandshakesPermutationCycleCounts given below.

Links

Crossrefs

Inverse of A057501 and the car/cdr-flipped conjugate of A069774, i.e. A057502(n) = A057163(A069774(A057163(n))). Cf. also A057507, A057510, A057513, A069771, A069772.

Programs

  • Maple
    map(CatalanRankGlobal,map(RotateHandshakesR, A014486));
RotateHandshakesR := n -> pars2binexp(deepreverse(RotateHandshakesP(deepreverse(binexp2pars(n)))));
deepreverse := proc(a) if 0 = nops(a) or list <> whattype(a) then (a) else [op(deepreverse(cdr(a))), deepreverse(a[1])]; fi; end;
with(group); CountCycles := b -> (nops(convert(b,'disjcyc')) + (nops(b)-convert(map(nops,convert(b,'disjcyc')),`+`)));
RotHandshakesPermutationCycleCounts := proc(upto_n) local u,n,a,r,b; a := []; for n from 0 to upto_n do b := []; u := (binomial(2*n,n)/(n+1)); for r from 0 to u-1 do b := [op(b),1+CatalanRank(n,RotateHandshakes(CatalanUnrank(n,r)))]; od; a := [op(a),CountCycles(b)]; od; RETURN(a); end;
# For other procedures, follow A057501.

A057511 Permutation of natural numbers: rotations of all branches of the rooted plane trees encoded by A014486.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 5, 7, 8, 9, 11, 14, 16, 19, 10, 15, 12, 17, 20, 13, 18, 21, 22, 23, 25, 28, 30, 33, 37, 39, 42, 44, 53, 51, 47, 56, 60, 24, 29, 38, 43, 52, 26, 40, 31, 45, 48, 34, 54, 57, 61, 27, 41, 32, 46, 55, 35, 49, 58, 62, 36, 50, 59, 63, 64, 65, 67, 70, 72, 75, 79, 81
Offset: 0

Views

Author

Antti Karttunen, Sep 03 2000

Keywords

Links

Crossrefs

Inverse permutation: A057512. Cycle counts: A057513. Number of fixed objects: A057546. Max. cycle lengths are given by Landau's function A000793.

Programs

  • Maple
    # See A057509 for rotateL, A057501 for other procedures.
map(CatalanRankGlobal,map(DeepRotateL, A014486));
DeepRotateL := n -> pars2binexp(deeprotateL(binexp2pars(n)));
deeprotateL := proc(a) if 0 = nops(a) or list <> whattype(a) then (a) else rotateL(map(deeprotateL,a)); fi; end;

A057512 Permutation of natural numbers: rotations of all branches of the rooted plane trees encoded by A014486. (to opposite direction of A057511).

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 5, 7, 8, 9, 14, 10, 16, 19, 11, 15, 12, 17, 20, 13, 18, 21, 22, 23, 37, 24, 42, 51, 25, 38, 26, 44, 53, 27, 47, 56, 60, 28, 39, 29, 43, 52, 30, 40, 31, 45, 54, 34, 46, 57, 61, 33, 41, 32, 48, 55, 35, 49, 58, 62, 36, 50, 59, 63, 64, 65, 107, 66, 121, 149, 67
Offset: 0

Views

Author

Antti Karttunen, Sep 03 2000

Keywords

Links

Crossrefs

Inverse permutation: A057511. Cycle counts: A057513.

Programs

  • Maple
    # See A057510 for rotateR, A057501 for other procedures.
map(CatalanRankGlobal,map(DeepRotateR, A014486));
DeepRotateR := n -> pars2binexp(deeprotateR(binexp2pars(n)));
deeprotateR := proc(a) if 0 = nops(a) or list <> whattype(a) then (a) else rotateR(map(deeprotateR,a)); fi; end;

A057546 Number of Catalan objects of size n fixed by Catalan Automorphism A057511/A057512 (deep rotation of general parenthesizations/plane trees).

Original entry on oeis.org

1, 1, 2, 3, 5, 6, 10, 11, 18, 21, 34, 35, 68, 69, 137, 148, 316, 317, 759, 760, 1869, 1915, 4833, 4834, 12796, 12802, 34108, 34384, 92792, 92793, 254752, 254753, 703083, 704956, 1958210, 1958231, 5485330, 5485331, 15427026, 15440591, 43618394, 43618395, 123807695, 123807696, 352561832, 352664217, 1007481494, 1007481495, 2887387009
Offset: 0

Views

Author

Antti Karttunen, Sep 07 2000

Keywords

Comments

Greater than A003238 because there exists also parenthesizations like ((() (())) ((()) ())) and (((()) ()) (() (()))) which are fixed by recursive deep rotation, corresponding to Catalan mountain ranges below:
...../\..../\............................./\......../\
../\/\../\/\.....and.its."dual"....../\/\../\/\
./____\/____\......................./____\/____\
It's obvious that a(p) = a(p-1)+1 for all primes p.

Links

Crossrefs

The first row of A079216. The leftmost edge of the triangle A079217 and also its row sums shifted by one. Occurs for first time in A073202 as row 12. Cf. A057513, A079223-A079227, A034731, A003238.

Programs

  • Maple
    with(numtheory,divisors); A057546 := proc(n) local d; if(0=n) then RETURN(1); else RETURN(add(A079216bi(d-1,n/d),d=divisors(n))); fi; end;

Formula

a(0)=1, a(n) = A079216(n, 1) = Sum_{d|n} A079216(d-1, n/d). - Antti Karttunen, Jan 03 2003

A057510 Permutation of natural numbers: rotations of the bottom branches of the rooted plane trees encoded by A014486. (to opposite direction of A057509).

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 5, 7, 8, 9, 14, 10, 16, 19, 11, 15, 12, 17, 18, 13, 20, 21, 22, 23, 37, 24, 42, 51, 25, 38, 26, 44, 47, 27, 53, 56, 60, 28, 39, 29, 43, 52, 30, 40, 31, 45, 46, 32, 48, 49, 50, 33, 41, 34, 54, 55, 35, 57, 58, 59, 36, 61, 62, 63, 64, 65, 107, 66, 121, 149, 67
Offset: 0

Views

Author

Antti Karttunen, Sep 03 2000

Keywords

Links

Crossrefs

Inverse of A057509 and the car/cdr-flipped conjugate of A069776 and also composition of A057502 & A069770, i.e. A057510(n) = A057163(A069776(A057163(n))) = A069770(A057502(n)).
Cycle counts given by A003239. Cf. also A057512, A057513.

Programs

  • Maple
    # reverse given in A057508, for CountCycles, see A057502, for other procedures, follow A057501.
map(CatalanRankGlobal,map(RotateBottomBranchesR, A014486));
RotateBottomBranchesR := n -> pars2binexp(rotateR(binexp2pars(n)));
rotateR := a -> reverse(rotateL(reverse(a)));
RotBBPermutationCycleCounts := proc(upto_n) local u,n,a,r,b; a := []; for n from 0 to upto_n do b := []; u := (binomial(2*n,n)/(n+1)); for r from 0 to u-1 do b := [op(b),1+CatalanRank(n,RotateBottomBranchesL(CatalanUnrank(n,r)))]; od; a := [op(a),CountCycles(b)]; od; RETURN(a); end;
A003239 := RotBBPermutationCycleCounts(some_value); (e.g. 9. Cf. A057502, A057162)

A073201 Array of cycle count sequences for the table A073200.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 3, 2, 1, 1, 7, 4, 1, 1, 1, 22, 11, 3, 1, 1, 1, 66, 31, 7, 2, 1, 1, 1, 217, 96, 22, 4, 3, 1, 1, 1, 715, 305, 66, 11, 7, 2, 1, 1, 1, 2438, 1007, 217, 30, 22, 4, 2, 2, 1, 1, 8398, 3389, 715, 93, 66, 11, 3, 5, 1, 1, 1, 29414, 11636, 2438, 292, 217, 30, 6, 14, 2, 2, 1, 1
Offset: 0

Views

Author

Antti Karttunen, Jun 25 2002

Keywords

Comments

Each row of this table gives the counts of separate orbits/cycles to which the Catalan bijection given in the corresponding row of A073200 partitions each A000108(n) structures encoded in the range [A014137(n-1)..A014138(n-1)] of the sequence A014486/A063171.
Note that for involutions (self-inverse Catalan bijections) this is always (A000108(n)+Affffff(n))/2, where Affffff is the corresponding "fix-count sequence" from the table A073202.

Links

  • A. Karttunen, Gatomorphisms (With the complete source and explanation)

Crossrefs

Only the first known occurrence(s) given (marked with ? if not yet proved/unclear): rows 0, 2, 4, etc.: A007595, Row 1: A073191, Rows 6 (& 8): A073431, Row 7: A000108, Rows 12, 14, 20, ...: A057513, Rows 16, 18, ...: A003239, Row 57, ..., 164: A007123, Row 168: A073193, Row 261: A002995, Row 2614: A057507, Row 2618 (?), row 17517: A001683.

A082325 Permutation of natural numbers: A057163-conjugate of A057511.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Apr 17 2003

Keywords

Links

Crossrefs

Inverse of A082326. a(n) = A069787(A082326(A069787(n))). a(n) = A082327(A082853(n))+A082852(n). Occurs in A073200 as row 1792. Cf. also A082337-A082338.
Differs from A082342 first time at n=39: a(39)=49, while A082342(39)=48.
Number of cycles: A057513. Number of fixed-points: A057546. Max. cycle size: A000793. LCM of cycle sizes: A003418. (In range [A014137(n-1)..A014138(n-1)] of this permutation, possibly shifted one term left or right).

Formula

a(n) = A057163(A057511(A057163(n)))

A082326 Permutation of natural numbers: A057163-conjugate of A057512.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Apr 17 2003

Keywords

Links

Crossrefs

Inverse of A082325. a(n) = A069787(A082325(A069787(n))). a(n) = A082328(A082853(n))+A082852(n). Occurs in A073200 as row 1794. Cf. also A082337-A082338.
Differs from A082341 first time at n=39: a(39)=49, while A082341(39)=48.
Number of cycles: A057513. Number of fixed-points: A057546. Max. cycle size: A000793. LCM of cycle sizes: A003418. (In range [A014137(n-1)..A014138(n-1)] of this permutation, possibly shifted one term left or right).

Formula

a(n) = A057163(A057512(A057163(n)))

A082341 Permutation of natural numbers induced by the Catalan bijection gma082341 acting on the parenthesizations encoded by A014486/A063171.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Apr 17 2003

Keywords

Comments

This is A057163-conjugate of A073285.

Links

Crossrefs

Inverse of A082342. a(n) = A057163(A073285(A057163(n))). Occurs in A073200 as row 1800. Cf. also A072797, A082337-A082339.
Differs from A082326 first time at n=39: a(39)=48, while A082326(39)=49.
Number of cycles: A057513. Number of fixed-points: A057546. Max. cycle size: A000793. LCM of cycle sizes: A003418. (In range [A014137(n-1)..A014138(n-1)] of this permutation, possibly shifted one term left or right).

A038775 a(n) is the number of cycles of the permutation that converts forest(n) of depth-first planar planted binary trees into breadth-first representation.

Original entry on oeis.org

1, 2, 3, 6, 10, 12, 17, 26, 34, 50, 56, 68, 82, 94, 113
Offset: 1

Views

Author

Wouter Meeussen, May 04 2000

Keywords

Comments

The first a(n) terms of A038774 add up to Catalan(n) = A000108(n).

Examples

			a(5)=10 since there are 10 cycles in this permutation of forest(5), with lengths 1, 1, 3, 4, 3, 2, 16, 8, 2, 2 summing up to 42=Catalan(5).

Links

Crossrefs

Cf. A000108, A038774.
Similarly generated sequences: A001683, A002995, A003239, A057507, A057513.

Extensions

a(13)-a(15) from Sean A. Irvine, May 22 2022
Showing 1-10 of 12 results. Next

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