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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.

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A074684 Permutation of natural numbers induced by Catalan Automorphism *A074684 acting on the parenthesizations encoded by A014486/A063171.

Original entry on oeis.org

0, 1, 3, 2, 8, 7, 5, 4, 6, 22, 21, 18, 17, 20, 13, 12, 10, 9, 11, 15, 14, 19, 16, 64, 63, 59, 58, 62, 50, 49, 46, 45, 48, 55, 54, 61, 57, 36, 35, 32, 31, 34, 27, 26, 24, 23, 25, 29, 28, 33, 30, 41, 40, 38, 37, 39, 52, 51, 60, 56, 43, 42, 47, 44, 53, 196, 195, 190, 189, 194
Offset: 0

Views

Author

Antti Karttunen, Sep 11 2002

Keywords

Comments

This bijection maps between the "standard" ordering of binary trees as encoded by A014486 and "variant A quaternary encoding" as explained in the sequence A085184.
This is a rare example of a simply defined Catalan Automorphism where the cycle count sequence (A089411) is not monotone. (See A127296 for a much more complex example.)

Links

Crossrefs

Row 17 of A122201. Inverse of A074683. a(n) = A057163(A074681(A057163(n))).
The number of cycles, maximum cycle sizes and LCM's of all cycle sizes in subpermutations limited by A014137 and A014138 are given by A089411, A086586 and A089412.

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)

A089854 Involution of natural numbers induced by Catalan automorphism *A089854 acting on the binary trees/parenthesizations encoded by A014486/A063171.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 8, 7, 9, 10, 11, 12, 13, 14, 15, 19, 21, 22, 16, 20, 17, 18, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 51, 52, 56, 58, 59, 60, 62, 63, 64, 42, 43, 53, 57, 61, 44, 54, 45, 46, 47, 55, 48, 49, 50, 65, 66, 67, 68, 69, 70, 71, 72
Offset: 0

Views

Author

Antti Karttunen, Nov 29 2003

Keywords

Comments

This automorphism effects the following transformation on the unlabeled rooted plane binary trees (letters A, B, C refer to arbitrary subtrees located on those nodes and () stands for an implied terminal node).
.A...B...........B...A
..\./.............\./
...x...C....-->....x...C...............()..A.........()..A..
....\./.............\./.................\./....-->....\./...
.....x...............x...................x.............x....
((a . b) . c) -> ((b . a) . c) ___ (() . a) ---> (() . a)
In terms of S-expressions, this automorphism swaps caar and cdar of an S-exp if its first element is not ().
See the Karttunen OEIS-Wiki link for a detailed explanation of how to obtain a given integer sequence from this definition.

Links

Crossrefs

a(n) = A089859(A069770(n)) = A069770(A089863(n)) = A057163(A089850(A057163(n))). Row 7 of A089840. Cf. A122282.
Number of cycles: A073191. Number of fixed points: A073190. Max. cycle size & LCM of all cycle sizes: A046698 (in each range limited by A014137 and A014138).

Extensions

Further comments and constructive implementation of Scheme-function (*A089854) added by Antti Karttunen, Jun 04 2011

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.

A089853 Permutation of natural numbers induced by Catalan automorphism *A089853 acting on the binary trees/parenthesizations encoded by A014486/A063171.

Original entry on oeis.org

0, 1, 2, 3, 6, 4, 5, 7, 8, 16, 19, 14, 9, 10, 15, 11, 12, 17, 18, 13, 20, 21, 22, 44, 47, 53, 56, 60, 42, 51, 37, 23, 24, 38, 25, 26, 27, 43, 52, 39, 28, 29, 40, 30, 31, 45, 46, 32, 48, 49, 50, 41, 33, 34, 54, 55, 35, 57, 58, 59, 36, 61, 62, 63, 64, 128, 131, 137, 140, 144
Offset: 0

Views

Author

Antti Karttunen, Nov 29 2003

Keywords

Comments

This automorphism effects the following transformation on the unlabeled rooted plane binary trees (letters A, B, C refer to arbitrary subtrees located on those nodes and () stands for an implied terminal node).
...B...C...........A...B
....\./.............\./
.A...x....-->....C...x.................A..().........A...()..
..\./.............\./...................\./....-->....\./...
...x...............x.....................x.............x....
(a . (b . c)) -> (c . (a . b)) ____ (a . ()) ---> (a . ())
In terms of S-expressions, this automorphism rotates car, cadr and cddr of an S-exp if its length > 1.
See the Karttunen OEIS-Wiki link for a detailed explanation of how to obtain a given integer sequence from this definition.

Links

Crossrefs

Inverse of A089851. a(n) = A072796(A089850(n)) = A057163(A089855(A057163(n))). Row 6 of A089840.
Number of cycles: A089847. Number of fixed-points: A089848 (in each range limited by A014137 and A014138).

Extensions

Further comments and constructive implementation of Scheme-function (*A089853) added by Antti Karttunen, Jun 04 2011

A127302 Matula-Goebel signatures for plane binary trees encoded by A014486.

Original entry on oeis.org

1, 4, 14, 14, 86, 86, 49, 86, 86, 886, 886, 454, 886, 886, 301, 301, 301, 886, 886, 301, 454, 886, 886, 13766, 13766, 6418, 13766, 13766, 3986, 3986, 3986, 13766, 13766, 3986, 6418, 13766, 13766, 3101, 3101, 1589, 3101, 3101, 1849, 1849, 3101, 13766
Offset: 0

Views

Author

Antti Karttunen, Jan 16 2007

Keywords

Comments

This sequence maps A000108(n) oriented (plane) rooted binary trees encoded in range [A014137(n-1)..A014138(n-1)] of A014486 to A001190(n+1) non-oriented rooted binary trees, encoded by their Matula-Goebel numbers (when viewed as a subset of non-oriented rooted general trees). See also the comments at A127301.
If the signature-permutation of a Catalan automorphism SP satisfies the condition A127302(SP(n)) = A127302(n) for all n, then it preserves the non-oriented form of a binary tree. Examples of such automorphisms include A069770, A057163, A122351, A069767/A069768, A073286-A073289, A089854, A089859/A089863, A089864, A122282, A123492-A123494, A123715/A123716, A127377-A127380, A127387 and A127388.
A153835 divides natural numbers to same equivalence classes, i.e. a(i) = a(j) <=> A153835(i) = A153835(j) - Antti Karttunen, Jan 03 2013

Examples

			A001190(n+1) distinct values occur each range [A014137(n-1)..A014138(n-1)]. As an example, terms A014486(4..8) encode the following five plane binary trees:
........\/.....\/.................\/.....\/...
.......\/.......\/.....\/.\/.....\/.......\/..
......\/.......\/.......\_/.......\/.......\/.
n=.....4........5........6........7........8..
The trees in positions 4, 5, 7 and 8 all produce Matula-Goebel number A000040(1)*A000040(A000040(1)*A000040(A000040(1)*A000040(1))) = 2*A000040(2*A000040(2*2)) = 2*A000040(14) = 2*43 = 86, as they are just different planar representations of the one and same non-oriented tree. The tree in position 6 produces Matula-Goebel number A000040(A000040(1)*A000040(1)) * A000040(A000040(1)*A000040(1)) = A000040(2*2) * A000040(2*2) = 7*7 = 49. Thus a(4..8) = 86,86,49,86,86.

Links

Crossrefs

Cf A127301, A153835, A153829.

Formula

a(n) = A127301(A057123(n)).
Can be also computed directly as a fold, see the Scheme-program. - Antti Karttunen, Jan 03 2013

A071661 Permutation A057505 applied twice ("squared").

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 7, 5, 8, 9, 14, 16, 11, 19, 17, 20, 12, 10, 15, 21, 18, 13, 22, 23, 37, 42, 28, 51, 44, 53, 30, 25, 39, 56, 47, 33, 60, 45, 54, 57, 48, 61, 31, 34, 26, 24, 38, 40, 43, 29, 52, 58, 62, 49, 46, 55, 35, 32, 27, 41, 63, 59, 50, 36, 64, 65, 107, 121, 79, 149, 126
Offset: 0

Views

Author

Antti Karttunen, May 30 2002

Keywords

Links

Crossrefs

Inverse permutation: A071662 and also its car/cdr-flipped conjugate, i.e. A071661(n) = A057163(A071662(A057163(n))) = A057505(A057505(n)). Cf. also A071663, A071665, A071667, A071669.

A089850 Involution of natural numbers induced by Catalan automorphism *A089850 acting on the binary trees/parenthesizations encoded by A014486/A063171.

Original entry on oeis.org

0, 1, 2, 3, 5, 4, 6, 7, 8, 12, 13, 11, 9, 10, 15, 14, 16, 17, 18, 19, 20, 21, 22, 31, 32, 34, 35, 36, 30, 33, 28, 23, 24, 29, 25, 26, 27, 40, 41, 39, 37, 38, 43, 42, 44, 45, 46, 47, 48, 49, 50, 52, 51, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 87, 88, 90, 91, 92, 96, 97, 99
Offset: 0

Views

Author

Antti Karttunen, Nov 29 2003

Keywords

Comments

This automorphism effects the following transformation on the unlabeled rooted plane binary trees (letters A, B, C refer to arbitrary subtrees located on those nodes and () stands for an implied terminal node.)
...B...C...........C...B
....\./.............\./
.A...x....-->....A...x.................A..().........A...()..
..\./.............\./...................\./....-->....\./...
...x...............x.....................x.............x....
(a . (b . c)) -> (a . (c . b)) ____ (a . ()) ---> (a . ())
In terms of S-expressions, this automorphism swaps cadr and cddr of an S-exp if its length > 1.
Look at the example in A069770 to see how this will produce the given sequence of integers.

Links

Crossrefs

a(n) = A069770(A089859(n)) = A089863(A069770(n)) = A057163(A089854(A057163(n))). Row 3 of A089840. Row 3771 of A122203 and row 3677 of A122204.
Number of cycles: A073191. Number of fixed points: A073190. Max. cycle size & LCM of all cycle sizes: A046698 (in each range limited by A014137 and A014138).

Extensions

The new mail-address, a graphical explanation and constructive implementation of Scheme-function (*A089850) added by Antti Karttunen, Jun 04 2011

A057161 Signature-permutation of a Catalan Automorphism: rotate one step counterclockwise the triangulations of polygons encoded by A014486.

Original entry on oeis.org

0, 1, 3, 2, 7, 8, 5, 6, 4, 17, 18, 20, 21, 22, 12, 13, 15, 16, 19, 10, 11, 14, 9, 45, 46, 48, 49, 50, 54, 55, 57, 58, 59, 61, 62, 63, 64, 31, 32, 34, 35, 36, 40, 41, 43, 44, 47, 52, 53, 56, 60, 26, 27, 29, 30, 33, 38, 39, 42, 51, 24, 25, 28, 37, 23, 129, 130, 132, 133, 134
Offset: 0

Views

Author

Antti Karttunen, Aug 18 2000; entry revised Jun 06 2014

Keywords

Comments

This is a permutation of natural numbers induced when Euler's triangulation of convex polygons, encoded by the sequence A014486 in a straightforward way (via binary trees, cf. the illustration of the rotation of a triangulated pentagon, given in the Links section) are rotated counterclockwise.
The number of cycles in range [A014137(n-1)..A014138(n)] of this permutation is given by A001683(n+2), otherwise the same sequence as for Catalan bijections *A074679/*A074680, but shifted once left (for an explanation, see the related notes in OEIS Wiki).
E.g., in range [A014137(0)..A014138(1)] = [1,1] there is one cycle (as a(1)=1), in range [A014137(1)..A014138(2)] = [2,3] there is one cycle (as a(2)=3 and a(3)=2), in range [A014137(2)..A014138(3)] = [4,8] there is also one cycle (as a(4) = 7, a(7) = 6, a(6) = 5, a(5) = 8 and a(8) = 4), and in range [A014137(3)..A014138(4)] = [9,22] there are A001683(4+2) = 4 cycles.
From the recursive forms of A057161 and A057503 it is seen that both can be viewed as a convergent limits of a process where either the left or right side argument of A085201 in formula for A057501 is "iteratively recursivized", and on the other hand, both of these can then in turn be made to converge towards A057505 by the same method, when the other side of the formula is also "recursivized".

Links

Crossrefs

Inverse: A057162.
Also, a "SPINE"-transform of A069774, and thus occurs as row 12 of A130403.
Other related permutations: A057163, A057164, A057501, A057504, A057505.
Cf. A001683 (cycle counts), A057544 (max cycle lengths).

Programs

  • Maple
    a(n) = CatalanRankGlobal(RotateTriangularization(A014486[n]))
CatalanRankGlobal given in A057117 and the other Maple procedures in A038776.
NextSubBinTree := proc(nn) local n,z,c; n := nn; c := 0; z := 0; while(c < 1) do z := 2*z + (n mod 2); c := c + (-1)^n; n := floor(n/2); od; RETURN(z); end;
BinTreeLeftBranch := n -> NextSubBinTree(floor(n/2));
BinTreeRightBranch := n -> NextSubBinTree(floor(n/(2^(1+binwidth(BinTreeLeftBranch(n))))));
RotateTriangularization := proc(nn) local n,s,z,w; n := binrev(nn); z := 0; w := 0; while(1 = (n mod 2)) do s := BinTreeRightBranch(n); z := z + (2^w)*s; w := w + binwidth(s); z := z + (2^w); w := w + 1; n := floor(n/2); od; RETURN(z); end;

Formula

a(0) = 0, and for n>=1, a(n) = A085201(a(A072771(n)), A057548(A072772(n))). [This formula reflects the S-expression implementation given first in the Program section: A085201 is a 2-ary function corresponding to 'append', A072771 and A072772 correspond to 'car' and 'cdr' (known also as first/rest or head/tail in some languages), and A057548 corresponds to unary form of function 'list'.]
As a composition of related permutations:
a(n) = A069767(A069769(n)).
a(n) = A057163(A057162(A057163(n))).
a(n) = A057164(A057504(A057164(n))). [For a proof, see pp. 53-54 in the "Introductory survey ..." draft]

A069787 Self-inverse permutation of natural numbers induced by the automorphism DeepRev1CarSide! acting on the parenthesizations encoded by A014486.

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, 20, 19, 22, 23, 24, 26, 25, 27, 31, 32, 30, 28, 29, 35, 34, 33, 36, 45, 46, 49, 48, 50, 44, 47, 42, 37, 38, 43, 40, 39, 41, 58, 59, 57, 54, 55, 56, 53, 51, 52, 63, 62, 61, 60, 64, 65, 66, 68, 67, 69, 73, 74
Offset: 0

Views

Author

Antti Karttunen, Apr 16 2002

Keywords

Links

Crossrefs

The car/cdr-flipped conjugate of A057164.

Formula

a(n) = A057163(A057164(A057163(n))).
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