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.

Previous Showing 21-30 of 90 results. Next

A130402 Signature permutations of ENIPS-transformations of A057163-conjugates of Catalan automorphisms in table A122203.

Original entry on oeis.org

0, 1, 0, 2, 1, 0, 3, 3, 1, 0, 4, 2, 2, 1, 0, 5, 7, 3, 2, 1, 0, 6, 8, 4, 3, 2, 1, 0, 7, 6, 5, 4, 3, 2, 1, 0, 8, 4, 7, 5, 4, 3, 2, 1, 0, 9, 5, 6, 6, 5, 4, 3, 2, 1, 0, 10, 17, 8, 8, 8, 5, 4, 3, 2, 1, 0, 11, 18, 9, 7, 6, 8, 5, 5, 3, 2, 1, 0, 12, 20, 10, 9, 7, 7, 7, 4, 4, 3, 2, 1, 0, 13, 22, 12, 10, 9, 6
Offset: 0

Views

Author

Antti Karttunen, Jun 11 2007

Keywords

Comments

Row n is the signature permutation of the Catalan automorphism which is obtained from A057163-conjugate of the n-th automorphism in the table A122203 with the recursion scheme "ENIPS", i.e. row n is obtained as ENIPS(A057163 o SPINE(A089840[n]) o A057163). See A122203 and A122204 for the description of SPINE and ENIPS. Each row occurs only once in this table. Inverses of these permutations can be found in table A130403. This table contains also all the rows of A122204 and A089840.

Crossrefs

Cf. The first 22 rows of this table: row 0 (identity permutation): A001477, 1: A082346, 2: A130935, 3: A073289, 4: A130937, 5: A130939, 6: A130941, 7: A130943, 8: A130945, 9: A130947, 10: A130949, 11: A130951, 12: A074687, 13: A130953, 14: A130955, 15: A130957, 16: A130959, 17: A057162, 18: A130961, 19: A130963, 20: A130965, 21: A069768. Other rows: 251: A069770, 3613: A082340, 65352: A057502.
Cf. also tables A089840, A122201-A122204, A122285-A122286, A130400-A130401.
Cf. As a sequence differs from A130403 for the first time at n=92, where a(n)=22, while A130403(n)=21.

A089839 Array A(x,y): (read as A(0,0), A(1,0), A(0,1), A(2,0), A(1,1), A(0,2), A(3,0), A(2,1), A(1,2), A(0,3),...) Position of the composition A089840[y] o A089840[x] in the table A089840.

Original entry on oeis.org

0, 1, 1, 2, 0, 2, 3, 14, 19, 3, 4, 15, 0, 21, 4, 5, 16, 6, 4, 17, 5, 6, 13, 5, 0, 3, 18, 6, 7, 12, 4, 2, 5, 6, 20, 7, 8, 21, 3, 6, 6, 4, 5, 15, 8, 9, 18, 1654606, 5, 2, 3, 2, 1654137, 13, 9, 10, 17, 1655095, 1654694, 0, 0, 0, 1654694, 1654255, 16, 10
Offset: 0

Views

Author

Antti Karttunen, Dec 05 2003

Keywords

Comments

This is a "multiplication table" of an infinite enumerable group. Each row and column is a permutation of A001477.

Examples

			A(2,1)=14 because A089840[2] = A072796, A089840[1] = A069770 and the composition A069770 o A072796 (here the right hand side permutation acts first) yields A073269 = A089840[14]. Similarly A(2,2)=0, as A089840[2] = A072796, which being an involution, yields A001477 (= A089840[0]) when "squared".

Links

Crossrefs

Column 1: A089837, row 1: A089838, the main diagonal: A089841.

A089863 Permutation of natural numbers induced by Catalan Automorphism *A089863 acting on the binary trees/parenthesizations encoded by A014486/A063171.

Original entry on oeis.org

0, 1, 3, 2, 7, 8, 6, 5, 4, 17, 18, 20, 21, 22, 16, 19, 15, 12, 13, 14, 11, 9, 10, 45, 46, 48, 49, 50, 54, 55, 57, 58, 59, 61, 62, 63, 64, 44, 47, 53, 56, 60, 43, 52, 40, 31, 32, 41, 34, 35, 36, 42, 51, 39, 30, 33, 37, 28, 23, 24, 38, 29, 25, 26, 27, 129, 130, 132, 133, 134
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...-->.....C...x...............()..A.........A..()..
....\./.............\./.................\./....-->....\./...
.....x...............x...................x.............x....
((a . b) . c) --> (c . (b . a)) __ (() . a) ----> (a . ())
See the Karttunen OEIS-Wiki link for a detailed explanation of how to obtain a given integer sequence from this definition.

Links

Crossrefs

Row 21 of A089840. Inverse of A089859. a(n) = A089850(A069770(n)) = A069770(A089854(n)). A089864 is the "square" of this permutation.
Number of cycles: A089407. Max. cycle size & LCM of all cycle sizes: A040002 (in each range limited by A014137 and A014138).

Extensions

A graphical description and constructive version of Scheme-implementation added by Antti Karttunen, Jun 04 2011

A057509 Permutation of natural numbers: rotations of the bottom 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, 18, 13, 20, 21, 22, 23, 25, 28, 30, 33, 37, 39, 42, 44, 47, 51, 53, 56, 60, 24, 29, 38, 43, 52, 26, 40, 31, 45, 46, 32, 48, 49, 50, 27, 41, 34, 54, 55, 35, 57, 58, 59, 36, 61, 62, 63, 64, 65, 67, 70, 72, 75, 79, 81
Offset: 0

Views

Author

Antti Karttunen, Sep 03 2000

Keywords

Comments

The number of objects (rooted planar trees, mountain ranges, parenthesizations) fixed by this permutation can be computed with procedure fixedcount, which gives A034731.

Links

Crossrefs

Inverse of A057510 and the car/cdr-flipped conjugate of A069775 and also composition of A069770 & A057501, i.e. A057509(n) = A057163(A069775(A057163(n))) = A057501(A069770(n)).
Cycle counts given by A003239. Cf. also A057511.

Programs

  • Maple
    map(CatalanRankGlobal,map(RotateBottomBranchesL, A014486));
RotateBottomBranchesL := n -> pars2binexp(rotateL(binexp2pars(n)));
rotateL := proc(a) if 0 = nops(a) then (a) else [op(cdr(a)), a[1]]; fi; end;
fixedcount := proc(n) local d,z; z := 0; for d in divisors(n) do z := z+C(d-1); od; RETURN(z); end;

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)

A073269 Permutation of natural numbers induced by Catalan Automorphism *A073269 acting on the binary trees/parenthesizations encoded by A014486/A063171.

Original entry on oeis.org

0, 1, 3, 2, 7, 6, 8, 4, 5, 17, 18, 16, 14, 15, 20, 19, 21, 9, 10, 22, 11, 12, 13, 45, 46, 48, 49, 50, 44, 47, 42, 37, 38, 43, 39, 40, 41, 54, 55, 53, 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, 130, 132, 133, 134
Offset: 0

Views

Author

Antti Karttunen, Jun 25 2002

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...C
......\./.........\./
...A...x...-->... .x...B...............A..().........()..A..
....\./.............\./.................\./....-->....\./...
.....x...............x...................x.............x....
(a . (b . c)) --> ((a . c) . b) ___ (a . ()) --> (() . a)
See the Karttunen OEIS-Wiki link for a detailed explanation of how to obtain a given integer sequence from this definition.

Links

Crossrefs

Row 14 of A089840. Inverse permutation: A073270. a(n) = A069770(A072796(n)).

Extensions

Further comments, a mail-address and Scheme-implementation of this automorphism added by Antti Karttunen, Jun 04 2011

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.

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

A130401 Signature permutations of REDRONI-transformations of non-recursive Catalan automorphisms in table A089840.

Original entry on oeis.org

0, 1, 0, 2, 1, 0, 3, 3, 1, 0, 4, 2, 2, 1, 0, 5, 7, 3, 2, 1, 0, 6, 8, 4, 3, 2, 1, 0, 7, 6, 6, 5, 3, 2, 1, 0, 8, 4, 5, 4, 5, 3, 2, 1, 0, 9, 5, 7, 6, 6, 6, 3, 2, 1, 0, 10, 17, 8, 7, 4, 5, 6, 3, 2, 1, 0, 11, 18, 9, 8, 7, 4, 4, 4, 3, 2, 1, 0, 12, 20, 14, 13, 8, 7, 5, 5, 4, 3, 2, 1, 0, 13, 21, 10, 12, 13
Offset: 0

Views

Author

Antti Karttunen, Jun 11 2007

Keywords

Comments

Row n is the signature permutation of the Catalan automorphism which is obtained from the n-th nonrecursive automorphism in the table A089840 with the recursion scheme "REDRONI". In this recursion scheme the given automorphism is applied at the root of binary tree after the algorithm has recursed down the cdr-branch (the right hand side tree in the context of binary trees), but before the algorithm recurses down to the car-branch (the left hand side of the binary tree, with respect to the new orientation of branches, possibly changed by the applied automorphism). I.e. this corresponds to the reversed depth-first in-order traversal of a Catalan structure, when it is interpreted as a binary tree. The associated Scheme-procedures REDRONI and !REDRONI can be used to obtain such a transformed automorphism from any constructively (or respectively: destructively) implemented automorphism. Each row occurs only once in this table and similar notes as given e.g. for table A122202 apply here, e.g. the rows of A089840 all occur here as well. This transformation has many fixed points besides the trivial identity automorphism *A001477: at least *A069770, *A089859 and *A129604 stay as they are. Inverses of these permutations can be found in table A130400.

Crossrefs

Cf. The first 22 rows of this table: row 0 (identity permutation): A001477, 1: A069770, 2: A073285, 3: A122342, 4: A130386, 5: A130384, 6: A130382, 7: A122349, 8: A082342, 9: A130392, 10: A130390, 11: A130388, 12: A071658, 13: A130930, 14: A130932, 15: A089859, 16: A130934, 18: A130394, 19: A130396, 20: A130928, 21: A130398. Other rows: row 1654694: A073280, row 1654720: A129604.
Cf. also tables A089840, A122201-A122204, A130402-A130403.
Cf. As a sequence differs from A130400 for the first time at n=80, where a(n)=14, while A130401(n)=11.
Previous Showing 21-30 of 90 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.