A014486 -id:A014486 - cp's OEIS frontend

A073270 Permutation of natural numbers induced by Catalan Automorphism *A073270 acting on the binary trees/parenthesizations encoded by A014486/A063171.

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

Offset: 0

Antti Karttunen, Jun 25 2002

nonn

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...............C...B
..\./.................\./
...x...C...-->.....A...x...............()..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.

Row 19 of A089840. Inverse permutation: A073269. a(n) = A072796(A069770(n)).

A graphical description and Scheme-implementations of automorphism added by Antti Karttunen, Jun 04 2011

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

Antti Karttunen, Nov 29 2003

nonn

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.

A. Karttunen, Catalan Automorphisms
A. Karttunen, C-program for computing this sequence
Index entries for signature-permutations induced by Catalan automorphisms

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).

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

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

nonn

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".

A. Karttunen, Table of n, a(n) for n = 0..2055
A. Karttunen, Introductory Survey of Catalan Automorphisms and Bijections (an unfinished draft), pp. 51-54.
A. Karttunen, Notes on the orbits of the related permutations A074679/A074680, OEIS Wiki.
A. Karttunen, Illustration of how the five triangulations of a pentagon will rotate, and the corresponding changes it induces in the binary trees
Index entries for signature-permutations of Catalan automorphisms

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).

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;

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]

A069772 Self-inverse permutation of natural numbers induced by the automorphism xReflectHandshakes acting on the parenthesizations encoded by A014486.

Original entry on oeis.org

0, 1, 2, 3, 7, 6, 5, 4, 8, 9, 10, 21, 20, 19, 14, 15, 18, 17, 16, 13, 12, 11, 22, 45, 46, 44, 42, 43, 31, 32, 30, 28, 29, 63, 62, 61, 60, 54, 55, 53, 51, 52, 26, 27, 25, 23, 24, 59, 58, 57, 56, 40, 41, 39, 37, 38, 50, 49, 48, 47, 36, 35, 34, 33, 64, 65, 67, 66, 68, 69, 170

Offset: 0

Antti Karttunen, Apr 16 2002

nonn

This automorphism reflects over the x-axis the interpretation n (the non-crossing handshakes) of Stanley's exercise 19.
Note that DeepRev (A057164) reflects over y-axis.
This transformation keeps palindromic parenthesizations/Dyck paths/rooted planar trees palindromic, but not necessarily same, meaning that this induces a permutation on the sequence A061855 (= A069766).

A. Karttunen, Gatomorphisms (Includes the complete Scheme program for computing this sequence)
R. P. Stanley, Exercises on Catalan and Related Numbers
Index entries for sequences that are permutations of the natural numbers

Composition of A057164 and A069771 in either order, i.e. A069772(n) = A057164(A069771(n)) = A069771(A057164(n)). Cf. also A061855, A069766, A057501, A069888, A069889.

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

Antti Karttunen, Apr 16 2002

nonn

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

The car/cdr-flipped conjugate of A057164.

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

A071655 Permutation of natural numbers induced by the automorphism robr_car_cdr! acting on the parenthesizations encoded by A014486.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, May 30 2002

nonn

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

Inverse permutation: A071656. The car/cdr-flipped conjugate of A071660, i.e. A071655(n) = A057163(A071660(A057163(n))). Cf. also A071657, A071658.

A072764 Tabular N X N -> N bijection induced by Lisp/Scheme function 'cons' combining the two planar binary trees/general trees/parenthesizations encoded by A014486(X) and A014486(Y).

Original entry on oeis.org

0, 1, 3, 2, 7, 6, 4, 8, 16, 14, 5, 17, 19, 42, 15, 9, 18, 44, 51, 43, 37, 10, 20, 47, 126, 52, 121, 38, 11, 21, 53, 135, 127, 149, 122, 39, 12, 22, 56, 154, 136, 385, 150, 123, 40, 13, 45, 60, 163, 155, 413, 386, 151, 124, 41, 23, 46, 128, 177, 164, 475, 414, 387, 152

Offset: 0

Antti Karttunen Jun 12 2002

Index entries for the sequences induced by list functions of Lisp
Index entries for sequences that are permutations of the natural numbers
A. Karttunen, Gatomorphisms (with the complete Scheme source)

Inverse permutation: A072765. a(n) = A069770(A072766(n)). Also transpose of A072766, i.e. a(n) = A072766(A038722(n)). The upper triangular region: A072773. Projection functions are A072771 ('car') & A072772 ('cdr'). The sizes of the corresponding Catalan structures: A072768. The first row: A057548, the first column: A072795, diagonal: A083938. Cf. also A080300, A025581, A002262.

a(0)=0 prepended by Sean A. Irvine, Oct 25 2024

A218787 a(n) = A014486-index for the n-th tendril of infinite beanstalk (A213730(n)), with the "lesser numbers to the left side" construction.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 1, 2, 0, 0, 0, 1, 2, 0, 8, 0, 0, 1, 0, 0, 1, 2, 0, 8, 0, 0, 1, 8, 0, 0, 3, 0, 2, 1, 0, 0, 0, 1, 2, 0, 8, 0, 0, 1, 8, 0, 0, 3, 0, 2, 1, 0, 8, 0, 0, 3, 0, 60, 0, 0, 172, 0, 2, 0, 1, 0, 0, 1, 2, 0, 8, 0, 0, 1, 8, 0, 0, 3, 0, 2, 1, 0, 8, 0, 0

Offset: 1

Antti Karttunen, Nov 11 2012

nonn

"Tendrils" of the beanstalk are the finite side-trees sprouting from its infinite trunk (see A179016) at the numbers given by A213730.

			A213730(9)=22, and from that branches 24 and 25 (as both A011371(24)=A011371(25)=22) and while 24 is a leaf (in A055938) the other branch 25 further branches to two leaves (as both A011371(28)=A011371(29)=25).
When we construct a binary tree from this in such a fashion that the lesser numbers go to the left, we obtain:
...........
...28...29.
.....\./...
..24..25...
...\ /.....
....22.....
...........
and the binary tree
........
...\./..
....*...
.\./....
..*.....
........
is located as A014486(2) in the normal encoding order of binary trees, thus a(9)=2.

A. Karttunen, Table of n, a(n) for n = 1..8727
A. Karttunen, Illustration of how binary trees (the second rightmost column) are encoded by A014486

These are the mirror-images of binary trees given in A218788, i.e. a(n) = A057163(A218788(n)). A218786 gives the sizes of these trees. Cf. A072764, A218609, A218611.

A085169 Permutation of natural numbers induced by the Catalan bijection gma085169 acting on symbolless S-expressions encoded by A014486/A063171.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Jun 23 2003

nonn

A parenthesization is fixed by the Catalan bijections A085169/A085170 if and only if no other elements than () and (()) occur at its top-level: (); ()(),(()); ()()(),()(()),(())(); ()()()(),()()(()),()(())(),(())()(),(())(()); ... There is a simple bijection between these and Zeckendorf-expansions, explaining why Fibonacci numbers gives the number of fixed points of this permutation.

In addition to "rising slope" and "descending slope" mappings from Dyck paths to noncrossing Murasaki-diagrams as illustrated in A085161 and A086431 there is also a mapping where we insert a vertical stick after every second parenthesis and connect those that are on the same level without any intermediate points below. This Catalan bijection converts between these two mappings. See the illustration at example lines.

			.........................
..._____....________.....
..|.....|..|.....|..|....
..|..|..|..|..|..|..|....
..|..|..|..|..|..|..|....
..|..|..|..|..|..|..|....
..|..|..|..|..|..|..|....
..1((2))3((4((5))6()7))..
...(())(((())()))........
...11001111001000=13256=A014486(368)
To obtain the same Murasaki diagram using the "rising slope mapping" illustrated in A085161, we should use the following Dyck path, encoded by 360th binary string in A014486/A063171:
....___.._____...........
...|...||...|.|..........
...||..|||..|.|..........
...||..|||..|.|..........
...||..||/\.|.|..........
...|/\.|/..\/\/\.........
.../..\/........\........
...11001110010100=13204=A014486(360)
So we have A085169(368)=360 and A085170(360)=368.

A. Karttunen, Gatomorphisms (With the complete Scheme source)
Index entries for signature-permutations induced by Catalan automorphisms

Inverse: A085170. a(n) = A086433(A082853(n))+A082852(n). A074684 = A083925(A085169(A057548(n))). Cf. also A085159, A085160, A085175.

Number of cycles: A086585. Number of fixed points: A000045. Max. cycle size: A086586. LCM of cycle sizes: A086587. (In range [A014137(n-1)..A014138(n-1)] of this permutation).

A089852 Involution of natural numbers induced by Catalan automorphism *A089852 acting on the binary trees/parenthesizations encoded by A014486/A063171.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Nov 29 2003

nonn

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...........B...A
....\./.............\./
.A...x....-->....C...x.................A..().........A...()..
..\./.............\./...................\./....-->....\./...
...x...............x.....................x.............x....
(a . (b . c)) -> (c . (b . a)) ____ (a . ()) ---> (a . ())
In terms of S-expressions, this automorphism swaps car and cddr of an S-exp if its length > 1, if possible, otherwise keeps it intact.
See the Karttunen OEIS-Wiki link for a detailed explanation of how to obtain a given integer sequence from this definition.

A. Karttunen, Catalan Automorphisms
A. Karttunen, C-program for computing this sequence
Index entries for signature-permutations induced by Catalan automorphisms

a(n) = A069770(A089858(n)) = A089861(A069770(n)) = A057163(A089856(A057163(n))). Row 5 of A089840.

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).

Further comments and constructive implementation of Scheme-function (*A089852) added by Antti Karttunen, Jun 04 2011

cp's OEIS Frontend

A073270 Permutation of natural numbers induced by Catalan Automorphism *A073270 acting on the binary trees/parenthesizations encoded by A014486/A063171.

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

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

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A069772 Self-inverse permutation of natural numbers induced by the automorphism xReflectHandshakes acting on the parenthesizations encoded by A014486.

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

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A071655 Permutation of natural numbers induced by the automorphism robr_car_cdr! acting on the parenthesizations encoded by A014486.

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A072764 Tabular N X N -> N bijection induced by Lisp/Scheme function 'cons' combining the two planar binary trees/general trees/parenthesizations encoded by A014486(X) and A014486(Y).

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A218787 a(n) = A014486-index for the n-th tendril of infinite beanstalk (A213730(n)), with the "lesser numbers to the left side" construction.

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A085169 Permutation of natural numbers induced by the Catalan bijection gma085169 acting on symbolless S-expressions encoded by A014486/A063171.

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A089852 Involution of natural numbers induced by Catalan automorphism *A089852 acting on the binary trees/parenthesizations encoded by A014486/A063171.

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