A063171 -id:A063171 - cp's OEIS frontend

A072643 Half of the binary width of the terms of A014486, the number of digits in A063171(n)/2.

0, 1, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6

Offset: 0

Antti Karttunen, Jun 02 2002

Paolo Xausa, Table of n, a(n) for n = 0..10000

Each value v occurs A000108(v) times. The maximum position for value v to occur is A014138(v). Permutations: A071673, A072644, A072645, A072660. Cf. also A002024, A072649.

a[n_] := Module[{i, c, a}, i = c = 0; a = 1; While[n>c, a *= (4*i+2)/(i+2); i++; c += a]; i];
Table[a[n], {n, 0, 104}] (* Jean-François Alcover, Dec 26 2017, from Sage code *)
Flatten[Array[Table[#, CatalanNumber[#]]&, 7, 0]] (* Paolo Xausa, Feb 13 2024 *)

def A072643(n) :
    i = c = 0; a = 1
    while n > c :
        a *= (4*i+2)/(2+i)
        i += 1; c += a
    return i
[A072643(n) for n in (0..100)] # Peter Luschny, Sep 07 2012

Sum_{n>=1} (-1)^(n+1)/a(n) = Sum_{n>=1} (-1)^(n+1)/(2^n-1) = 0.76449978034844420919... . - Amiram Eldar, Feb 18 2024

A057508 Self-inverse permutation of natural numbers induced by the function 'reverse' (present in programming languages like Lisp, Scheme, Prolog and Haskell) when it acts on symbolless S-expressions encoded by A014486/A063171.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 5, 7, 8, 9, 14, 11, 16, 19, 10, 15, 12, 17, 18, 13, 20, 21, 22, 23, 37, 28, 42, 51, 25, 39, 30, 44, 47, 33, 53, 56, 60, 24, 38, 29, 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, 107, 79, 121, 149, 70

Offset: 0

Antti Karttunen Sep 03 2000

nonn

A. Karttunen, Gatomorphisms (Includes the complete Scheme program for computing this sequence)
Index entries for the sequences induced by list functions of Lisp
Index entries for signature-permutations induced by Catalan automorphisms

The car/cdr-flipped conjugate of A069769, i.e. A057508(n) = A057163(A069769(A057163(n))). Cf. also A057164, A057509, A057510, A033538.

Similar function for Maple lists can be implemented as: reverse := proc(a) if 0 = nops(a) then (a) else [op(reverse(cdr(a))),a[1]]; fi; end;

A089859 Permutation of natural numbers induced by Catalan Automorphism *A089859 acting on the binary trees/parenthesizations encoded by A014486/A063171.

Original entry on oeis.org

0, 1, 3, 2, 8, 7, 6, 4, 5, 21, 22, 20, 17, 18, 19, 16, 14, 9, 10, 15, 11, 12, 13, 58, 59, 62, 63, 64, 57, 61, 54, 45, 46, 55, 48, 49, 50, 56, 60, 53, 44, 47, 51, 42, 37, 23, 24, 38, 25, 26, 27, 52, 43, 39, 28, 29, 40, 30, 31, 32, 41, 33, 34, 35, 36, 170, 171, 174, 175, 176, 184

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...-->... .x...A...............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.

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

Row 15 of A089840. Inverse of A089863. a(n) = A089854(A069770(n)) = A069770(A089850(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).

A graphical description and constructive implementation of Scheme-function (*A089859) added by Antti Karttunen, Jun 04 2011

A089851 Permutation of natural numbers induced by Catalan automorphism *A089851 acting on the binary trees/parenthesizations encoded by A014486/A063171.

Original entry on oeis.org

0, 1, 2, 3, 5, 6, 4, 7, 8, 12, 13, 15, 16, 19, 11, 14, 9, 17, 18, 10, 20, 21, 22, 31, 32, 34, 35, 36, 40, 41, 43, 44, 47, 52, 53, 56, 60, 30, 33, 39, 42, 51, 28, 37, 23, 45, 46, 24, 48, 49, 50, 29, 38, 25, 54, 55, 26, 57, 58, 59, 27, 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...A
....\./.............\./
.A...x....-->....B...x.................A..().........A...()..
..\./.............\./...................\./....-->....\./...
...x...............x.....................x.............x....
(a . (b . c)) -> (b . (c . a)) ____ (a . ()) ---> (a . ())
In terms of S-expressions, this rotates car, cadr and cddr of an S-exp
if its length > 1, otherwise keeps it intact.
Note that the first clause corresponds to generator C of Thompson's groups T and V.
(Cf. also A072796, A074679 and A154121 for other related generators).
See "Catalan Automorphisms" OEIS-Wiki page for a detailed explanation 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

Inverse of A089853. a(n) = A089850(A072796(n)) = A057163(A089857(A057163(n))). Row 4 of A089840.

Number of cycles: A089847. Number of fixed-points: A089848 (in each range limited by A014137 and A014138).

The new mail-address, further comments and constructive implementation of Scheme-function (*A089851) added by Antti Karttunen, Jun 04 2011

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

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).
.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.

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

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).

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

A074683 Permutation of natural numbers induced by the Catalan Automorphism *A074683 acting on parenthesizations as encoded and ordered by A014486/A063171.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Sep 11 2002

nonn

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 Catalan Automorphism (with simple definition) where the cycle count sequence (A089411) is not monotone. (See A127296 for more complex example.)

Row 12 of A122202. Inverse of A074684. a(n) = A057163(A074682(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.

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

Antti Karttunen, Sep 11 2002

nonn

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.)

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.

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

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).
.....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.

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

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

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).
.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.

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

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).

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

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

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...........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.

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

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).

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

cp's OEIS Frontend

A072643 Half of the binary width of the terms of A014486, the number of digits in A063171(n)/2.

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A057508 Self-inverse permutation of natural numbers induced by the function 'reverse' (present in programming languages like Lisp, Scheme, Prolog and Haskell) when it acts on symbolless S-expressions encoded by A014486/A063171.

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Maple

A089859 Permutation of natural numbers induced by Catalan Automorphism *A089859 acting on the binary trees/parenthesizations encoded by A014486/A063171.

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A089851 Permutation of natural numbers induced by Catalan automorphism *A089851 acting on the binary trees/parenthesizations encoded by A014486/A063171.

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

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A074683 Permutation of natural numbers induced by the Catalan Automorphism *A074683 acting on parenthesizations as encoded and ordered by A014486/A063171.

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

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

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

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

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