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A072797 Self-inverse permutation of natural numbers induced by Catalan Automorphism *A072797 acting on the parenthesizations encoded by A014486. 33
0, 1, 2, 3, 4, 5, 7, 6, 8, 9, 10, 11, 12, 13, 17, 18, 16, 14, 15, 20, 19, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 45, 46, 48, 49, 50, 44, 47, 42, 37, 38, 43, 39, 40, 41, 54, 55, 53, 51, 52, 57, 56, 58, 59, 61, 60, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

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...........A...C

..\./.............\./

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

....\./.............\./.................\./....-->....\./...

.....x...............x...................x.............x....

((a . b) . c) --> ((a . c) . b) ____ (() . a) ---> (() . a)

In terms of S-expressions, this automorphism swaps cdar and cdr of an S-exp if its car-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

Table of n, a(n) for n=0..71.

A. Karttunen, Catalan Automorphisms

Index entries for signature-permutations induced by Catalan automorphisms

PROG

(Scheme function implementing this automorphism on list-structures/S-expressions, both constructive (*A072797) and destructive (*A072797!) version:)(define (*A072797 s) (if (and (pair? s) (pair? (car s))) (cons (cons (caar s) (cdr s)) (cdar s)) s))

(define (*A072797! s) (cond ((not (pair? s)) s) ((not (pair? (car s))) s) (else (swap! s) (robl! s) (swap! (car s)) s)))

(define (robl! s) (let ((ex-car (car s))) (set-car! s (cddr s)) (set-cdr! (cdr s) ex-car) (swap! (cdr s)) (swap! s) s))

(define (swap! s) (let ((ex-car (car s))) (set-car! s (cdr s)) (set-cdr! s ex-car) s))

CROSSREFS

Row 8 of A089840. A left/right-flipped conjugate of A072796, i.e. A072797(n) = A057163(A072796(A057163(n))). Counts for fixed points and counts for distinct cycles (in each range limited by A014137 and A014138) are given in A073190 and A073191.

Sequence in context: A131424 A222249 A230565 * A131169 A131170 A082338

Adjacent sequences:  A072794 A072795 A072796 * A072798 A072799 A072800

KEYWORD

nonn

AUTHOR

Antti Karttunen, Jun 12 2002

EXTENSIONS

Further comments and constructive implementation of Scheme-function (*A072797) added by Antti Karttunen, Jun 04 2011

STATUS

approved

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Last modified January 18 13:09 EST 2019. Contains 319271 sequences. (Running on oeis4.)