A089857 - OEIS

%I #21 May 01 2018 02:42:51

%S 0,1,2,3,4,5,8,6,7,9,10,11,12,13,21,22,19,14,15,20,16,17,18,23,24,25,

%T 26,27,28,29,30,31,32,33,34,35,36,58,59,62,63,64,56,60,51,37,38,52,39,

%U 40,41,57,61,53,42,43,54,44,45,46,55,47,48,49,50,65,66,67,68,69,70,71,72

%N Permutation of natural numbers induced by Catalan Automorphism *A089857 acting on the binary trees/parenthesizations encoded by A014486/A063171.

%C .A...B...........C...A

%C ..\./.............\./

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

%C ....\./.............\./.................\./....-->....\./...

%C .....x...............x...................x.............x....

%C ((a . b) . c) -> ((c . a) . b) _____ (() . a) ---> (() . a)

%C In terms of S-expressions, this automorphism rotates caar, cdar and cdr of an S-exp, i.e., if car-side is not ().

%C See the Karttunen OEIS-Wiki link for a detailed explanation how to obtain a given integer sequence from this definition.

%H A. Karttunen, <a href="http://oeis.org/wiki/Catalan_Automorphisms">Catalan Automorphisms</a>

%H A. Karttunen, <a href="/A089408/a089408.c.txt">C-program for computing this sequence</a>

%H <a href="/index/Per#IntegerPermutationCatAuto">Index entries for signature-permutations induced by Catalan automorphisms</a>

%o (Scheme functions implementing this automorphism on list-structures/S-expressions, both constructive (*A089857) and destructive (*A089857!) versions:)

%o (define (*A089857 s) (if (and (pair? s) (pair? (car s))) (cons (cons (cdr s) (caar s)) (cdar s)) s))

%o (define (*A089857! s) (cond ((not (pair? s)) s) ((not (pair? (car s))) s) (else (swap! s) (robl! s) s)))

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

%o (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))

%Y Row 11 of A089840. Inverse of A089855. a(n) = A074679(A069770(n)) = A069770(A089862(n)) = A057163(A089851(A057163(n))).

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

%K nonn

%O 0,3

%A _Antti Karttunen_, Nov 29 2003

%E Further comments and constructive implementation of Scheme-function (*A089857) added by _Antti Karttunen_, Jun 04 2011