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A072797 Self-inverse permutation of natural numbers induced by Catalan Automorphism *A072797 acting on the parenthesizations encoded by A014486. 33

%I #19 May 01 2018 02:59:57

%S 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,

%T 26,27,28,29,30,31,32,33,34,35,36,45,46,48,49,50,44,47,42,37,38,43,39,

%U 40,41,54,55,53,51,52,57,56,58,59,61,60,62,63,64,65,66,67,68,69,70,71

%N Self-inverse permutation of natural numbers induced by Catalan Automorphism *A072797 acting on the parenthesizations encoded by A014486.

%C 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).

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

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

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

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

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

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

%C In terms of S-expressions, this automorphism swaps cdar and cdr of an S-exp if its car-element is not ().

%C See the Karttunen OEIS-Wiki link for a detailed explanation of 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 href="/index/Per#IntegerPermutationCatAuto">Index entries for signature-permutations induced by Catalan automorphisms</a>

%o (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))

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

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

%Y 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.

%K nonn

%O 0,3

%A _Antti Karttunen_, Jun 12 2002

%E Further comments and constructive implementation of Scheme-function (*A072797) added by _Antti Karttunen_, Jun 04 2011

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