login
Permutation of natural numbers induced by Catalan automorphism *A089853 acting on the binary trees/parenthesizations encoded by A014486/A063171.
17

%I #22 May 01 2018 02:56:39

%S 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,

%T 56,60,42,51,37,23,24,38,25,26,27,43,52,39,28,29,40,30,31,45,46,32,48,

%U 49,50,41,33,34,54,55,35,57,58,59,36,61,62,63,64,128,131,137,140,144

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

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

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

%C .A...x....-->....C...x.................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 car, cadr and cddr of an S-exp if its length > 1.

%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. 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,

%o both constructive (*A089853) and destructive (*A089853!) versions:)

%o (define (*A089853 s) (if (and (pair? s) (pair? (cdr s))) (cons (cddr s) (cons (car s) (cadr s))) s))

%o (define (*A089853! s) (cond ((not (pair? s)) s) ((not (pair? (cdr s))) s) (else (let ((org_cadr (cadr s))) (set-car! (cdr s) (car s)) (set-car! s (cddr s)) (set-cdr! (cdr s) org_cadr) s))))

%Y Inverse of A089851. a(n) = A072796(A089850(n)) = A057163(A089855(A057163(n))). Row 6 of A089840.

%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 (*A089853) added by _Antti Karttunen_, Jun 04 2011